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    <link:label id="lab_gemi_ServiceBasedCommonUnitsMember_documentation_en-US" xlink:label="lab_gemi_ServiceBasedCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Service Based Common Units</link:label>
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    <link:label id="lab_gemi_ShareBasedPaymentArrangementByShareBasedPaymentAwardExercisePricePercentageOfFairValue_documentation_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementByShareBasedPaymentAwardExercisePricePercentageOfFairValue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement By Share-Based Payment Award, Exercise Price, Percentage Of Fair Value</link:label>
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    <link:label id="lab_us-gaap_AdditionalPaidInCapitalMember_c40246fa-204b-49a1-9044-dbca6fed5b09_terseLabel_en-US" xlink:label="lab_us-gaap_AdditionalPaidInCapitalMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Additional Paid-In Capital</link:label>
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    <link:label id="lab_us-gaap_IPOMember_17990a02-3a02-4e9e-95f6-a1f53398dab4_terseLabel_en-US" xlink:label="lab_us-gaap_IPOMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">IPO</link:label>
    <link:label id="lab_us-gaap_IPOMember_label_en-US" xlink:label="lab_us-gaap_IPOMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">IPO [Member]</link:label>
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    <link:label id="lab_gemi_CommodityFuturesTradingCommissionLitigationMember_952b9718-d0bb-4232-83b9-3b8de7d8e867_terseLabel_en-US" xlink:label="lab_gemi_CommodityFuturesTradingCommissionLitigationMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Commodity Futures Trading Commission Litigation</link:label>
    <link:label id="lab_gemi_CommodityFuturesTradingCommissionLitigationMember_label_en-US" xlink:label="lab_gemi_CommodityFuturesTradingCommissionLitigationMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Commodity Futures Trading Commission Litigation [Member]</link:label>
    <link:label id="lab_gemi_CommodityFuturesTradingCommissionLitigationMember_documentation_en-US" xlink:label="lab_gemi_CommodityFuturesTradingCommissionLitigationMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Commodity Futures Trading Commission Litigation</link:label>
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    <link:label id="lab_gemi_CommonClassBIncentiveUnitsHeldByFoundersMember_f7672f66-c8f2-4bb0-a252-562d0ee7b05f_terseLabel_en-US" xlink:label="lab_gemi_CommonClassBIncentiveUnitsHeldByFoundersMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common Class B, Incentive Units Held By Founders</link:label>
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    <link:label id="lab_gemi_CommonClassBIncentiveUnitsHeldByFoundersMember_documentation_en-US" xlink:label="lab_gemi_CommonClassBIncentiveUnitsHeldByFoundersMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Class B, Incentive Units Held By Founders</link:label>
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    <link:label id="lab_gemi_DebtConversionTriggeringEventPercentageOfTheLowestPricePerUnitPaid_e43e4795-6400-4047-9c2e-d8d2246c8ccc_terseLabel_en-US" xlink:label="lab_gemi_DebtConversionTriggeringEventPercentageOfTheLowestPricePerUnitPaid" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Qualified financing event, conversion price as a percentage of the lowest price per unit paid by the other purchasers of the convertible preferred units</link:label>
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    <link:label id="lab_us-gaap_StockCompensationPlanMember_7abea962-31b5-42ee-a8f4-6ea70372fc49_terseLabel_en-US" xlink:label="lab_us-gaap_StockCompensationPlanMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity awards</link:label>
    <link:label id="lab_us-gaap_StockCompensationPlanMember_label_en-US" xlink:label="lab_us-gaap_StockCompensationPlanMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement [Member]</link:label>
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    <link:label id="lab_gemi_Loan13Member_2095edb9-a9a1-4b31-9234-179ac5633831_terseLabel_en-US" xlink:label="lab_gemi_Loan13Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 13</link:label>
    <link:label id="lab_gemi_Loan13Member_label_en-US" xlink:label="lab_gemi_Loan13Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 13 [Member]</link:label>
    <link:label id="lab_gemi_Loan13Member_documentation_en-US" xlink:label="lab_gemi_Loan13Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 13</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan13Member" xlink:href="gemi-20250930.xsd#gemi_Loan13Member"/>
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    <link:label id="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock_f22acc6b-ddd7-432e-91b8-a24a44d1cc84_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transactions</link:label>
    <link:label id="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transactions Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyTransactionsDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:to="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NamedExecutiveOfficersFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_NamedExecutiveOfficersFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Named Executive Officers, Footnote</link:label>
    <link:label id="lab_ecd_NamedExecutiveOfficersFnTextBlock_label_en-US" xlink:label="lab_ecd_NamedExecutiveOfficersFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Named Executive Officers, Footnote [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NamedExecutiveOfficersFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NamedExecutiveOfficersFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NamedExecutiveOfficersFnTextBlock" xlink:to="lab_ecd_NamedExecutiveOfficersFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OverAllotmentOptionStockholdersMember_4c489b68-93d1-4f50-9eaa-603dd96e079a_terseLabel_en-US" xlink:label="lab_gemi_OverAllotmentOptionStockholdersMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Over-Allotment Option, Stockholders</link:label>
    <link:label id="lab_gemi_OverAllotmentOptionStockholdersMember_label_en-US" xlink:label="lab_gemi_OverAllotmentOptionStockholdersMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Over-Allotment Option, Stockholders [Member]</link:label>
    <link:label id="lab_gemi_OverAllotmentOptionStockholdersMember_documentation_en-US" xlink:label="lab_gemi_OverAllotmentOptionStockholdersMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Over-Allotment Option, Stockholders</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OverAllotmentOptionStockholdersMember" xlink:href="gemi-20250930.xsd#gemi_OverAllotmentOptionStockholdersMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OverAllotmentOptionStockholdersMember" xlink:to="lab_gemi_OverAllotmentOptionStockholdersMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RevenueFromContractWithCustomerTextBlock_eeae0c2e-ff10-413b-a2b7-ecf255e6f276_terseLabel_en-US" xlink:label="lab_us-gaap_RevenueFromContractWithCustomerTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Revenue</link:label>
    <link:label id="lab_us-gaap_RevenueFromContractWithCustomerTextBlock_label_en-US" xlink:label="lab_us-gaap_RevenueFromContractWithCustomerTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenue from Contract with Customer [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RevenueFromContractWithCustomerTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RevenueFromContractWithCustomerTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RevenueFromContractWithCustomerTextBlock" xlink:to="lab_us-gaap_RevenueFromContractWithCustomerTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_NumberOfClassesOfStock_8679be01-7463-4e6b-9937-48551ea7c150_terseLabel_en-US" xlink:label="lab_gemi_NumberOfClassesOfStock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of classes of stock</link:label>
    <link:label id="lab_gemi_NumberOfClassesOfStock_label_en-US" xlink:label="lab_gemi_NumberOfClassesOfStock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Number Of Classes Of Stock</link:label>
    <link:label id="lab_gemi_NumberOfClassesOfStock_documentation_en-US" xlink:label="lab_gemi_NumberOfClassesOfStock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Number Of Classes Of Stock</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_NumberOfClassesOfStock" xlink:href="gemi-20250930.xsd#gemi_NumberOfClassesOfStock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_NumberOfClassesOfStock" xlink:to="lab_gemi_NumberOfClassesOfStock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RecognitionCriteriaNotMetMember_e90b17cc-59a7-4409-a0bd-8271aa1b7662_terseLabel_en-US" xlink:label="lab_gemi_RecognitionCriteriaNotMetMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Recognition Criteria, Not Met</link:label>
    <link:label id="lab_gemi_RecognitionCriteriaNotMetMember_label_en-US" xlink:label="lab_gemi_RecognitionCriteriaNotMetMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Recognition Criteria, Not Met [Member]</link:label>
    <link:label id="lab_gemi_RecognitionCriteriaNotMetMember_documentation_en-US" xlink:label="lab_gemi_RecognitionCriteriaNotMetMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Recognition Criteria, Not Met</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RecognitionCriteriaNotMetMember" xlink:href="gemi-20250930.xsd#gemi_RecognitionCriteriaNotMetMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RecognitionCriteriaNotMetMember" xlink:to="lab_gemi_RecognitionCriteriaNotMetMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfStockByClassTable_99b57177-d46c-4f90-911f-9386131ce3f7_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfStockByClassTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock, Class of Stock [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfStockByClassTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfStockByClassTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock, Class of Stock [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfStockByClassTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfStockByClassTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfStockByClassTable" xlink:to="lab_us-gaap_ScheduleOfStockByClassTable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid_45648214-8940-4e5c-9e52-99d625157087_terseLabel_en-US" xlink:label="lab_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Repayment of third party loan interest denominated in crypto</link:label>
    <link:label id="lab_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid_label_en-US" xlink:label="lab_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Third Party Debt, Denominated In Crypto Asset, Repaid</link:label>
    <link:label id="lab_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid_documentation_en-US" xlink:label="lab_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Third Party Debt, Denominated In Crypto Asset, Repaid</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid" xlink:href="gemi-20250930.xsd#gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid" xlink:to="lab_gemi_ThirdPartyDebtDenominatedInCryptoAssetRepaid" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures_5d78260a-bc39-4f67-bfd0-478c950d22c4_terseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Issuance of common stock upon settlement of RSUs, net of shares withheld (in shares)</link:label>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures_label_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Issued During Period, Shares, Restricted Stock Award, Net of Forfeitures</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures" xlink:to="lab_us-gaap_StockIssuedDuringPeriodSharesRestrictedStockAwardNetOfForfeitures" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CustodyFeesReceivableMember_610d2853-997f-466c-9f8f-0fda658e6bb8_terseLabel_en-US" xlink:label="lab_gemi_CustodyFeesReceivableMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Custody fees receivable</link:label>
    <link:label id="lab_gemi_CustodyFeesReceivableMember_label_en-US" xlink:label="lab_gemi_CustodyFeesReceivableMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Custody Fees Receivable [Member]</link:label>
    <link:label id="lab_gemi_CustodyFeesReceivableMember_documentation_en-US" xlink:label="lab_gemi_CustodyFeesReceivableMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Custody Fees Receivable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CustodyFeesReceivableMember" xlink:href="gemi-20250930.xsd#gemi_CustodyFeesReceivableMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CustodyFeesReceivableMember" xlink:to="lab_gemi_CustodyFeesReceivableMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherAssetsMiscellaneousNoncurrent_261f6ebe-bcba-43b8-82a6-18b62a5accfb_terseLabel_en-US" xlink:label="lab_us-gaap_OtherAssetsMiscellaneousNoncurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other</link:label>
    <link:label id="lab_us-gaap_OtherAssetsMiscellaneousNoncurrent_label_en-US" xlink:label="lab_us-gaap_OtherAssetsMiscellaneousNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Assets, Miscellaneous, Noncurrent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherAssetsMiscellaneousNoncurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherAssetsMiscellaneousNoncurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherAssetsMiscellaneousNoncurrent" xlink:to="lab_us-gaap_OtherAssetsMiscellaneousNoncurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AdjToCompAmt_terseLabel_en-US" xlink:label="lab_ecd_AdjToCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustment to Compensation, Amount</link:label>
    <link:label id="lab_ecd_AdjToCompAmt_label_en-US" xlink:label="lab_ecd_AdjToCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Adjustment to Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AdjToCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AdjToCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AdjToCompAmt" xlink:to="lab_ecd_AdjToCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice_745e3b56-3206-42e7-8c1c-39e5c441f94f_terseLabel_en-US" xlink:label="lab_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percentage of IPO price, resulting from converted debt</link:label>
    <link:label id="lab_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice_label_en-US" xlink:label="lab_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Convertible, Conversion Price, Percentage Of Initial Public Offering Price</link:label>
    <link:label id="lab_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice_documentation_en-US" xlink:label="lab_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Instrument, Convertible, Conversion Price, Percentage Of Initial Public Offering Price</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice" xlink:href="gemi-20250930.xsd#gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice" xlink:to="lab_gemi_DebtInstrumentConvertibleConversionPricePercentageOfInitialPublicOfferingPrice" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetActivityLineItems_4a5e3e62-4b6d-4fd3-850f-f20d05eea40f_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetActivityLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto Asset, Activity [Line Items]</link:label>
    <link:label id="lab_us-gaap_CryptoAssetActivityLineItems_label_en-US" xlink:label="lab_us-gaap_CryptoAssetActivityLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Activity [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetActivityLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetActivityLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetActivityLineItems" xlink:to="lab_us-gaap_CryptoAssetActivityLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets_16225fb8-454a-4e16-abb3-117e07df7b16_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Other assets</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Other Operating Assets</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInOtherOperatingAssets"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:to="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConsolidationPolicyTextBlock_025ff0b0-d32e-4918-875f-be811e62d0c2_terseLabel_en-US" xlink:label="lab_us-gaap_ConsolidationPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Consolidation</link:label>
    <link:label id="lab_us-gaap_ConsolidationPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_ConsolidationPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Consolidation, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConsolidationPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConsolidationPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConsolidationPolicyTextBlock" xlink:to="lab_us-gaap_ConsolidationPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensation_859a6190-cc27-47a9-ae02-1c7797a0e09c_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensation" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock-based compensation</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensation_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensation" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Noncash Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensation" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensation"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensation" xlink:to="lab_us-gaap_ShareBasedCompensation" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EmployeeStockOptionMember_7426ec34-d541-4153-893b-e5f93835baa0_verboseLabel_en-US" xlink:label="lab_us-gaap_EmployeeStockOptionMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">NSOs</link:label>
    <link:label id="lab_us-gaap_EmployeeStockOptionMember_d4aef294-5841-4884-9626-8c597ffa7f08_terseLabel_en-US" xlink:label="lab_us-gaap_EmployeeStockOptionMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Outstanding stock options</link:label>
    <link:label id="lab_us-gaap_EmployeeStockOptionMember_label_en-US" xlink:label="lab_us-gaap_EmployeeStockOptionMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Option [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EmployeeStockOptionMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EmployeeStockOptionMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EmployeeStockOptionMember" xlink:to="lab_us-gaap_EmployeeStockOptionMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfPropertyPlantAndEquipmentTable_9ca8ab94-c684-4a43-8bb8-00d8a51224b5_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfPropertyPlantAndEquipmentTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfPropertyPlantAndEquipmentTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfPropertyPlantAndEquipmentTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfPropertyPlantAndEquipmentTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfPropertyPlantAndEquipmentTable"/>
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    <link:label id="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis_9794718b-cc22-4fa2-9d2e-d476488c8207_terseLabel_en-US" xlink:label="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Antidilutive Securities [Axis]</link:label>
    <link:label id="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis_label_en-US" xlink:label="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Antidilutive Securities [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xlink:to="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareByAntidilutiveSecuritiesAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockholdersEquity_2b65b691-34d7-4cc8-b983-bd47d9ebc585_totalLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total stockholders&#8217; equity (deficit)</link:label>
    <link:label id="lab_us-gaap_StockholdersEquity_c8e0dda4-3c20-417a-afc6-4dadce8430cc_terseLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Beginning balance</link:label>
    <link:label id="lab_us-gaap_StockholdersEquity_d2d83d8b-82d8-4dc7-82d9-1869f6e51b83_periodEndLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Ending balance</link:label>
    <link:label id="lab_us-gaap_StockholdersEquity_label_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity, Attributable to Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockholdersEquity" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockholdersEquity"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockholdersEquity" xlink:to="lab_us-gaap_StockholdersEquity" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetHoldingTableTextBlock_ba7b2cc0-2dcc-48bf-b338-b7ac16158781_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetHoldingTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Crypto Assets Held At Fair Value</link:label>
    <link:label id="lab_us-gaap_CryptoAssetHoldingTableTextBlock_label_en-US" xlink:label="lab_us-gaap_CryptoAssetHoldingTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Holding [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetHoldingTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetHoldingTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetHoldingTableTextBlock" xlink:to="lab_us-gaap_CryptoAssetHoldingTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_InsiderTradingPoliciesProcLineItems_label_en-US" xlink:label="lab_ecd_InsiderTradingPoliciesProcLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_InsiderTradingPoliciesProcLineItems" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_InsiderTradingPoliciesProcLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_InsiderTradingPoliciesProcLineItems" xlink:to="lab_ecd_InsiderTradingPoliciesProcLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentTextBlock_abed9224-7492-40bb-8ef0-c669ef3dc0a7_terseLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Software, Property and Equipment, Net</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentTextBlock_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentTextBlock" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonStockSharesAuthorized_5b3d7361-2210-489f-9384-e9e119ed23f7_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common stock, shares authorized (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonStockSharesAuthorized_label_en-US" xlink:label="lab_us-gaap_CommonStockSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Shares Authorized</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockSharesAuthorized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockSharesAuthorized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockSharesAuthorized" xlink:to="lab_us-gaap_CommonStockSharesAuthorized" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesNewIssues_501cb7fb-251f-495c-a866-69e7dcff5c0b_terseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesNewIssues" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Issuance of common stock (in shares)</link:label>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesNewIssues_label_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesNewIssues" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Issued During Period, Shares, New Issues</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockIssuedDuringPeriodSharesNewIssues" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockIssuedDuringPeriodSharesNewIssues"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockIssuedDuringPeriodSharesNewIssues" xlink:to="lab_us-gaap_StockIssuedDuringPeriodSharesNewIssues" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod_bf236f73-df44-4770-b162-97da7bbbbbec_terseLabel_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash received for issuance of long term debt, utilized during the period</link:label>
    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod_label_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds From Issuance Of Long-Term Debt, Monetary, Used During Period</link:label>
    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod_documentation_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Proceeds From Issuance Of Long-Term Debt, Monetary, Used During Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod" xlink:href="gemi-20250930.xsd#gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod" xlink:to="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetaryUsedDuringPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConcentrationRiskCreditRisk_4f82bbca-6921-4a26-9b01-54a3957d41ea_terseLabel_en-US" xlink:label="lab_us-gaap_ConcentrationRiskCreditRisk" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Concentration of credit risk</link:label>
    <link:label id="lab_us-gaap_ConcentrationRiskCreditRisk_label_en-US" xlink:label="lab_us-gaap_ConcentrationRiskCreditRisk" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Concentration Risk, Credit Risk, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConcentrationRiskCreditRisk" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConcentrationRiskCreditRisk"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConcentrationRiskCreditRisk" xlink:to="lab_us-gaap_ConcentrationRiskCreditRisk" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue_68d2f3de-bc18-4203-bc7b-c2a78b4c5466_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Conversion upon IPO (in dollars per share)</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue_label_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Equity Instruments Other Than Options, Initial Public Offering Conversion, Weighted-Average Grant Date Fair Value</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue_documentation_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Equity Instruments Other Than Options, Weighted-Average Grant Date Fair Value, Initial Public Offering Conversion</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue" xlink:href="gemi-20250930.xsd#gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue" xlink:to="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversionWeightedAverageGrantDateFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A7500ETHAugust52024Member_84b5a117-1a0e-403a-bea6-472575ea78f8_terseLabel_en-US" xlink:label="lab_gemi_A7500ETHAugust52024Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">7,500 ETH, August 5, 2024</link:label>
    <link:label id="lab_gemi_A7500ETHAugust52024Member_label_en-US" xlink:label="lab_gemi_A7500ETHAugust52024Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">7,500 ETH, August 5, 2024 [Member]</link:label>
    <link:label id="lab_gemi_A7500ETHAugust52024Member_documentation_en-US" xlink:label="lab_gemi_A7500ETHAugust52024Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">7,500 ETH, August 5, 2024</link:label>
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    <link:label id="lab_us-gaap_LiabilitiesAndStockholdersEquityAbstract_16fa1272-9083-4324-ba75-8233addd3545_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesAndStockholdersEquityAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Liabilities, Convertible Units, and Members' Deficit</link:label>
    <link:label id="lab_us-gaap_LiabilitiesAndStockholdersEquityAbstract_label_en-US" xlink:label="lab_us-gaap_LiabilitiesAndStockholdersEquityAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liabilities and Equity [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesAndStockholdersEquityAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesAndStockholdersEquityAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilitiesAndStockholdersEquityAbstract" xlink:to="lab_us-gaap_LiabilitiesAndStockholdersEquityAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_CoverAbstract_label_en-US" xlink:label="lab_dei_CoverAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cover [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_CoverAbstract" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_CoverAbstract"/>
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    <link:label id="lab_us-gaap_CashAndCashEquivalentsFairValueDisclosure_f304a5ab-c1dc-4ff3-b8ab-21318983274e_terseLabel_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents:</link:label>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsFairValueDisclosure_b3bc6df8-8dbc-4682-919d-2f83ae15dca9_verboseLabel_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsFairValueDisclosure_label_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash and Cash Equivalents, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashAndCashEquivalentsFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:to="lab_us-gaap_CashAndCashEquivalentsFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConversionOfStockNameDomain_ae404a4e-8d19-4d31-a0bb-3a5046ca1d27_terseLabel_en-US" xlink:label="lab_us-gaap_ConversionOfStockNameDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Conversion of Stock, Name [Domain]</link:label>
    <link:label id="lab_us-gaap_ConversionOfStockNameDomain_label_en-US" xlink:label="lab_us-gaap_ConversionOfStockNameDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Conversion of Stock, Name [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConversionOfStockNameDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConversionOfStockNameDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConversionOfStockNameDomain" xlink:to="lab_us-gaap_ConversionOfStockNameDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember_53e537a6-7b01-4cd0-aa47-9b42f0ca23f8_terseLabel_en-US" xlink:label="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Portion at Fair Value Measurement</link:label>
    <link:label id="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember_label_en-US" xlink:label="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Portion at Fair Value Measurement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PortionAtFairValueFairValueDisclosureMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:to="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonUnitOutstanding_ee0f7088-0722-46fc-a755-d3a17be29318_terseLabel_en-US" xlink:label="lab_us-gaap_CommonUnitOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common units, outstanding (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonUnitOutstanding_label_en-US" xlink:label="lab_us-gaap_CommonUnitOutstanding" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Unit, Outstanding</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonUnitOutstanding" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonUnitOutstanding"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonUnitOutstanding" xlink:to="lab_us-gaap_CommonUnitOutstanding" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_MaximumMember_f420bd1f-caee-4b2c-a6e7-13a3ff26ca07_terseLabel_en-US" xlink:label="lab_srt_MaximumMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Maximum</link:label>
    <link:label id="lab_srt_MaximumMember_label_en-US" xlink:label="lab_srt_MaximumMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Maximum [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_MaximumMember" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_MaximumMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_MaximumMember" xlink:to="lab_srt_MaximumMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice_ed7fa8e3-9c63-45e6-89bc-7c8ea36e9a5f_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Weighted-average exercise price exercisable (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Weighted Average Exercise Price</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableWeightedAverageExercisePrice" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsNet_5f222018-0541-4c9d-a1d3-4133c5d5e277_terseLabel_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Intangible assets, net</link:label>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsNet_75e04c8e-98c0-4a80-bf92-4ef3743bd32e_totalLabel_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsNet" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Intangible Assets, Net</link:label>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsNet_label_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Finite-Lived Intangible Assets, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsNet" xlink:to="lab_us-gaap_FiniteLivedIntangibleAssetsNet" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable_b24048a7-c82c-4003-ab1c-cbe1781e2015_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Repurchase Agreement, Counterparty [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreement, Counterparty [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable" xlink:to="lab_us-gaap_ScheduleOfRepurchaseAgreementCounterpartyTable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetReceivedAsRevenue_bf8aa0d6-f3bf-4d58-a52f-7952af75f526_negatedTerseLabel_en-US" xlink:label="lab_gemi_CryptoAssetReceivedAsRevenue" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets received as revenue</link:label>
    <link:label id="lab_gemi_CryptoAssetReceivedAsRevenue_label_en-US" xlink:label="lab_gemi_CryptoAssetReceivedAsRevenue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Received As Revenue</link:label>
    <link:label id="lab_gemi_CryptoAssetReceivedAsRevenue_documentation_en-US" xlink:label="lab_gemi_CryptoAssetReceivedAsRevenue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Received As Revenue</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetReceivedAsRevenue" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetReceivedAsRevenue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetReceivedAsRevenue" xlink:to="lab_gemi_CryptoAssetReceivedAsRevenue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentCollateralAmount_e1fadf7d-aae8-44b7-ad0f-74f336d7d399_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentCollateralAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit card receivables</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentCollateralAmount_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentCollateralAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Collateral Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentCollateralAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentCollateralAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentCollateralAmount" xlink:to="lab_us-gaap_DebtInstrumentCollateralAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RestrictedStockMember_d269f13d-a939-4024-b5ff-590c9147e622_terseLabel_en-US" xlink:label="lab_us-gaap_RestrictedStockMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">RSAs</link:label>
    <link:label id="lab_us-gaap_RestrictedStockMember_c5ad260b-ce75-4cc9-a1d1-0f31a9536040_verboseLabel_en-US" xlink:label="lab_us-gaap_RestrictedStockMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Unvested RSAs</link:label>
    <link:label id="lab_us-gaap_RestrictedStockMember_label_en-US" xlink:label="lab_us-gaap_RestrictedStockMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Stock [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RestrictedStockMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RestrictedStockMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RestrictedStockMember" xlink:to="lab_us-gaap_RestrictedStockMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonPeoNeoAvgTotalCompAmt_terseLabel_en-US" xlink:label="lab_ecd_NonPeoNeoAvgTotalCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Total Compensation Amount</link:label>
    <link:label id="lab_ecd_NonPeoNeoAvgTotalCompAmt_label_en-US" xlink:label="lab_ecd_NonPeoNeoAvgTotalCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Total Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonPeoNeoAvgTotalCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonPeoNeoAvgTotalCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonPeoNeoAvgTotalCompAmt" xlink:to="lab_ecd_NonPeoNeoAvgTotalCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentLineItems_0a56e43a-e3f1-4fcc-9c19-a0d086c5955f_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Instrument [Line Items]</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentLineItems_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentLineItems" xlink:to="lab_us-gaap_DebtInstrumentLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable_0760e0af-e543-4aad-90df-57de746b27e9_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Share-Based Compensation Arrangements by Share-Based Payment Award [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Share-Based Compensation Arrangements by Share-Based Payment Award [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xlink:to="lab_us-gaap_ScheduleOfShareBasedCompensationArrangementsByShareBasedPaymentAwardTable" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AdjToNonPeoNeoCompFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustment to Non-PEO NEO Compensation Footnote</link:label>
    <link:label id="lab_ecd_AdjToNonPeoNeoCompFnTextBlock_label_en-US" xlink:label="lab_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Adjustment to Non-PEO NEO Compensation Footnote [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AdjToNonPeoNeoCompFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:to="lab_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent_96a7dfd8-f860-4aa1-a981-c253f850d427_terseLabel_en-US" xlink:label="lab_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Public company event, conversion price as a percentage of the price per common unit implied in the public company event</link:label>
    <link:label id="lab_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent_label_en-US" xlink:label="lab_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Conversion, Public Company Event, Price Per Unit As A Percentage Of Price Per Unit Implied In The Public Event</link:label>
    <link:label id="lab_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent_documentation_en-US" xlink:label="lab_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Conversion, Public Company Event, Price Per Unit As A Percentage Of Price Per Unit Implied In The Public Event</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent" xlink:href="gemi-20250930.xsd#gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent" xlink:to="lab_gemi_DebtConversionPublicCompanyEventPricePerUnitAsAPercentageOfPricePerUnitImpliedInThePublicEvent" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Forgone Recovery due to Disqualification of Tax Benefits, Amount</link:label>
    <link:label id="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt_label_en-US" xlink:label="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Forgone Recovery due to Disqualification of Tax Benefits, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:to="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfPreferredUnitsTable_3c9d956e-415e-4a05-b9b2-866d9327bd25_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfPreferredUnitsTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Partners' Capital, Master Limited Partnership, Preferred Unit [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfPreferredUnitsTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfPreferredUnitsTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Partners' Capital, Master Limited Partnership, Preferred Unit [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfPreferredUnitsTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfPreferredUnitsTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfPreferredUnitsTable" xlink:to="lab_us-gaap_ScheduleOfPreferredUnitsTable" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonRule10b51ArrTrmntdFlag_terseLabel_en-US" xlink:label="lab_ecd_NonRule10b51ArrTrmntdFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-Rule 10b5-1 Arrangement Terminated</link:label>
    <link:label id="lab_ecd_NonRule10b51ArrTrmntdFlag_label_en-US" xlink:label="lab_ecd_NonRule10b51ArrTrmntdFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-Rule 10b5-1 Arrangement Terminated [Flag]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonRule10b51ArrTrmntdFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonRule10b51ArrTrmntdFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonRule10b51ArrTrmntdFlag" xlink:to="lab_ecd_NonRule10b51ArrTrmntdFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetFuturesLiabilityMember_ef06e08b-ae8e-4230-93b4-820d94c7454b_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetFuturesLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset futures</link:label>
    <link:label id="lab_gemi_CryptoAssetFuturesLiabilityMember_label_en-US" xlink:label="lab_gemi_CryptoAssetFuturesLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Futures, Liability [Member]</link:label>
    <link:label id="lab_gemi_CryptoAssetFuturesLiabilityMember_documentation_en-US" xlink:label="lab_gemi_CryptoAssetFuturesLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Futures, Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetFuturesLiabilityMember" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetFuturesLiabilityMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetFuturesLiabilityMember" xlink:to="lab_gemi_CryptoAssetFuturesLiabilityMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_78a8b61d-75c6-47f2-a37d-8552f5149f00_totalLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net loss before income taxes</link:label>
    <link:label id="lab_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_label_en-US" xlink:label="lab_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income (Loss) from Continuing Operations before Income Taxes, Noncontrolling Interest</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xlink:to="lab_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityShellCompany_18045d54-ec65-4c17-9503-3adbb0cd4414_terseLabel_en-US" xlink:label="lab_dei_EntityShellCompany" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Shell Company</link:label>
    <link:label id="lab_dei_EntityShellCompany_label_en-US" xlink:label="lab_dei_EntityShellCompany" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Shell Company</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityShellCompany" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityShellCompany"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityShellCompany" xlink:to="lab_dei_EntityShellCompany" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage_6a56a850-ebb6-4c0b-aff1-b57d715a5e26_terseLabel_en-US" xlink:label="lab_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Exercisable increments percentage</link:label>
    <link:label id="lab_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage_label_en-US" xlink:label="lab_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class Of Warrant Or Right, Vesting And Exercisable Increments, Percentage</link:label>
    <link:label id="lab_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage_documentation_en-US" xlink:label="lab_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Class Of Warrant Or Right, Vesting And Exercisable Increments, Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage" xlink:href="gemi-20250930.xsd#gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage" xlink:to="lab_gemi_ClassOfWarrantOrRightVestingAndExercisableIncrementsPercentage" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentTable_11902aef-c6fe-4c93-aecb-721a839a79d1_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Long-Term Debt Instruments [Table]</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentTable_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Long-Term Debt Instruments [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentTable" xlink:to="lab_us-gaap_DebtInstrumentTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis_1fa1fbd1-6f6a-428c-9159-12325e702cda_terseLabel_en-US" xlink:label="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement of Financial Position Location, Activity, Accrual [Axis]</link:label>
    <link:label id="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis_label_en-US" xlink:label="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Financial Position Location, Activity, Accrual [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis" xlink:to="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementOfStockholdersEquityAbstract_label_en-US" xlink:label="lab_us-gaap_StatementOfStockholdersEquityAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Stockholders' Equity [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfStockholdersEquityAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfStockholdersEquityAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfStockholdersEquityAbstract" xlink:to="lab_us-gaap_StatementOfStockholdersEquityAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseRemainingLeaseTerm_47ef47ea-3303-456f-bafa-b5282d993533_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseRemainingLeaseTerm" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Remaining lease term</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseRemainingLeaseTerm_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseRemainingLeaseTerm" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Remaining Lease Term</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseRemainingLeaseTerm" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseRemainingLeaseTerm"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseRemainingLeaseTerm" xlink:to="lab_us-gaap_LesseeOperatingLeaseRemainingLeaseTerm" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardTmgMnpiCnsdrdFlag_terseLabel_en-US" xlink:label="lab_ecd_AwardTmgMnpiCnsdrdFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Timing MNPI Considered</link:label>
    <link:label id="lab_ecd_AwardTmgMnpiCnsdrdFlag_label_en-US" xlink:label="lab_ecd_AwardTmgMnpiCnsdrdFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Timing MNPI Considered [Flag]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardTmgMnpiCnsdrdFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardTmgMnpiCnsdrdFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AwardTmgMnpiCnsdrdFlag" xlink:to="lab_ecd_AwardTmgMnpiCnsdrdFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_FoundersAndExecutiveOfficersMember_b07b02ca-957b-4942-bd4b-69995d90760c_terseLabel_en-US" xlink:label="lab_gemi_FoundersAndExecutiveOfficersMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Founders And Executive Officers</link:label>
    <link:label id="lab_gemi_FoundersAndExecutiveOfficersMember_label_en-US" xlink:label="lab_gemi_FoundersAndExecutiveOfficersMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Founders And Executive Officers [Member]</link:label>
    <link:label id="lab_gemi_FoundersAndExecutiveOfficersMember_documentation_en-US" xlink:label="lab_gemi_FoundersAndExecutiveOfficersMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Founders And Executive Officers</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_FoundersAndExecutiveOfficersMember" xlink:href="gemi-20250930.xsd#gemi_FoundersAndExecutiveOfficersMember"/>
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    <link:label id="lab_us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract_d6bf4ebb-2928-4c44-ba9c-877631f23309_terseLabel_en-US" xlink:label="lab_us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Prepaid expenses and other current assets</link:label>
    <link:label id="lab_us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract_label_en-US" xlink:label="lab_us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Prepaid Expense and Other Assets, Current [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract" xlink:to="lab_us-gaap_PrepaidExpenseAndOtherAssetsCurrentAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PreferredUnitsAuthorized_95e9ad57-66c9-478b-8b69-df5dc0cc41ed_verboseLabel_en-US" xlink:label="lab_us-gaap_PreferredUnitsAuthorized" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Units Authorized (in shares)</link:label>
    <link:label id="lab_us-gaap_PreferredUnitsAuthorized_label_en-US" xlink:label="lab_us-gaap_PreferredUnitsAuthorized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Units, Authorized</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredUnitsAuthorized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredUnitsAuthorized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PreferredUnitsAuthorized" xlink:to="lab_us-gaap_PreferredUnitsAuthorized" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseLiabilityCurrent_026bd376-4cd0-4bd6-b844-499e761ab3de_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiabilityCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Current lease liabilities</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseLiabilityCurrent_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiabilityCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Liability, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseLiabilityCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseLiabilityCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseLiabilityCurrent" xlink:to="lab_us-gaap_OperatingLeaseLiabilityCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A86BTCMarch112025Member_ba97d154-e6f7-4cab-a975-cb206cb4163d_terseLabel_en-US" xlink:label="lab_gemi_A86BTCMarch112025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">86 BTC, March 11, 2025</link:label>
    <link:label id="lab_gemi_A86BTCMarch112025Member_label_en-US" xlink:label="lab_gemi_A86BTCMarch112025Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">86 BTC, March 11, 2025 [Member]</link:label>
    <link:label id="lab_gemi_A86BTCMarch112025Member_documentation_en-US" xlink:label="lab_gemi_A86BTCMarch112025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">86 BTC, March 11, 2025</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A86BTCMarch112025Member" xlink:href="gemi-20250930.xsd#gemi_A86BTCMarch112025Member"/>
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    <link:label id="lab_gemi_Loan4Member_15d044eb-940d-4241-b7d2-c6d15ede5af0_terseLabel_en-US" xlink:label="lab_gemi_Loan4Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 4</link:label>
    <link:label id="lab_gemi_Loan4Member_label_en-US" xlink:label="lab_gemi_Loan4Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 4 [Member]</link:label>
    <link:label id="lab_gemi_Loan4Member_documentation_en-US" xlink:label="lab_gemi_Loan4Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 4</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan4Member" xlink:href="gemi-20250930.xsd#gemi_Loan4Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan4Member" xlink:to="lab_gemi_Loan4Member" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_DocumentFiscalPeriodFocus_470bbb7d-54bf-4dad-9618-ebc74d70fcd9_terseLabel_en-US" xlink:label="lab_dei_DocumentFiscalPeriodFocus" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Document Fiscal Period Focus</link:label>
    <link:label id="lab_dei_DocumentFiscalPeriodFocus_label_en-US" xlink:label="lab_dei_DocumentFiscalPeriodFocus" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Document Fiscal Period Focus</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_DocumentFiscalPeriodFocus" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_DocumentFiscalPeriodFocus"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_DocumentFiscalPeriodFocus" xlink:to="lab_dei_DocumentFiscalPeriodFocus" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion_506d4dc7-4299-41d7-9b60-ac6de45072a4_terseLabel_en-US" xlink:label="lab_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Minimum percentage of outstanding common stock held, triggering automatic conversion of stock</link:label>
    <link:label id="lab_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion_label_en-US" xlink:label="lab_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Convertible, Minimum Ownership Percentage Triggering Conversion</link:label>
    <link:label id="lab_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion_documentation_en-US" xlink:label="lab_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Stock, Convertible, Minimum Ownership Percentage Triggering Conversion</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion" xlink:to="lab_gemi_CommonStockConvertibleMinimumOwnershipPercentageTriggeringConversion" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativesFairValueLineItems_044dc31f-c720-4f38-b308-5cd2387d8c49_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativesFairValueLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivatives, Fair Value [Line Items]</link:label>
    <link:label id="lab_us-gaap_DerivativesFairValueLineItems_label_en-US" xlink:label="lab_us-gaap_DerivativesFairValueLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivatives, Fair Value [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativesFairValueLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativesFairValueLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativesFairValueLineItems" xlink:to="lab_us-gaap_DerivativesFairValueLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TemporaryEquitySharesAuthorized_27dedb39-1f67-4b54-b74c-2c3dbeb6b1d5_terseLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible preferred units, shares authorized (in shares)</link:label>
    <link:label id="lab_us-gaap_TemporaryEquitySharesAuthorized_label_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Temporary Equity, Shares Authorized</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TemporaryEquitySharesAuthorized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TemporaryEquitySharesAuthorized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TemporaryEquitySharesAuthorized" xlink:to="lab_us-gaap_TemporaryEquitySharesAuthorized" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetRealizedGainOperating_828f99dc-7880-4548-8aba-f21ae3e21c70_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetRealizedGainOperating" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Realized gain on crypto assets</link:label>
    <link:label id="lab_us-gaap_CryptoAssetRealizedGainOperating_label_en-US" xlink:label="lab_us-gaap_CryptoAssetRealizedGainOperating" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Realized Gain, Operating</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetRealizedGainOperating" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetRealizedGainOperating"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetRealizedGainOperating" xlink:to="lab_us-gaap_CryptoAssetRealizedGainOperating" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardTmgMethodTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AwardTmgMethodTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Timing Method</link:label>
    <link:label id="lab_ecd_AwardTmgMethodTextBlock_label_en-US" xlink:label="lab_ecd_AwardTmgMethodTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Timing Method [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardTmgMethodTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardTmgMethodTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AwardTmgMethodTextBlock" xlink:to="lab_ecd_AwardTmgMethodTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_LitigationCaseTypeDomain_7dac72f4-67a0-4abf-b2d9-416345d95d92_terseLabel_en-US" xlink:label="lab_srt_LitigationCaseTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Litigation Case [Domain]</link:label>
    <link:label id="lab_srt_LitigationCaseTypeDomain_label_en-US" xlink:label="lab_srt_LitigationCaseTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Litigation Case [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_LitigationCaseTypeDomain" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_LitigationCaseTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_LitigationCaseTypeDomain" xlink:to="lab_srt_LitigationCaseTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetsCollateralPayableCurrent_898d80ba-f993-4bce-b5df-497c15e900c6_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetsCollateralPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset collateral payable</link:label>
    <link:label id="lab_gemi_CryptoAssetsCollateralPayableCurrent_label_en-US" xlink:label="lab_gemi_CryptoAssetsCollateralPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Assets, Collateral Payable, Current</link:label>
    <link:label id="lab_gemi_CryptoAssetsCollateralPayableCurrent_documentation_en-US" xlink:label="lab_gemi_CryptoAssetsCollateralPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Assets, Collateral Payable, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetsCollateralPayableCurrent" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetsCollateralPayableCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetsCollateralPayableCurrent" xlink:to="lab_gemi_CryptoAssetsCollateralPayableCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FurnitureAndFixturesMember_780f0454-ba43-4f4d-9bb8-20c81842fd32_terseLabel_en-US" xlink:label="lab_us-gaap_FurnitureAndFixturesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Furniture and fixtures</link:label>
    <link:label id="lab_us-gaap_FurnitureAndFixturesMember_label_en-US" xlink:label="lab_us-gaap_FurnitureAndFixturesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Furniture and Fixtures [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FurnitureAndFixturesMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FurnitureAndFixturesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FurnitureAndFixturesMember" xlink:to="lab_us-gaap_FurnitureAndFixturesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AwardTypeAxis_b7c04566-82f3-439b-896c-83e1cc7bc461_terseLabel_en-US" xlink:label="lab_us-gaap_AwardTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Type [Axis]</link:label>
    <link:label id="lab_us-gaap_AwardTypeAxis_label_en-US" xlink:label="lab_us-gaap_AwardTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Type [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AwardTypeAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AwardTypeAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AwardTypeAxis" xlink:to="lab_us-gaap_AwardTypeAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetSaleNumberOfUnits_c59da095-4d35-4716-add6-bd7892ed431a_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetSaleNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of crypto assets disposed (in units)</link:label>
    <link:label id="lab_gemi_CryptoAssetSaleNumberOfUnits_label_en-US" xlink:label="lab_gemi_CryptoAssetSaleNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Sale, Number Of Units</link:label>
    <link:label id="lab_gemi_CryptoAssetSaleNumberOfUnits_documentation_en-US" xlink:label="lab_gemi_CryptoAssetSaleNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset, Disposition, Number Of Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetSaleNumberOfUnits" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetSaleNumberOfUnits"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetSaleNumberOfUnits" xlink:to="lab_gemi_CryptoAssetSaleNumberOfUnits" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy_9b262db4-7e5f-4fc2-8549-d5bb685c24ec_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock-based compensation</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:to="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_TradingSymbol_ba6a1cfb-1b2c-44f7-9c4c-d5de49d660bc_terseLabel_en-US" xlink:label="lab_dei_TradingSymbol" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Trading Symbol</link:label>
    <link:label id="lab_dei_TradingSymbol_label_en-US" xlink:label="lab_dei_TradingSymbol" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Symbol</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_TradingSymbol" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_TradingSymbol"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_TradingSymbol" xlink:to="lab_dei_TradingSymbol" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_2c7e6dce-3697-4cf8-a7ee-0465e68d633a_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2029</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year Four</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetDisposition_64677e44-ee33-47b0-8964-771136a83ac6_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetDisposition" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Disposition of crypto assets</link:label>
    <link:label id="lab_us-gaap_CryptoAssetDisposition_label_en-US" xlink:label="lab_us-gaap_CryptoAssetDisposition" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Disposition</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetDisposition" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetDisposition"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetDisposition" xlink:to="lab_us-gaap_CryptoAssetDisposition" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation_26aa3801-065c-45ed-9cb3-9fc955eb1a71_terseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Issuance of common stock upon settlement of stock awards, net of shares withheld (in shares)</link:label>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation_label_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Shares Issued, Shares, Share-Based Payment Arrangement, after Forfeiture</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation" xlink:to="lab_us-gaap_StockIssuedDuringPeriodSharesShareBasedCompensation" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityAddressCityOrTown_37710773-ba43-4144-a315-c4884e16f3d1_terseLabel_en-US" xlink:label="lab_dei_EntityAddressCityOrTown" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, City or Town</link:label>
    <link:label id="lab_dei_EntityAddressCityOrTown_label_en-US" xlink:label="lab_dei_EntityAddressCityOrTown" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Address, City or Town</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressCityOrTown" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressCityOrTown"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityAddressCityOrTown" xlink:to="lab_dei_EntityAddressCityOrTown" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RepurchaseAgreementsDiscount_74b34a45-35c0-4941-8408-1688d93b7574_negatedTerseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsDiscount" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Discount</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsDiscount_label_en-US" xlink:label="lab_gemi_RepurchaseAgreementsDiscount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Discount</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsDiscount_documentation_en-US" xlink:label="lab_gemi_RepurchaseAgreementsDiscount" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Discount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RepurchaseAgreementsDiscount" xlink:href="gemi-20250930.xsd#gemi_RepurchaseAgreementsDiscount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RepurchaseAgreementsDiscount" xlink:to="lab_gemi_RepurchaseAgreementsDiscount" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShortTermDebtTypeDomain_5b208d7f-eccc-47f0-b02b-d09f58435aa3_terseLabel_en-US" xlink:label="lab_us-gaap_ShortTermDebtTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Short-Term Debt, Type [Domain]</link:label>
    <link:label id="lab_us-gaap_ShortTermDebtTypeDomain_label_en-US" xlink:label="lab_us-gaap_ShortTermDebtTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Short-Term Debt, Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShortTermDebtTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShortTermDebtTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShortTermDebtTypeDomain" xlink:to="lab_us-gaap_ShortTermDebtTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance_83137983-6798-4288-a7f6-1eabff8af4eb_terseLabel_en-US" xlink:label="lab_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Recognized revenue related to contract liabilities</link:label>
    <link:label id="lab_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance_label_en-US" xlink:label="lab_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Contract With Customer, Liability, Revenue Recognized, Excluding Opening Balance</link:label>
    <link:label id="lab_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance_documentation_en-US" xlink:label="lab_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Contract With Customer, Liability, Revenue Recognized, Excluding Opening Balance</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance" xlink:href="gemi-20250930.xsd#gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance" xlink:to="lab_gemi_ContractWithCustomerLiabilityRevenueRecognizedExcludingOpeningBalance" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PreferredUnitsByNameAxis_23d679a7-f657-4d99-a591-44ae63988861_terseLabel_en-US" xlink:label="lab_us-gaap_PreferredUnitsByNameAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Preferred Units by Name [Axis]</link:label>
    <link:label id="lab_us-gaap_PreferredUnitsByNameAxis_label_en-US" xlink:label="lab_us-gaap_PreferredUnitsByNameAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Units by Name [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredUnitsByNameAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredUnitsByNameAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PreferredUnitsByNameAxis" xlink:to="lab_us-gaap_PreferredUnitsByNameAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IntangibleAssetsDisclosureTextBlock_a314a16a-0625-4e98-ab7e-707de2067af8_terseLabel_en-US" xlink:label="lab_us-gaap_IntangibleAssetsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Intangible Assets, Net</link:label>
    <link:label id="lab_us-gaap_IntangibleAssetsDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_IntangibleAssetsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Intangible Assets Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IntangibleAssetsDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IntangibleAssetsDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IntangibleAssetsDisclosureTextBlock" xlink:to="lab_us-gaap_IntangibleAssetsDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CreditCardReceivablesPledgedCurrent_fc853894-d43d-4382-9e4d-0494d57d3bf2_terseLabel_en-US" xlink:label="lab_gemi_CreditCardReceivablesPledgedCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit card receivables pledged</link:label>
    <link:label id="lab_gemi_CreditCardReceivablesPledgedCurrent_label_en-US" xlink:label="lab_gemi_CreditCardReceivablesPledgedCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Card Receivables Pledged, Current</link:label>
    <link:label id="lab_gemi_CreditCardReceivablesPledgedCurrent_documentation_en-US" xlink:label="lab_gemi_CreditCardReceivablesPledgedCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Credit Card Receivables Pledged, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CreditCardReceivablesPledgedCurrent" xlink:href="gemi-20250930.xsd#gemi_CreditCardReceivablesPledgedCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CreditCardReceivablesPledgedCurrent" xlink:to="lab_gemi_CreditCardReceivablesPledgedCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt_terseLabel_en-US" xlink:label="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Compensation Actually Paid Amount</link:label>
    <link:label id="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt_label_en-US" xlink:label="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Compensation Actually Paid Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonPeoNeoAvgCompActuallyPaidAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:to="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RestrictedStockUnitsRSUMember_710d0cb1-16d5-49a0-a4fd-d5d40fbe8fae_terseLabel_en-US" xlink:label="lab_us-gaap_RestrictedStockUnitsRSUMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">RSUs</link:label>
    <link:label id="lab_us-gaap_RestrictedStockUnitsRSUMember_7320e561-2328-4e11-8dc4-818233846f7d_verboseLabel_en-US" xlink:label="lab_us-gaap_RestrictedStockUnitsRSUMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Unvested RSUs</link:label>
    <link:label id="lab_us-gaap_RestrictedStockUnitsRSUMember_label_en-US" xlink:label="lab_us-gaap_RestrictedStockUnitsRSUMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Stock Units (RSUs) [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RestrictedStockUnitsRSUMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RestrictedStockUnitsRSUMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RestrictedStockUnitsRSUMember" xlink:to="lab_us-gaap_RestrictedStockUnitsRSUMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock_terseLabel_en-US" xlink:label="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Other Measure</link:label>
    <link:label id="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock_label_en-US" xlink:label="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Other Measure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CompActuallyPaidVsOtherMeasureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:to="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:type="arc" order="1"/>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PreferredStockSharesIssued" xlink:to="lab_us-gaap_PreferredStockSharesIssued" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PnsnAdjsSvcCstMember_terseLabel_en-US" xlink:label="lab_ecd_PnsnAdjsSvcCstMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pension Adjustments Service Cost</link:label>
    <link:label id="lab_ecd_PnsnAdjsSvcCstMember_label_en-US" xlink:label="lab_ecd_PnsnAdjsSvcCstMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pension Adjustments Service Cost [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PnsnAdjsSvcCstMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PnsnAdjsSvcCstMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PnsnAdjsSvcCstMember" xlink:to="lab_ecd_PnsnAdjsSvcCstMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoLoansPayableRealizedGainLossNonoperating_77981e29-9f8d-402e-8b20-7458a8ac811a_terseLabel_en-US" xlink:label="lab_gemi_CryptoLoansPayableRealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Realized Gain (Loss)</link:label>
    <link:label id="lab_gemi_CryptoLoansPayableRealizedGainLossNonoperating_label_en-US" xlink:label="lab_gemi_CryptoLoansPayableRealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Loans Payable, Realized Gain (Loss), Nonoperating</link:label>
    <link:label id="lab_gemi_CryptoLoansPayableRealizedGainLossNonoperating_documentation_en-US" xlink:label="lab_gemi_CryptoLoansPayableRealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Loans Payable, Realized Gain (Loss), Nonoperating</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoLoansPayableRealizedGainLossNonoperating" xlink:href="gemi-20250930.xsd#gemi_CryptoLoansPayableRealizedGainLossNonoperating"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoLoansPayableRealizedGainLossNonoperating" xlink:to="lab_gemi_CryptoLoansPayableRealizedGainLossNonoperating" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis_5708e9e2-094b-4d72-b4a5-a17a25078ce8_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value Hierarchy and NAV [Axis]</link:label>
    <link:label id="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis_label_en-US" xlink:label="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Hierarchy and NAV [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueByFairValueHierarchyLevelAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:to="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetOptionsLiabilityMember_4af75420-08b4-4f60-bfb3-f4f0023dde45_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetOptionsLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset options</link:label>
    <link:label id="lab_gemi_CryptoAssetOptionsLiabilityMember_label_en-US" xlink:label="lab_gemi_CryptoAssetOptionsLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Options, Liability [Member]</link:label>
    <link:label id="lab_gemi_CryptoAssetOptionsLiabilityMember_documentation_en-US" xlink:label="lab_gemi_CryptoAssetOptionsLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Options, Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetOptionsLiabilityMember" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetOptionsLiabilityMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetOptionsLiabilityMember" xlink:to="lab_gemi_CryptoAssetOptionsLiabilityMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions_f125c06d-21a1-4817-80f1-01fac4824a65_negatedTerseLabel_en-US" xlink:label="lab_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Custodial funds due to customers, net of redemptions</link:label>
    <link:label id="lab_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions_label_en-US" xlink:label="lab_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payments For Custodial Funds Due To Customers, Net Of Redemptions</link:label>
    <link:label id="lab_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions_documentation_en-US" xlink:label="lab_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Payments For Custodial Funds Due To Customers, Net Of Redemptions</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions" xlink:href="gemi-20250930.xsd#gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions" xlink:to="lab_gemi_PaymentsForCustodialFundsDueToCustomersNetOfRedemptions" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ReceivableCryptoAssetsPledgedCurrent_319d2e52-5b07-4926-978f-9b7b55476930_terseLabel_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetsPledgedCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Receivable, crypto assets pledged</link:label>
    <link:label id="lab_gemi_ReceivableCryptoAssetsPledgedCurrent_label_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetsPledgedCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Receivable, Crypto Assets Pledged, Current</link:label>
    <link:label id="lab_gemi_ReceivableCryptoAssetsPledgedCurrent_documentation_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetsPledgedCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Receivable, Crypto Assets Pledged, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ReceivableCryptoAssetsPledgedCurrent" xlink:href="gemi-20250930.xsd#gemi_ReceivableCryptoAssetsPledgedCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ReceivableCryptoAssetsPledgedCurrent" xlink:to="lab_gemi_ReceivableCryptoAssetsPledgedCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DebtInstrumentCollateralRate_1eb3427e-d951-43c6-8b16-7d6ea6d7c240_terseLabel_en-US" xlink:label="lab_gemi_DebtInstrumentCollateralRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collateral Rate (Initial Collateral Level)</link:label>
    <link:label id="lab_gemi_DebtInstrumentCollateralRate_label_en-US" xlink:label="lab_gemi_DebtInstrumentCollateralRate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Collateral Rate</link:label>
    <link:label id="lab_gemi_DebtInstrumentCollateralRate_documentation_en-US" xlink:label="lab_gemi_DebtInstrumentCollateralRate" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Instrument, Collateral Rate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DebtInstrumentCollateralRate" xlink:href="gemi-20250930.xsd#gemi_DebtInstrumentCollateralRate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtInstrumentCollateralRate" xlink:to="lab_gemi_DebtInstrumentCollateralRate" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentGross_c8c433f5-5576-40fc-aef0-9d0d7dca0803_terseLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentGross" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Software, property and equipment, gross</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentGross_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentGross" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment, Gross</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentGross" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentGross"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentGross" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentGross" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent_efb3e655-1c49-4080-82a6-9ddf4399f653_terseLabel_en-US" xlink:label="lab_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Third party loans</link:label>
    <link:label id="lab_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent_label_en-US" xlink:label="lab_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Third Party Debt, Including Accrued Interest, Current</link:label>
    <link:label id="lab_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent_documentation_en-US" xlink:label="lab_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Third Party Debt, Including Accrued Interest, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent" xlink:href="gemi-20250930.xsd#gemi_ThirdPartyDebtIncludingAccruedInterestCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent" xlink:to="lab_gemi_ThirdPartyDebtIncludingAccruedInterestCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetHoldingLineItems_0f8659ea-f765-4588-b909-7e235096b5b0_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetHoldingLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto Asset, Holding [Line Items]</link:label>
    <link:label id="lab_us-gaap_CryptoAssetHoldingLineItems_label_en-US" xlink:label="lab_us-gaap_CryptoAssetHoldingLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Holding [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetHoldingLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetHoldingLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetHoldingLineItems" xlink:to="lab_us-gaap_CryptoAssetHoldingLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock_terseLabel_en-US" xlink:label="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Company Selected Measure</link:label>
    <link:label id="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock_label_en-US" xlink:label="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Company Selected Measure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:to="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ProceedsFromIssuanceOfPrivatePlacement_cef0b26d-5a48-45eb-9ea3-23d43d7d0523_terseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from issuance of common stock in connection with concurrent private placement with Nasdaq</link:label>
    <link:label id="lab_us-gaap_ProceedsFromIssuanceOfPrivatePlacement_7f934757-1b56-45f1-94e6-c373c5dee0cc_verboseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Gross proceeds from stock issued and sold</link:label>
    <link:label id="lab_us-gaap_ProceedsFromIssuanceOfPrivatePlacement_label_en-US" xlink:label="lab_us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds from Issuance of Private Placement</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromIssuanceOfPrivatePlacement"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:to="lab_us-gaap_ProceedsFromIssuanceOfPrivatePlacement" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AllTradingArrangementsMember_terseLabel_en-US" xlink:label="lab_ecd_AllTradingArrangementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Trading Arrangements</link:label>
    <link:label id="lab_ecd_AllTradingArrangementsMember_label_en-US" xlink:label="lab_ecd_AllTradingArrangementsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">All Trading Arrangements [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AllTradingArrangementsMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AllTradingArrangementsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AllTradingArrangementsMember" xlink:to="lab_ecd_AllTradingArrangementsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_EqtyAwrdsAdjFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments, Footnote</link:label>
    <link:label id="lab_ecd_EqtyAwrdsAdjFnTextBlock_label_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments, Footnote [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_EqtyAwrdsAdjFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_EqtyAwrdsAdjFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_EqtyAwrdsAdjFnTextBlock" xlink:to="lab_ecd_EqtyAwrdsAdjFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TotalShareholderRtnVsPeerGroupTextBlock_terseLabel_en-US" xlink:label="lab_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total Shareholder Return Vs Peer Group</link:label>
    <link:label id="lab_ecd_TotalShareholderRtnVsPeerGroupTextBlock_label_en-US" xlink:label="lab_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Total Shareholder Return Vs Peer Group [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TotalShareholderRtnVsPeerGroupTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:to="lab_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AccruedLiabilitiesCurrent_b33de26b-a51e-42e7-883f-28491746e6b4_terseLabel_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued expenses</link:label>
    <link:label id="lab_us-gaap_AccruedLiabilitiesCurrent_95299356-46e0-44cf-bda9-530a28bf5370_totalLabel_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total accrued expenses</link:label>
    <link:label id="lab_us-gaap_AccruedLiabilitiesCurrent_label_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accrued Liabilities, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccruedLiabilitiesCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccruedLiabilitiesCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccruedLiabilitiesCurrent" xlink:to="lab_us-gaap_AccruedLiabilitiesCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PaymentsToDevelopSoftware_b0678299-2b30-408e-a0d1-791cd1b67d8a_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_PaymentsToDevelopSoftware" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Capitalization of internally developed software costs</link:label>
    <link:label id="lab_us-gaap_PaymentsToDevelopSoftware_label_en-US" xlink:label="lab_us-gaap_PaymentsToDevelopSoftware" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payments to Develop Software</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsToDevelopSoftware" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsToDevelopSoftware"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PaymentsToDevelopSoftware" xlink:to="lab_us-gaap_PaymentsToDevelopSoftware" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber_6c3c4405-7c4a-4b1e-bf88-0cf1db7d742b_periodStartLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Unvested beginning balance</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber_f994d584-af91-4355-b700-5a0b73d6280b_periodEndLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Unvested ending balance</link:label>
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    <link:label id="lab_us-gaap_CommitmentsAndContingenciesDisclosureAbstract_label_en-US" xlink:label="lab_us-gaap_CommitmentsAndContingenciesDisclosureAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Commitments and Contingencies Disclosure [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommitmentsAndContingenciesDisclosureAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommitmentsAndContingenciesDisclosureAbstract"/>
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    <link:label id="lab_gemi_OfficeSpaceInMiamiFloridaMember_7cc75a16-e664-4a8d-9fc9-59b6556766fc_terseLabel_en-US" xlink:label="lab_gemi_OfficeSpaceInMiamiFloridaMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Office Space In Miami, Florida</link:label>
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    <link:label id="lab_us-gaap_NetCashProvidedByUsedInOperatingActivities_1384b951-bf5e-4b1a-95a7-e0b8349d03a6_totalLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInOperatingActivities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net cash used in operating activities</link:label>
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    <link:label id="lab_gemi_OfficeSpaceInNewYorkCityMember_7f60a261-bac9-4939-9633-7d831f6c8864_terseLabel_en-US" xlink:label="lab_gemi_OfficeSpaceInNewYorkCityMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Office Space In New York City</link:label>
    <link:label id="lab_gemi_OfficeSpaceInNewYorkCityMember_label_en-US" xlink:label="lab_gemi_OfficeSpaceInNewYorkCityMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Office Space In New York City [Member]</link:label>
    <link:label id="lab_gemi_OfficeSpaceInNewYorkCityMember_documentation_en-US" xlink:label="lab_gemi_OfficeSpaceInNewYorkCityMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Office Space In New York City</link:label>
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    <link:label id="lab_gemi_Loan3Member_73f16626-5ea0-4f35-9fc7-09d190c211e5_terseLabel_en-US" xlink:label="lab_gemi_Loan3Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 3</link:label>
    <link:label id="lab_gemi_Loan3Member_label_en-US" xlink:label="lab_gemi_Loan3Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 3 [Member]</link:label>
    <link:label id="lab_gemi_Loan3Member_documentation_en-US" xlink:label="lab_gemi_Loan3Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 3</link:label>
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    <link:label id="lab_us-gaap_PreferredStockParOrStatedValuePerShare_cacdd28b-a01b-4b71-a361-28a95a7785fc_verboseLabel_en-US" xlink:label="lab_us-gaap_PreferredStockParOrStatedValuePerShare" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Preferred stock, par value (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_PreferredStockParOrStatedValuePerShare_label_en-US" xlink:label="lab_us-gaap_PreferredStockParOrStatedValuePerShare" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Stock, Par or Stated Value Per Share</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredStockParOrStatedValuePerShare" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredStockParOrStatedValuePerShare"/>
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    <link:label id="lab_us-gaap_SubsequentEventMember_6211de9c-c6f9-42f5-88ac-23994d44a8bc_terseLabel_en-US" xlink:label="lab_us-gaap_SubsequentEventMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Subsequent Event</link:label>
    <link:label id="lab_us-gaap_SubsequentEventMember_label_en-US" xlink:label="lab_us-gaap_SubsequentEventMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Subsequent Event [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubsequentEventMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubsequentEventMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SubsequentEventMember" xlink:to="lab_us-gaap_SubsequentEventMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfShareBasedPaymentAwardStockOptionsValuationAssumptionsTableTextBlock_715c48fd-e239-4e73-808c-d898a802b75c_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfShareBasedPaymentAwardStockOptionsValuationAssumptionsTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Share-Based Payment Award, Stock Options, Valuation Assumptions</link:label>
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    <link:label id="lab_us-gaap_RepurchaseAgreementsInterestExpenseAmount_554261f8-d49b-4485-a3ac-4c85e69907ff_terseLabel_en-US" xlink:label="lab_us-gaap_RepurchaseAgreementsInterestExpenseAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest Expense</link:label>
    <link:label id="lab_us-gaap_RepurchaseAgreementsInterestExpenseAmount_label_en-US" xlink:label="lab_us-gaap_RepurchaseAgreementsInterestExpenseAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Interest Expense Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RepurchaseAgreementsInterestExpenseAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RepurchaseAgreementsInterestExpenseAmount"/>
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    <link:label id="lab_ecd_PeoTotalCompAmt_terseLabel_en-US" xlink:label="lab_ecd_PeoTotalCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">PEO Total Compensation Amount</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeoTotalCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeoTotalCompAmt"/>
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    <link:label id="lab_us-gaap_LongtermDebtTypeAxis_74161786-6376-4772-af48-095511ed79fc_terseLabel_en-US" xlink:label="lab_us-gaap_LongtermDebtTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Type [Axis]</link:label>
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    <link:label id="lab_us-gaap_Depreciation_9f51930c-93f5-4b05-a041-3dcf95a968d6_terseLabel_en-US" xlink:label="lab_us-gaap_Depreciation" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Depreciation expense</link:label>
    <link:label id="lab_us-gaap_Depreciation_label_en-US" xlink:label="lab_us-gaap_Depreciation" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Depreciation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Depreciation" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Depreciation"/>
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    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRemainderOfFiscalYear_c8017374-5972-4d8e-a9f5-8dc01f23a595_terseLabel_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRemainderOfFiscalYear" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Remaining 2025</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_gemi_DebtInstrumentCovenantRestrictedDrawdownsStablecoinHeldMinimumValue" xlink:href="gemi-20250930.xsd#gemi_DebtInstrumentCovenantRestrictedDrawdownsStablecoinHeldMinimumValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtInstrumentCovenantRestrictedDrawdownsStablecoinHeldMinimumValue" xlink:to="lab_gemi_DebtInstrumentCovenantRestrictedDrawdownsStablecoinHeldMinimumValue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementEquityComponentsAxis_f283d449-9888-4279-9638-ca8b9fd3c183_terseLabel_en-US" xlink:label="lab_us-gaap_StatementEquityComponentsAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Components [Axis]</link:label>
    <link:label id="lab_us-gaap_StatementEquityComponentsAxis_label_en-US" xlink:label="lab_us-gaap_StatementEquityComponentsAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Components [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementEquityComponentsAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementEquityComponentsAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementEquityComponentsAxis" xlink:to="lab_us-gaap_StatementEquityComponentsAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityExTransitionPeriod_ec8d153d-7c29-4b06-9655-200e28dedbc8_terseLabel_en-US" xlink:label="lab_dei_EntityExTransitionPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Ex Transition Period</link:label>
    <link:label id="lab_dei_EntityExTransitionPeriod_label_en-US" xlink:label="lab_dei_EntityExTransitionPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Ex Transition Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityExTransitionPeriod" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityExTransitionPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityExTransitionPeriod" xlink:to="lab_dei_EntityExTransitionPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentLineItems_46558875-f6b7-4c07-8f30-5468c881e90f_terseLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment [Line Items]</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentLineItems_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentLineItems" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SubleaseIncome_9982b863-3d8c-4d97-9797-158ee65be48c_terseLabel_en-US" xlink:label="lab_us-gaap_SubleaseIncome" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Sublease income</link:label>
    <link:label id="lab_us-gaap_SubleaseIncome_label_en-US" xlink:label="lab_us-gaap_SubleaseIncome" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Sublease Income</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubleaseIncome" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubleaseIncome"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SubleaseIncome" xlink:to="lab_us-gaap_SubleaseIncome" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConvertibleDebtMember_fc9bc7e3-fb22-486b-bc3c-990a36f533cb_terseLabel_en-US" xlink:label="lab_us-gaap_ConvertibleDebtMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible Notes</link:label>
    <link:label id="lab_us-gaap_ConvertibleDebtMember_label_en-US" xlink:label="lab_us-gaap_ConvertibleDebtMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Convertible Debt [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConvertibleDebtMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConvertibleDebtMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConvertibleDebtMember" xlink:to="lab_us-gaap_ConvertibleDebtMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonRule10b51ArrAdoptedFlag_terseLabel_en-US" xlink:label="lab_ecd_NonRule10b51ArrAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-Rule 10b5-1 Arrangement Adopted</link:label>
    <link:label id="lab_ecd_NonRule10b51ArrAdoptedFlag_label_en-US" xlink:label="lab_ecd_NonRule10b51ArrAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-Rule 10b5-1 Arrangement Adopted [Flag]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonRule10b51ArrAdoptedFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonRule10b51ArrAdoptedFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonRule10b51ArrAdoptedFlag" xlink:to="lab_ecd_NonRule10b51ArrAdoptedFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod_2b1ffead-6dfd-4c60-99b6-1ee6c179823a_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Expired (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Expirations in Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExpirationsInPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LoansPayable_d1c719e5-9807-48e3-8d5b-3757c217edee_terseLabel_en-US" xlink:label="lab_us-gaap_LoansPayable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Principal Outstanding</link:label>
    <link:label id="lab_us-gaap_LoansPayable_label_en-US" xlink:label="lab_us-gaap_LoansPayable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loans Payable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LoansPayable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LoansPayable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LoansPayable" xlink:to="lab_us-gaap_LoansPayable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A500BTCMay112023Member_283a2831-b405-422c-b031-30742cf88ef5_terseLabel_en-US" xlink:label="lab_gemi_A500BTCMay112023Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">500 BTC May 11, 2023</link:label>
    <link:label id="lab_gemi_A500BTCMay112023Member_label_en-US" xlink:label="lab_gemi_A500BTCMay112023Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">500 BTC May 11, 2023 [Member]</link:label>
    <link:label id="lab_gemi_A500BTCMay112023Member_documentation_en-US" xlink:label="lab_gemi_A500BTCMay112023Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">500 BTC May 11, 2023</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A500BTCMay112023Member" xlink:href="gemi-20250930.xsd#gemi_A500BTCMay112023Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A500BTCMay112023Member" xlink:to="lab_gemi_A500BTCMay112023Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk_cf1b07c7-a13b-4553-ba4c-1f4e93614bb3_terseLabel_en-US" xlink:label="lab_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest payable, fair value adjustments related to instrument-specific credit risk gain</link:label>
    <link:label id="lab_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk_label_en-US" xlink:label="lab_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Payable, Current, Fair Value Adjustment For Instrument-Specific Credit Risk</link:label>
    <link:label id="lab_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk_documentation_en-US" xlink:label="lab_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Interest Payable, Current, Fair Value Adjustment For Instrument-Specific Credit Risk</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk" xlink:href="gemi-20250930.xsd#gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk" xlink:to="lab_gemi_InterestPayableCurrentFairValueAdjustmentForInstrumentSpecificCreditRisk" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdvertisingCostsPolicyTextBlock_5d57eaa5-6b70-4788-afc0-a8c17cc19312_terseLabel_en-US" xlink:label="lab_us-gaap_AdvertisingCostsPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Sales and marketing</link:label>
    <link:label id="lab_us-gaap_AdvertisingCostsPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_AdvertisingCostsPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Advertising Cost [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdvertisingCostsPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdvertisingCostsPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdvertisingCostsPolicyTextBlock" xlink:to="lab_us-gaap_AdvertisingCostsPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CustomerCustodialFundsCurrent_010ae91c-6650-458e-b6b2-c9f0b18dae04_terseLabel_en-US" xlink:label="lab_gemi_CustomerCustodialFundsCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Customer custodial funds</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsCurrent_label_en-US" xlink:label="lab_gemi_CustomerCustodialFundsCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds, Current</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsCurrent_documentation_en-US" xlink:label="lab_gemi_CustomerCustodialFundsCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CustomerCustodialFundsCurrent" xlink:href="gemi-20250930.xsd#gemi_CustomerCustodialFundsCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CustomerCustodialFundsCurrent" xlink:to="lab_gemi_CustomerCustodialFundsCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityAddressStateOrProvince_815f212f-df08-4367-8caa-e4139b8dcd5e_terseLabel_en-US" xlink:label="lab_dei_EntityAddressStateOrProvince" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, State or Province</link:label>
    <link:label id="lab_dei_EntityAddressStateOrProvince_label_en-US" xlink:label="lab_dei_EntityAddressStateOrProvince" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Address, State or Province</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressStateOrProvince" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressStateOrProvince"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityAddressStateOrProvince" xlink:to="lab_dei_EntityAddressStateOrProvince" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FinancingReceivablesPeriodPastDueDomain_2f10f42e-0d02-4f95-9e23-96fc55156ea8_terseLabel_en-US" xlink:label="lab_us-gaap_FinancingReceivablesPeriodPastDueDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Financial Asset, Aging [Domain]</link:label>
    <link:label id="lab_us-gaap_FinancingReceivablesPeriodPastDueDomain_label_en-US" xlink:label="lab_us-gaap_FinancingReceivablesPeriodPastDueDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Financial Asset, Aging [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivablesPeriodPastDueDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivablesPeriodPastDueDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FinancingReceivablesPeriodPastDueDomain" xlink:to="lab_us-gaap_FinancingReceivablesPeriodPastDueDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable_5ab84484-a71f-4827-9777-b02319090b63_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Antidilutive Security, Excluded EPS Calculation [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Antidilutive Security, Excluded EPS Calculation [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable" xlink:to="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ProceedsFromDisposalCryptoAsset_1289e3e4-5891-4ce6-9cf7-f8daef49a427_terseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromDisposalCryptoAsset" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from disposal of crypto assets</link:label>
    <link:label id="lab_us-gaap_ProceedsFromDisposalCryptoAsset_label_en-US" xlink:label="lab_us-gaap_ProceedsFromDisposalCryptoAsset" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds from Disposal, Crypto Asset</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromDisposalCryptoAsset" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromDisposalCryptoAsset"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ProceedsFromDisposalCryptoAsset" xlink:to="lab_us-gaap_ProceedsFromDisposalCryptoAsset" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LiabilitiesCurrent_352b725b-cfb5-403f-846e-ea14e3279b7a_totalLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total current liabilities</link:label>
    <link:label id="lab_us-gaap_LiabilitiesCurrent_label_en-US" xlink:label="lab_us-gaap_LiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liabilities, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilitiesCurrent" xlink:to="lab_us-gaap_LiabilitiesCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain_29d9763d-69a9-4953-a372-131a81368475_terseLabel_en-US" xlink:label="lab_us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Long-Term Purchase Commitment, Category of Item Purchased [Domain]</link:label>
    <link:label id="lab_us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain_label_en-US" xlink:label="lab_us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Purchase Commitment, Category of Item Purchased [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain" xlink:to="lab_us-gaap_LongTermPurchaseCommitmentCategoryOfItemPurchasedDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract_643ee245-b753-4429-9774-f98e8b70359c_terseLabel_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other comprehensive income (loss):</link:label>
    <link:label id="lab_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract_label_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Comprehensive Income (Loss), before Tax, Portion Attributable to Parent [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract" xlink:to="lab_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParentAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_IndividualAxis_terseLabel_en-US" xlink:label="lab_ecd_IndividualAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Individual:</link:label>
    <link:label id="lab_ecd_IndividualAxis_label_en-US" xlink:label="lab_ecd_IndividualAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Individual [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_IndividualAxis" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_IndividualAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_IndividualAxis" xlink:to="lab_ecd_IndividualAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_TransactionRevenueExchangeMember_7c1b02ad-75b0-4c35-b0fe-34ad8031c224_terseLabel_en-US" xlink:label="lab_gemi_TransactionRevenueExchangeMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Exchange</link:label>
    <link:label id="lab_gemi_TransactionRevenueExchangeMember_label_en-US" xlink:label="lab_gemi_TransactionRevenueExchangeMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Transaction Revenue, Exchange [Member]</link:label>
    <link:label id="lab_gemi_TransactionRevenueExchangeMember_documentation_en-US" xlink:label="lab_gemi_TransactionRevenueExchangeMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Transaction Revenue, Exchange</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TransactionRevenueExchangeMember" xlink:href="gemi-20250930.xsd#gemi_TransactionRevenueExchangeMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_TransactionRevenueExchangeMember" xlink:to="lab_gemi_TransactionRevenueExchangeMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdjustmentForAmortization_bb883d02-0b08-4941-b07a-0e108cfffbef_terseLabel_en-US" xlink:label="lab_us-gaap_AdjustmentForAmortization" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amortization expenses</link:label>
    <link:label id="lab_us-gaap_AdjustmentForAmortization_label_en-US" xlink:label="lab_us-gaap_AdjustmentForAmortization" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Amortization</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdjustmentForAmortization" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdjustmentForAmortization"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdjustmentForAmortization" xlink:to="lab_us-gaap_AdjustmentForAmortization" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions_bacbf0f8-620c-4a58-b0c3-77cba7e1f87a_negatedTerseLabel_en-US" xlink:label="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Net reductions</link:label>
    <link:label id="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions_label_en-US" xlink:label="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Securities Without Readily Determinable Fair Value, Net Reductions</link:label>
    <link:label id="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions_documentation_en-US" xlink:label="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Equity Securities Without Readily Determinable Fair Value, Net Reductions</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions" xlink:href="gemi-20250930.xsd#gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions" xlink:to="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueNetReductions" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember_f40a0ecf-60af-4682-bd2a-dd3e39dda5f9_terseLabel_en-US" xlink:label="lab_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">NYDIG</link:label>
    <link:label id="lab_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember_label_en-US" xlink:label="lab_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">New York Digital Investment Group Funding LLC ("NYDIG") [Member]</link:label>
    <link:label id="lab_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember_documentation_en-US" xlink:label="lab_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">New York Digital Investment Group Funding LLC ("NYDIG")</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember" xlink:href="gemi-20250930.xsd#gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember" xlink:to="lab_gemi_NewYorkDigitalInvestmentGroupFundingLLCNYDIGMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Forgone Recovery, Explanation of Impracticability</link:label>
    <link:label id="lab_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock_label_en-US" xlink:label="lab_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Forgone Recovery, Explanation of Impracticability [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock" xlink:to="lab_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RepaymentsOfLongTermDebt_75aca942-6802-4439-880e-27fae1aa1cd1_terseLabel_en-US" xlink:label="lab_us-gaap_RepaymentsOfLongTermDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Repayment of aggregate principal amount of outstanding debt</link:label>
    <link:label id="lab_us-gaap_RepaymentsOfLongTermDebt_label_en-US" xlink:label="lab_us-gaap_RepaymentsOfLongTermDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repayments of Long-Term Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RepaymentsOfLongTermDebt" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RepaymentsOfLongTermDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RepaymentsOfLongTermDebt" xlink:to="lab_us-gaap_RepaymentsOfLongTermDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityInteractiveDataCurrent_712aba1a-341d-42dd-bdb5-7daeea0b64a2_terseLabel_en-US" xlink:label="lab_dei_EntityInteractiveDataCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Interactive Data Current</link:label>
    <link:label id="lab_dei_EntityInteractiveDataCurrent_label_en-US" xlink:label="lab_dei_EntityInteractiveDataCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Interactive Data Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityInteractiveDataCurrent" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityInteractiveDataCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityInteractiveDataCurrent" xlink:to="lab_dei_EntityInteractiveDataCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShortTermDebtAbstractRollForward_6715350b-79c5-48ad-9ae5-fe6c496c7b7f_terseLabel_en-US" xlink:label="lab_gemi_ShortTermDebtAbstractRollForward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Short Term Debt Abstract [Roll Forward]</link:label>
    <link:label id="lab_gemi_ShortTermDebtAbstractRollForward_label_en-US" xlink:label="lab_gemi_ShortTermDebtAbstractRollForward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Short Term Debt Abstract [Roll Forward]</link:label>
    <link:label id="lab_gemi_ShortTermDebtAbstractRollForward_documentation_en-US" xlink:label="lab_gemi_ShortTermDebtAbstractRollForward" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Short Term Debt Abstract</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShortTermDebtAbstractRollForward" xlink:href="gemi-20250930.xsd#gemi_ShortTermDebtAbstractRollForward"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShortTermDebtAbstractRollForward" xlink:to="lab_gemi_ShortTermDebtAbstractRollForward" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetLoansMember_92bd5db5-3e72-4072-9385-fdc0e20534d0_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetLoansMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto Asset Loans</link:label>
    <link:label id="lab_gemi_CryptoAssetLoansMember_label_en-US" xlink:label="lab_gemi_CryptoAssetLoansMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Loans [Member]</link:label>
    <link:label id="lab_gemi_CryptoAssetLoansMember_documentation_en-US" xlink:label="lab_gemi_CryptoAssetLoansMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Loans</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetLoansMember" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetLoansMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetLoansMember" xlink:to="lab_gemi_CryptoAssetLoansMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetFairValueRestricted_397eec62-66a0-4a63-a15c-14df40c878ce_verboseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValueRestricted" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Fair value of restricted crypto assets held</link:label>
    <link:label id="lab_us-gaap_CryptoAssetFairValueRestricted_label_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValueRestricted" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Fair Value, Restricted</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetFairValueRestricted" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetFairValueRestricted"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetFairValueRestricted" xlink:to="lab_us-gaap_CryptoAssetFairValueRestricted" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1_3870ef57-e2f0-4ee6-b85e-dc07ef720a95_terseLabel_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate intrinsic value of options exercisable</link:label>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1_label_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Intrinsic Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1" xlink:to="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsExercisableIntrinsicValue1" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_EmperyDigitalInc.WarrantsMember_17d4e870-91c6-43de-953c-401ba9800e95_terseLabel_en-US" xlink:label="lab_gemi_EmperyDigitalInc.WarrantsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Empery Common Stock</link:label>
    <link:label id="lab_gemi_EmperyDigitalInc.WarrantsMember_label_en-US" xlink:label="lab_gemi_EmperyDigitalInc.WarrantsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Empery Digital Inc. Warrants [Member]</link:label>
    <link:label id="lab_gemi_EmperyDigitalInc.WarrantsMember_documentation_en-US" xlink:label="lab_gemi_EmperyDigitalInc.WarrantsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Empery Digital Inc. Warrants</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_EmperyDigitalInc.WarrantsMember" xlink:href="gemi-20250930.xsd#gemi_EmperyDigitalInc.WarrantsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_EmperyDigitalInc.WarrantsMember" xlink:to="lab_gemi_EmperyDigitalInc.WarrantsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock_070a82c3-f5f0-44a1-8091-93d9a0f7931e_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Service-Based Incentive Units Activity</link:label>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock_label_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Service-Based Incentive Units, Activity [Table Text Block]</link:label>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock_documentation_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Service-Based Incentive Units, Activity</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock" xlink:href="gemi-20250930.xsd#gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock" xlink:to="lab_gemi_ShareBasedPaymentArrangementServiceBasedIncentiveUnitsActivityTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A1400BTCMay242023Member_bb06e674-017e-42d2-9552-2cb02e71b45f_terseLabel_en-US" xlink:label="lab_gemi_A1400BTCMay242023Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">1,400 BTC, May 24, 2023</link:label>
    <link:label id="lab_gemi_A1400BTCMay242023Member_label_en-US" xlink:label="lab_gemi_A1400BTCMay242023Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">1,400 BTC, May 24, 2023 [Member]</link:label>
    <link:label id="lab_gemi_A1400BTCMay242023Member_documentation_en-US" xlink:label="lab_gemi_A1400BTCMay242023Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">1,400 BTC, May 24, 2023</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A1400BTCMay242023Member" xlink:href="gemi-20250930.xsd#gemi_A1400BTCMay242023Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A1400BTCMay242023Member" xlink:to="lab_gemi_A1400BTCMay242023Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisaggregationOfRevenueTable_ee50c2ed-4160-4811-9ea7-4c110cfa2ff8_terseLabel_en-US" xlink:label="lab_us-gaap_DisaggregationOfRevenueTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Disaggregation of Revenue [Table]</link:label>
    <link:label id="lab_us-gaap_DisaggregationOfRevenueTable_label_en-US" xlink:label="lab_us-gaap_DisaggregationOfRevenueTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disaggregation of Revenue [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisaggregationOfRevenueTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisaggregationOfRevenueTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisaggregationOfRevenueTable" xlink:to="lab_us-gaap_DisaggregationOfRevenueTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentByTypeAxis_2fda8569-b13e-4a64-93c9-1219f307cd69_terseLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentByTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Long-Lived Tangible Asset [Axis]</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentByTypeAxis_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentByTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Lived Tangible Asset [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentByTypeAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentByTypeAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentByTypeAxis" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentByTypeAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetOptionsMember_2c9ddb1c-a34f-498f-8cf6-088c70cfca79_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetOptionsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset options</link:label>
    <link:label id="lab_gemi_CryptoAssetOptionsMember_label_en-US" xlink:label="lab_gemi_CryptoAssetOptionsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Options [Member]</link:label>
    <link:label id="lab_gemi_CryptoAssetOptionsMember_documentation_en-US" xlink:label="lab_gemi_CryptoAssetOptionsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Options</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetOptionsMember" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetOptionsMember"/>
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    <link:label id="lab_gemi_TransactionProcessingExpense_00a9006b-0e68-4e13-a7fb-2f2aad3c70f8_terseLabel_en-US" xlink:label="lab_gemi_TransactionProcessingExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Transaction processing</link:label>
    <link:label id="lab_gemi_TransactionProcessingExpense_label_en-US" xlink:label="lab_gemi_TransactionProcessingExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Transaction Processing Expense</link:label>
    <link:label id="lab_gemi_TransactionProcessingExpense_documentation_en-US" xlink:label="lab_gemi_TransactionProcessingExpense" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Transaction Processing Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TransactionProcessingExpense" xlink:href="gemi-20250930.xsd#gemi_TransactionProcessingExpense"/>
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    <link:label id="lab_gemi_ReceivableCryptoAssetPledgedUnrealizedGainLossOperating_c69f0cdb-eae5-4374-afa8-466db3d1696a_negatedTerseLabel_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetPledgedUnrealizedGainLossOperating" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Unrealized loss on receivable, crypto assets pledged</link:label>
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    <link:label id="lab_gemi_ReceivableCryptoAssetPledgedUnrealizedGainLossOperating_documentation_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetPledgedUnrealizedGainLossOperating" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Receivable Crypto Asset Pledged, Unrealized Gain (Loss), Operating</link:label>
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    <link:label id="lab_gemi_MarketBasedStockOptionsMember_9eaca930-a228-4ddd-9605-75cd06a29e9f_terseLabel_en-US" xlink:label="lab_gemi_MarketBasedStockOptionsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Market Based Stock Options</link:label>
    <link:label id="lab_gemi_MarketBasedStockOptionsMember_label_en-US" xlink:label="lab_gemi_MarketBasedStockOptionsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Market Based Stock Options [Member]</link:label>
    <link:label id="lab_gemi_MarketBasedStockOptionsMember_documentation_en-US" xlink:label="lab_gemi_MarketBasedStockOptionsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Market Based Stock Options</link:label>
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    <link:label id="lab_us-gaap_OtherAssetsNoncurrent_8d4b98e3-2a05-4a3b-9a6a-5b5cb391f8c7_verboseLabel_en-US" xlink:label="lab_us-gaap_OtherAssetsNoncurrent" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Other non-current assets</link:label>
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    <link:label id="lab_us-gaap_InterestPayableCurrent_ff1cd509-dbd8-49a7-a2b3-8131fdb90783_terseLabel_en-US" xlink:label="lab_us-gaap_InterestPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest Payable</link:label>
    <link:label id="lab_us-gaap_InterestPayableCurrent_label_en-US" xlink:label="lab_us-gaap_InterestPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Payable, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestPayableCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestPayableCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestPayableCurrent" xlink:to="lab_us-gaap_InterestPayableCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityCentralIndexKey_d019170a-81ec-43e2-8542-6d2b8d43e9f2_terseLabel_en-US" xlink:label="lab_dei_EntityCentralIndexKey" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Central Index Key</link:label>
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    <link:label id="lab_ecd_PeoName_terseLabel_en-US" xlink:label="lab_ecd_PeoName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">PEO Name</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeoName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeoName"/>
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    <link:label id="lab_us-gaap_DebtInstrumentConvertibleConversionPrice1_1043adab-ee18-4019-8b26-992420eaf3d2_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentConvertibleConversionPrice1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Price per share resulting from converted debt (in dollars per share)</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentConvertibleConversionPrice1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentConvertibleConversionPrice1"/>
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    <link:label id="lab_us-gaap_Revenues_ab13c8f9-3d38-42c8-a79b-170501921dc7_terseLabel_en-US" xlink:label="lab_us-gaap_Revenues" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total revenue</link:label>
    <link:label id="lab_us-gaap_Revenues_ca15cdbd-614f-484e-8a8a-451c0a3f9e85_totalLabel_en-US" xlink:label="lab_us-gaap_Revenues" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total revenue</link:label>
    <link:label id="lab_us-gaap_Revenues_label_en-US" xlink:label="lab_us-gaap_Revenues" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenues</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Revenues" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Revenues"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_Revenues" xlink:to="lab_us-gaap_Revenues" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PreferredUnitsLineItems_5c6f27a6-97ba-4cc4-b6fe-cbc1580ef6cd_terseLabel_en-US" xlink:label="lab_us-gaap_PreferredUnitsLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Preferred Units [Line Items]</link:label>
    <link:label id="lab_us-gaap_PreferredUnitsLineItems_label_en-US" xlink:label="lab_us-gaap_PreferredUnitsLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Units [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredUnitsLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredUnitsLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PreferredUnitsLineItems" xlink:to="lab_us-gaap_PreferredUnitsLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConversionOfStockByUniqueDescriptionAxis_54367ed5-7d28-4950-a6c8-ef874b248aee_terseLabel_en-US" xlink:label="lab_us-gaap_ConversionOfStockByUniqueDescriptionAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock Conversion Description [Axis]</link:label>
    <link:label id="lab_us-gaap_ConversionOfStockByUniqueDescriptionAxis_label_en-US" xlink:label="lab_us-gaap_ConversionOfStockByUniqueDescriptionAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Conversion Description [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConversionOfStockByUniqueDescriptionAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConversionOfStockByUniqueDescriptionAxis"/>
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    <link:label id="lab_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember_terseLabel_en-US" xlink:label="lab_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Prior Year End Fair Value of Equity Awards Granted in Any Prior Year that Fail to Meet Applicable Vesting Conditions During Covered Year</link:label>
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    <link:label id="lab_us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems_4965934d-a089-4f79-ae71-3eb001cecca7_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, Balance Sheet Grouping, Financial Statement Captions [Line Items]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueBalanceSheetGroupingFinancialStatementCaptionsLineItems"/>
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    <link:label id="lab_ecd_OutstandingAggtErrCompAmt_terseLabel_en-US" xlink:label="lab_ecd_OutstandingAggtErrCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Outstanding Aggregate Erroneous Compensation Amount</link:label>
    <link:label id="lab_ecd_OutstandingAggtErrCompAmt_label_en-US" xlink:label="lab_ecd_OutstandingAggtErrCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Outstanding Aggregate Erroneous Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_OutstandingAggtErrCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_OutstandingAggtErrCompAmt"/>
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    <link:label id="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueImpairmentsLossAndDownwardPriceAdjustmentAnnualAmount_f889ec2f-bd79-4597-86d1-36c22661b3f9_negatedTerseLabel_en-US" xlink:label="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueImpairmentsLossAndDownwardPriceAdjustmentAnnualAmount" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Impairments and downward adjustments</link:label>
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    <link:label id="lab_gemi_InterestExpenseRelatedPartyDebt_fc0d7265-eb26-40de-a899-e27c77a55758_negatedTerseLabel_en-US" xlink:label="lab_gemi_InterestExpenseRelatedPartyDebt" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Interest expense on related party loans</link:label>
    <link:label id="lab_gemi_InterestExpenseRelatedPartyDebt_label_en-US" xlink:label="lab_gemi_InterestExpenseRelatedPartyDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Expense, Related Party Debt</link:label>
    <link:label id="lab_gemi_InterestExpenseRelatedPartyDebt_documentation_en-US" xlink:label="lab_gemi_InterestExpenseRelatedPartyDebt" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Interest Expense, Debt, Related Party Loans</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_InterestExpenseRelatedPartyDebt" xlink:href="gemi-20250930.xsd#gemi_InterestExpenseRelatedPartyDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_InterestExpenseRelatedPartyDebt" xlink:to="lab_gemi_InterestExpenseRelatedPartyDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetFuturesMember_a0baa060-28e5-4c2e-8efc-ebd39c9df4ff_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetFuturesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset futures</link:label>
    <link:label id="lab_gemi_CryptoAssetFuturesMember_label_en-US" xlink:label="lab_gemi_CryptoAssetFuturesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Futures [Member]</link:label>
    <link:label id="lab_gemi_CryptoAssetFuturesMember_documentation_en-US" xlink:label="lab_gemi_CryptoAssetFuturesMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Futures</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetFuturesMember" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetFuturesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetFuturesMember" xlink:to="lab_gemi_CryptoAssetFuturesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationAwardTrancheOneMember_3b62777f-2f4d-488f-824b-750e23bc3981_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationAwardTrancheOneMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Tranche One</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationAwardTrancheOneMember_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationAwardTrancheOneMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Tranche One [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationAwardTrancheOneMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationAwardTrancheOneMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationAwardTrancheOneMember" xlink:to="lab_us-gaap_ShareBasedCompensationAwardTrancheOneMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ContractWithCustomerLiabilityCurrent_f13ec329-8510-4efa-8e7b-d1051b2360bd_terseLabel_en-US" xlink:label="lab_us-gaap_ContractWithCustomerLiabilityCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Unearned advisory fee revenue</link:label>
    <link:label id="lab_us-gaap_ContractWithCustomerLiabilityCurrent_label_en-US" xlink:label="lab_us-gaap_ContractWithCustomerLiabilityCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Contract with Customer, Liability, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ContractWithCustomerLiabilityCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ContractWithCustomerLiabilityCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ContractWithCustomerLiabilityCurrent" xlink:to="lab_us-gaap_ContractWithCustomerLiabilityCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_8e1d4d83-ac9b-4d3f-8ce2-960196c2f096_terseLabel_en-US" xlink:label="lab_us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Exercise price of warrants (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1_label_en-US" xlink:label="lab_us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class of Warrant or Right, Exercise Price of Warrants or Rights</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xlink:to="lab_us-gaap_ClassOfWarrantOrRightExercisePriceOfWarrantsOrRights1" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent_a967d1ff-d448-457b-8426-e31072dd8a8c_terseLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible preferred units: 30,955 Series A Units authorized, issued and outstanding with an aggregate liquidation preference of $223,022 as of December&#160;31, 2024; 8,910 Series B Units authorized as of December&#160;31, 2024; 8,887 Series B Units issued and outstanding as of December&#160;31, 2024; and the Series B Units had an aggregate liquidation preference of $423,903 as of December&#160;31, 2024. No shares authorized, issued, and outstanding as of September&#160;30, 2025.</link:label>
    <link:label id="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent_592e6eea-2ead-408e-8d62-255078a360be_periodStartLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Beginning balance</link:label>
    <link:label id="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent_617523e8-92b9-4369-aceb-eb78a0252c8c_periodEndLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Ending balance</link:label>
    <link:label id="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent_label_en-US" xlink:label="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Temporary Equity, Carrying Amount, Attributable to Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TemporaryEquityCarryingAmountAttributableToParent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:to="lab_us-gaap_TemporaryEquityCarryingAmountAttributableToParent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_TransactionRevenueMember_dacf2ed1-0c1a-45aa-b857-002413bdbdb0_terseLabel_en-US" xlink:label="lab_gemi_TransactionRevenueMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Transaction revenue</link:label>
    <link:label id="lab_gemi_TransactionRevenueMember_label_en-US" xlink:label="lab_gemi_TransactionRevenueMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Transaction Revenue [Member]</link:label>
    <link:label id="lab_gemi_TransactionRevenueMember_documentation_en-US" xlink:label="lab_gemi_TransactionRevenueMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Transaction Revenue</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TransactionRevenueMember" xlink:href="gemi-20250930.xsd#gemi_TransactionRevenueMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_TransactionRevenueMember" xlink:to="lab_gemi_TransactionRevenueMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardExrcPrice_terseLabel_en-US" xlink:label="lab_ecd_AwardExrcPrice" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Exercise Price</link:label>
    <link:label id="lab_ecd_AwardExrcPrice_label_en-US" xlink:label="lab_ecd_AwardExrcPrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Exercise Price</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardExrcPrice" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardExrcPrice"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AwardExrcPrice" xlink:to="lab_ecd_AwardExrcPrice" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityFilerCategory_588769e9-5f0a-447c-a8fa-cf61877dd698_terseLabel_en-US" xlink:label="lab_dei_EntityFilerCategory" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Filer Category</link:label>
    <link:label id="lab_dei_EntityFilerCategory_label_en-US" xlink:label="lab_dei_EntityFilerCategory" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Filer Category</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityFilerCategory" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityFilerCategory"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityFilerCategory" xlink:to="lab_dei_EntityFilerCategory" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_AccountsReceivablePercentPastDue_6804a5d7-4151-46c8-aa9b-09496082a1f1_terseLabel_en-US" xlink:label="lab_gemi_AccountsReceivablePercentPastDue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percentage of receivables current and past due</link:label>
    <link:label id="lab_gemi_AccountsReceivablePercentPastDue_label_en-US" xlink:label="lab_gemi_AccountsReceivablePercentPastDue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts Receivable, Percent Past Due</link:label>
    <link:label id="lab_gemi_AccountsReceivablePercentPastDue_documentation_en-US" xlink:label="lab_gemi_AccountsReceivablePercentPastDue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Accounts Receivable, Percent Past Due</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_AccountsReceivablePercentPastDue" xlink:href="gemi-20250930.xsd#gemi_AccountsReceivablePercentPastDue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_AccountsReceivablePercentPastDue" xlink:to="lab_gemi_AccountsReceivablePercentPastDue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ClassOfStockDomain_c5f7e2b6-430e-45bc-a2de-506371dc2e1b_terseLabel_en-US" xlink:label="lab_us-gaap_ClassOfStockDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Class of Stock [Domain]</link:label>
    <link:label id="lab_us-gaap_ClassOfStockDomain_label_en-US" xlink:label="lab_us-gaap_ClassOfStockDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class of Stock [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ClassOfStockDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ClassOfStockDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ClassOfStockDomain" xlink:to="lab_us-gaap_ClassOfStockDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DerivativeLiabilityMember_d5eadea7-3837-4c0b-8fea-feea1984f4a9_terseLabel_en-US" xlink:label="lab_gemi_DerivativeLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative liabilities</link:label>
    <link:label id="lab_gemi_DerivativeLiabilityMember_label_en-US" xlink:label="lab_gemi_DerivativeLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Liability [Member]</link:label>
    <link:label id="lab_gemi_DerivativeLiabilityMember_documentation_en-US" xlink:label="lab_gemi_DerivativeLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Derivative Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DerivativeLiabilityMember" xlink:href="gemi-20250930.xsd#gemi_DerivativeLiabilityMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DerivativeLiabilityMember" xlink:to="lab_gemi_DerivativeLiabilityMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SubsequentEventTypeAxis_13e0e848-b41e-4466-a506-cb7078f2bcda_terseLabel_en-US" xlink:label="lab_us-gaap_SubsequentEventTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Subsequent Event Type [Axis]</link:label>
    <link:label id="lab_us-gaap_SubsequentEventTypeAxis_label_en-US" xlink:label="lab_us-gaap_SubsequentEventTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Subsequent Event Type [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubsequentEventTypeAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubsequentEventTypeAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SubsequentEventTypeAxis" xlink:to="lab_us-gaap_SubsequentEventTypeAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan12Member_d7711e70-a5a0-4323-a3a6-daa31179fb3c_terseLabel_en-US" xlink:label="lab_gemi_Loan12Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 12</link:label>
    <link:label id="lab_gemi_Loan12Member_label_en-US" xlink:label="lab_gemi_Loan12Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 12 [Member]</link:label>
    <link:label id="lab_gemi_Loan12Member_documentation_en-US" xlink:label="lab_gemi_Loan12Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 12</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan12Member" xlink:href="gemi-20250930.xsd#gemi_Loan12Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan12Member" xlink:to="lab_gemi_Loan12Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DerivativeInstrumentRecognitionCriteriaDomain_1fa20b5e-5f05-4b03-a1f8-d5b53f420559_terseLabel_en-US" xlink:label="lab_gemi_DerivativeInstrumentRecognitionCriteriaDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative Instrument, Recognition Criteria [Domain]</link:label>
    <link:label id="lab_gemi_DerivativeInstrumentRecognitionCriteriaDomain_label_en-US" xlink:label="lab_gemi_DerivativeInstrumentRecognitionCriteriaDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Instrument, Recognition Criteria [Domain]</link:label>
    <link:label id="lab_gemi_DerivativeInstrumentRecognitionCriteriaDomain_documentation_en-US" xlink:label="lab_gemi_DerivativeInstrumentRecognitionCriteriaDomain" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Derivative Instrument, Recognition Criteria [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DerivativeInstrumentRecognitionCriteriaDomain" xlink:href="gemi-20250930.xsd#gemi_DerivativeInstrumentRecognitionCriteriaDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DerivativeInstrumentRecognitionCriteriaDomain" xlink:to="lab_gemi_DerivativeInstrumentRecognitionCriteriaDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfPreferredUnitsTextBlock_f0c3cd28-093e-4c28-a863-093df59226e4_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfPreferredUnitsTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Convertible Preferred Units</link:label>
    <link:label id="lab_us-gaap_ScheduleOfPreferredUnitsTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfPreferredUnitsTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Preferred Units [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfPreferredUnitsTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfPreferredUnitsTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfPreferredUnitsTextBlock" xlink:to="lab_us-gaap_ScheduleOfPreferredUnitsTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueOptionQuantitativeDisclosuresTable_85869f11-2d2d-4603-85fc-e93001ac8257_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueOptionQuantitativeDisclosuresTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value Option, Disclosures [Table]</link:label>
    <link:label id="lab_us-gaap_FairValueOptionQuantitativeDisclosuresTable_label_en-US" xlink:label="lab_us-gaap_FairValueOptionQuantitativeDisclosuresTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Option, Disclosures [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueOptionQuantitativeDisclosuresTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueOptionQuantitativeDisclosuresTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueOptionQuantitativeDisclosuresTable" xlink:to="lab_us-gaap_FairValueOptionQuantitativeDisclosuresTable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_TransactionRevenueOtherMember_e19b2fa2-7095-4aee-97a4-f463b50d778f_terseLabel_en-US" xlink:label="lab_gemi_TransactionRevenueOtherMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other transaction revenue</link:label>
    <link:label id="lab_gemi_TransactionRevenueOtherMember_label_en-US" xlink:label="lab_gemi_TransactionRevenueOtherMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Transaction Revenue, Other [Member]</link:label>
    <link:label id="lab_gemi_TransactionRevenueOtherMember_documentation_en-US" xlink:label="lab_gemi_TransactionRevenueOtherMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Transaction Revenue, Other</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TransactionRevenueOtherMember" xlink:href="gemi-20250930.xsd#gemi_TransactionRevenueOtherMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_TransactionRevenueOtherMember" xlink:to="lab_gemi_TransactionRevenueOtherMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearThree_53abd66a-8678-4587-b8aa-0e2cff6afde9_terseLabel_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearThree" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2028</link:label>
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    <link:label id="lab_us-gaap_ClassOfWarrantOrRightDomain_864499a5-577f-4329-9546-5410e5282bcc_terseLabel_en-US" xlink:label="lab_us-gaap_ClassOfWarrantOrRightDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Class of Warrant or Right [Domain]</link:label>
    <link:label id="lab_us-gaap_ClassOfWarrantOrRightDomain_label_en-US" xlink:label="lab_us-gaap_ClassOfWarrantOrRightDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class of Warrant or Right [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ClassOfWarrantOrRightDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ClassOfWarrantOrRightDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ClassOfWarrantOrRightDomain" xlink:to="lab_us-gaap_ClassOfWarrantOrRightDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsOutstanding_b4c87346-6a38-453c-b17d-daf2bc8ea065_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Units outstanding (in shares)</link:label>
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    <link:label id="lab_us-gaap_VestingDomain_label_en-US" xlink:label="lab_us-gaap_VestingDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Vesting [Domain]</link:label>
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    <link:label id="lab_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember_terseLabel_en-US" xlink:label="lab_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Vesting Date Fair Value of Equity Awards Granted and Vested in Covered Year</link:label>
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    <link:label id="lab_gemi_EarnProgramRelatedLitigationMember_1dc932e3-0a13-4596-8912-2558214fca8e_terseLabel_en-US" xlink:label="lab_gemi_EarnProgramRelatedLitigationMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Earn Program Related Litigation</link:label>
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    <link:label id="lab_dei_EntityAddressAddressLine1_202688ca-c148-48f4-8ade-1fe12457796a_terseLabel_en-US" xlink:label="lab_dei_EntityAddressAddressLine1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, Address Line One</link:label>
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    <link:label id="lab_gemi_AssembledWorkforceMember_documentation_en-US" xlink:label="lab_gemi_AssembledWorkforceMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Assembled Workforce</link:label>
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    <link:label id="lab_gemi_CommonUnitConversionMember_06cb8eaf-b911-4262-af2d-a2c43f81418c_terseLabel_en-US" xlink:label="lab_gemi_CommonUnitConversionMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Conversion of Common Units to common stock in connection with IPO</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_gemi_CommonUnitConversionMember" xlink:href="gemi-20250930.xsd#gemi_CommonUnitConversionMember"/>
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    <link:label id="lab_dei_EntityAddressAddressLine2_ac71ee24-f3a3-429e-a986-211d3f767e13_terseLabel_en-US" xlink:label="lab_dei_EntityAddressAddressLine2" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, Address Line Two</link:label>
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    <link:label id="lab_gemi_InterestPayableCurrentFairValueAdjustmentForUnrealizedLosses_6920e69e-347c-4120-adeb-cda57f726c55_terseLabel_en-US" xlink:label="lab_gemi_InterestPayableCurrentFairValueAdjustmentForUnrealizedLosses" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest payable, fair value adjustments related to unrealized losses</link:label>
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    <link:label id="lab_gemi_InterestPayableCurrentFairValueAdjustmentForUnrealizedLosses_documentation_en-US" xlink:label="lab_gemi_InterestPayableCurrentFairValueAdjustmentForUnrealizedLosses" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Interest Payable, Current, Fair Value Adjustment For Unrealized Losses</link:label>
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    <link:label id="lab_gemi_LesseeOperatingLeaseTermOfContractRentFreePeriod_9140b95b-7c2c-4552-929b-a88dc682d194_terseLabel_en-US" xlink:label="lab_gemi_LesseeOperatingLeaseTermOfContractRentFreePeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Rent-free period</link:label>
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    <link:label id="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueImpairmentLossAndDownwardPriceAdjustmentCumulativeAmount_28a4f1f7-36c5-41aa-8340-5ad190b6722c_terseLabel_en-US" xlink:label="lab_gemi_EquitySecuritiesWithoutReadilyDeterminableFairValueImpairmentLossAndDownwardPriceAdjustmentCumulativeAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cumulative impairments and downward adjustments</link:label>
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    <link:label id="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember_64bc4ebb-f625-46da-86f8-2eb398006e98_terseLabel_en-US" xlink:label="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accumulated Other Comprehensive Income</link:label>
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    <link:label id="lab_gemi_InterestPayableCurrentFairValueAdjustment_label_en-US" xlink:label="lab_gemi_InterestPayableCurrentFairValueAdjustment" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Payable, Current, Fair Value Adjustment</link:label>
    <link:label id="lab_gemi_InterestPayableCurrentFairValueAdjustment_documentation_en-US" xlink:label="lab_gemi_InterestPayableCurrentFairValueAdjustment" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Interest Payable, Current, Fair Value Adjustment</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_InterestPayableCurrentFairValueAdjustment" xlink:to="lab_gemi_InterestPayableCurrentFairValueAdjustment" xlink:type="arc" order="1"/>
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    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesConversionOfConvertibleSecurities_label_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesConversionOfConvertibleSecurities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Issued During Period, Shares, Conversion of Convertible Securities</link:label>
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    <link:label id="lab_gemi_SeriesA7PreferredUnitsMember_31cb750b-4ec9-4e34-9f7e-721f914ff412_terseLabel_en-US" xlink:label="lab_gemi_SeriesA7PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Series A-7</link:label>
    <link:label id="lab_gemi_SeriesA7PreferredUnitsMember_label_en-US" xlink:label="lab_gemi_SeriesA7PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Series A-7 Preferred Units [Member]</link:label>
    <link:label id="lab_gemi_SeriesA7PreferredUnitsMember_documentation_en-US" xlink:label="lab_gemi_SeriesA7PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Series A-7 Preferred Units</link:label>
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    <link:label id="lab_us-gaap_DerivativeInstrumentsAndHedgingActivitiesDisclosureTextBlock_6e727cf7-5ea0-4221-8099-d736471e76c1_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeInstrumentsAndHedgingActivitiesDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivatives</link:label>
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    <link:label id="lab_us-gaap_CryptoAssetNumberOfRestrictedUnits_9e2a8dcd-b097-497a-b5f3-e0b345f4615e_verboseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetNumberOfRestrictedUnits" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Number of restricted crypto assets held (in units)</link:label>
    <link:label id="lab_us-gaap_CryptoAssetNumberOfRestrictedUnits_label_en-US" xlink:label="lab_us-gaap_CryptoAssetNumberOfRestrictedUnits" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Number of Units, Restricted</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetNumberOfRestrictedUnits" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetNumberOfRestrictedUnits"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetNumberOfRestrictedUnits" xlink:to="lab_us-gaap_CryptoAssetNumberOfRestrictedUnits" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetNumberOfUnitsIssuedAsCollateral_019ac9dc-0d6b-4cd1-b8a0-6e91a6d8d21d_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetNumberOfUnitsIssuedAsCollateral" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of crypto assets designated to insurance providers (in units)</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetNumberOfUnitsIssuedAsCollateral" xlink:to="lab_gemi_CryptoAssetNumberOfUnitsIssuedAsCollateral" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardGrantDateFairValue_terseLabel_en-US" xlink:label="lab_ecd_AwardGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value as of Grant Date</link:label>
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    <link:label id="lab_gemi_LossContingencyUnsecuredClaimAmount_2841d275-6946-44cb-9655-227e90809799_terseLabel_en-US" xlink:label="lab_gemi_LossContingencyUnsecuredClaimAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">General unsecured creditor claim</link:label>
    <link:label id="lab_gemi_LossContingencyUnsecuredClaimAmount_label_en-US" xlink:label="lab_gemi_LossContingencyUnsecuredClaimAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loss Contingency, Unsecured Claim Amount</link:label>
    <link:label id="lab_gemi_LossContingencyUnsecuredClaimAmount_documentation_en-US" xlink:label="lab_gemi_LossContingencyUnsecuredClaimAmount" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US"></link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_LossContingencyUnsecuredClaimAmount" xlink:href="gemi-20250930.xsd#gemi_LossContingencyUnsecuredClaimAmount"/>
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    <link:label id="lab_gemi_EtherMember_cf43ceee-f291-4861-bfcf-8757d1b1a3ee_terseLabel_en-US" xlink:label="lab_gemi_EtherMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Ether</link:label>
    <link:label id="lab_gemi_EtherMember_d382f16f-0aa4-4d71-91c0-aca588c6fe79_verboseLabel_en-US" xlink:label="lab_gemi_EtherMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Ether</link:label>
    <link:label id="lab_gemi_EtherMember_label_en-US" xlink:label="lab_gemi_EtherMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Ether [Member]</link:label>
    <link:label id="lab_gemi_EtherMember_documentation_en-US" xlink:label="lab_gemi_EtherMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Ether</link:label>
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    <link:label id="lab_us-gaap_PreferredUnitsClassDomain_e41947ca-8b4b-4814-8389-30da75f56bfd_terseLabel_en-US" xlink:label="lab_us-gaap_PreferredUnitsClassDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Preferred Units, Class [Domain]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PreferredUnitsClassDomain" xlink:to="lab_us-gaap_PreferredUnitsClassDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock_e6826e80-dc68-41d3-a4a0-042cbcd7d7f4_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Term Loans</link:label>
    <link:label id="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock_7f63f5cf-8c54-4104-ae73-23e078ef010f_verboseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Schedule Third Party Loans</link:label>
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    <link:label id="lab_us-gaap_LaborAndRelatedExpense_599b0037-9207-4513-9e0f-f70c7eaf5b0d_terseLabel_en-US" xlink:label="lab_us-gaap_LaborAndRelatedExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Salaries and compensation</link:label>
    <link:label id="lab_us-gaap_LaborAndRelatedExpense_label_en-US" xlink:label="lab_us-gaap_LaborAndRelatedExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Labor and Related Expense</link:label>
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    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsWeightedAverageVolatilityRate_52cc6d7b-da3e-48ea-8f76-063459d17cdc_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsWeightedAverageVolatilityRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Volatility</link:label>
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    <link:label id="lab_gemi_CryptoAssetCollateralHeldCurrent_3c184897-2648-4331-823d-54b25d7db253_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetCollateralHeldCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset collateral held</link:label>
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    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriod_d9b5baac-2195-4870-a90b-7f0beeb02379_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsGrantsInPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Granted (in shares)</link:label>
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    <link:label id="lab_srt_TitleOfIndividualWithRelationshipToEntityDomain_label_en-US" xlink:label="lab_srt_TitleOfIndividualWithRelationshipToEntityDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Title and Position [Domain]</link:label>
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    <link:label id="lab_us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized_f176c3a0-6dd8-414d-bb14-d445607efca3_terseLabel_en-US" xlink:label="lab_us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Remaining unrecognized expense</link:label>
    <link:label id="lab_us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized_label_en-US" xlink:label="lab_us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Nonvested Award, Cost Not yet Recognized, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized" xlink:to="lab_us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod_00acd481-9c36-492f-bdf9-41f44f4baee0_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Forfeited (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod_1c77bea7-e051-49f3-9ce1-0629cda4e611_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Awards forfeited (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Forfeited in Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeitedInPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RelatedPartyTransactionLineItems_ccb9465e-428e-4c69-b2c9-49d262c14472_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transaction [Line Items]</link:label>
    <link:label id="lab_us-gaap_RelatedPartyTransactionLineItems_label_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyTransactionLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyTransactionLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyTransactionLineItems" xlink:to="lab_us-gaap_RelatedPartyTransactionLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CustodialFundsDueToCustomerFairValueDisclosure_a5627ef1-a6f0-4539-9c4e-b67c54a5bec5_terseLabel_en-US" xlink:label="lab_gemi_CustodialFundsDueToCustomerFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Custodial funds due to customer</link:label>
    <link:label id="lab_gemi_CustodialFundsDueToCustomerFairValueDisclosure_label_en-US" xlink:label="lab_gemi_CustodialFundsDueToCustomerFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Custodial Funds Due To Customer, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_CustodialFundsDueToCustomerFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_CustodialFundsDueToCustomerFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Custodial Funds Due To Customer, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CustodialFundsDueToCustomerFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_CustodialFundsDueToCustomerFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CustodialFundsDueToCustomerFairValueDisclosure" xlink:to="lab_gemi_CustodialFundsDueToCustomerFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_DocumentQuarterlyReport_1e82bfe0-905b-46a3-b0f3-cf14d4f14117_terseLabel_en-US" xlink:label="lab_dei_DocumentQuarterlyReport" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Document Quarterly Report</link:label>
    <link:label id="lab_dei_DocumentQuarterlyReport_label_en-US" xlink:label="lab_dei_DocumentQuarterlyReport" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Document Quarterly Report</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_DocumentQuarterlyReport" xlink:to="lab_dei_DocumentQuarterlyReport" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent_eddd43e5-015d-4a62-ac35-5227fe648af6_terseLabel_en-US" xlink:label="lab_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party loans</link:label>
    <link:label id="lab_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent_label_en-US" xlink:label="lab_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Loans, Including Accrued Interest, Noncurrent</link:label>
    <link:label id="lab_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent_documentation_en-US" xlink:label="lab_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Loans, Including Accrued Interest, Noncurrent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent" xlink:href="gemi-20250930.xsd#gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent" xlink:to="lab_gemi_RelatedPartyLoansIncludingAccruedInterestNoncurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_SeriesFFPreferredUnitsMember_efe9fe28-929c-4268-8de3-1034361e4772_terseLabel_en-US" xlink:label="lab_gemi_SeriesFFPreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Series FF Preferred Units</link:label>
    <link:label id="lab_gemi_SeriesFFPreferredUnitsMember_label_en-US" xlink:label="lab_gemi_SeriesFFPreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Series FF Preferred Units [Member]</link:label>
    <link:label id="lab_gemi_SeriesFFPreferredUnitsMember_documentation_en-US" xlink:label="lab_gemi_SeriesFFPreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Series FF Preferred Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_SeriesFFPreferredUnitsMember" xlink:href="gemi-20250930.xsd#gemi_SeriesFFPreferredUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_SeriesFFPreferredUnitsMember" xlink:to="lab_gemi_SeriesFFPreferredUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongTermDebtFairValue_8b2493d9-0a57-4fe5-bb06-4b3b29e39687_terseLabel_en-US" xlink:label="lab_us-gaap_LongTermDebtFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair value of Principal Outstanding</link:label>
    <link:label id="lab_us-gaap_LongTermDebtFairValue_a4bfc400-6411-4a1d-9b8b-30b6c72a6688_verboseLabel_en-US" xlink:label="lab_us-gaap_LongTermDebtFairValue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Fair value of debt that can be converted</link:label>
    <link:label id="lab_us-gaap_LongTermDebtFairValue_label_en-US" xlink:label="lab_us-gaap_LongTermDebtFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebtFairValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebtFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LongTermDebtFairValue" xlink:to="lab_us-gaap_LongTermDebtFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ChangedPeerGroupFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_ChangedPeerGroupFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Changed Peer Group, Footnote</link:label>
    <link:label id="lab_ecd_ChangedPeerGroupFnTextBlock_label_en-US" xlink:label="lab_ecd_ChangedPeerGroupFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Changed Peer Group, Footnote [Text Block]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ChangedPeerGroupFnTextBlock" xlink:to="lab_ecd_ChangedPeerGroupFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_RepurchaseAgreementCounterpartyNameDomain_a299dc75-f605-4c56-af7e-24052de349ec_terseLabel_en-US" xlink:label="lab_srt_RepurchaseAgreementCounterpartyNameDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Counterparty Name [Domain]</link:label>
    <link:label id="lab_srt_RepurchaseAgreementCounterpartyNameDomain_label_en-US" xlink:label="lab_srt_RepurchaseAgreementCounterpartyNameDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Counterparty Name [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_RepurchaseAgreementCounterpartyNameDomain" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_RepurchaseAgreementCounterpartyNameDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_RepurchaseAgreementCounterpartyNameDomain" xlink:to="lab_srt_RepurchaseAgreementCounterpartyNameDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid_8da91c62-2462-4bed-ace8-c56e6cb897b8_terseLabel_en-US" xlink:label="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Repayments of related party loans denominated in crypto assets</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid_36cfe264-234b-4c05-b06c-3aaa05b383a7_verboseLabel_en-US" xlink:label="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Amount Repaid</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid_label_en-US" xlink:label="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Debt, Denominated In Crypto Asset, Repaid</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid_documentation_en-US" xlink:label="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loans Payable, Denominated In Crypto Asset, Repaid</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid" xlink:href="gemi-20250930.xsd#gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid" xlink:to="lab_gemi_RelatedPartyDebtDenominatedInCryptoAssetRepaid" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_9f638874-e751-4952-8951-a29a868e4ac0_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2027</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year Two</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValuesDerivativesBalanceSheetLocationByDerivativeContractTypeByHedgingDesignationTable_8c714427-282a-492e-be41-1b74e1e0168f_terseLabel_en-US" xlink:label="lab_us-gaap_FairValuesDerivativesBalanceSheetLocationByDerivativeContractTypeByHedgingDesignationTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Values Derivatives, Balance Sheet Location, by Derivative Contract Type [Table]</link:label>
    <link:label id="lab_us-gaap_FairValuesDerivativesBalanceSheetLocationByDerivativeContractTypeByHedgingDesignationTable_label_en-US" xlink:label="lab_us-gaap_FairValuesDerivativesBalanceSheetLocationByDerivativeContractTypeByHedgingDesignationTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Values Derivatives, Balance Sheet Location, by Derivative Contract Type [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValuesDerivativesBalanceSheetLocationByDerivativeContractTypeByHedgingDesignationTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValuesDerivativesBalanceSheetLocationByDerivativeContractTypeByHedgingDesignationTable"/>
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    <link:label id="lab_ecd_AdjToPeoCompFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AdjToPeoCompFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustment To PEO Compensation, Footnote</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_AdjToPeoCompFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AdjToPeoCompFnTextBlock"/>
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    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2_26bb1c73-884d-4df6-99f5-3ebd912a8fc4_terseLabel_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Weighted-average remaining contractual life of options outstanding (in years)</link:label>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2_label_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Remaining Contractual Term</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2" xlink:to="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardOptionsOutstandingWeightedAverageRemainingContractualTerm2" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_FundingDebtFairValueDisclosure_9922cab8-08d8-4afe-8b2d-7de3d4c4518a_terseLabel_en-US" xlink:label="lab_gemi_FundingDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Funding debt</link:label>
    <link:label id="lab_gemi_FundingDebtFairValueDisclosure_label_en-US" xlink:label="lab_gemi_FundingDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Funding Debt, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_FundingDebtFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_FundingDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Funding Debt, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_FundingDebtFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_FundingDebtFairValueDisclosure"/>
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    <link:label id="lab_us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock_3b918b61-9b4d-4b5e-a23a-233219219e29_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Accrued Expenses and Other Current Liabilities</link:label>
    <link:label id="lab_us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Accounts Payable and Accrued Liabilities [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock" xlink:to="lab_us-gaap_ScheduleOfAccountsPayableAndAccruedLiabilitiesTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrIndTitle_terseLabel_en-US" xlink:label="lab_ecd_TrdArrIndTitle" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Title</link:label>
    <link:label id="lab_ecd_TrdArrIndTitle_label_en-US" xlink:label="lab_ecd_TrdArrIndTitle" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement, Individual Title</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrIndTitle" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrIndTitle"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TrdArrIndTitle" xlink:to="lab_ecd_TrdArrIndTitle" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PeerGroupTotalShareholderRtnAmt_terseLabel_en-US" xlink:label="lab_ecd_PeerGroupTotalShareholderRtnAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Peer Group Total Shareholder Return Amount</link:label>
    <link:label id="lab_ecd_PeerGroupTotalShareholderRtnAmt_label_en-US" xlink:label="lab_ecd_PeerGroupTotalShareholderRtnAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Peer Group Total Shareholder Return Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeerGroupTotalShareholderRtnAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeerGroupTotalShareholderRtnAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PeerGroupTotalShareholderRtnAmt" xlink:to="lab_ecd_PeerGroupTotalShareholderRtnAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OtherRevenueMember_3aab15bd-a84b-4593-b37b-bf952d21ff39_terseLabel_en-US" xlink:label="lab_gemi_OtherRevenueMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other revenue</link:label>
    <link:label id="lab_gemi_OtherRevenueMember_label_en-US" xlink:label="lab_gemi_OtherRevenueMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Revenue [Member]</link:label>
    <link:label id="lab_gemi_OtherRevenueMember_documentation_en-US" xlink:label="lab_gemi_OtherRevenueMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Other Revenue</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OtherRevenueMember" xlink:href="gemi-20250930.xsd#gemi_OtherRevenueMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OtherRevenueMember" xlink:to="lab_gemi_OtherRevenueMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward_04122a7a-fcfa-4566-9132-5b9a9a642b19_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Shares purchased (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Shares Purchased for Award</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesPurchasedForAward" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_BitcoinMember_0a48f683-7e9a-4a54-9937-df83e42acb23_terseLabel_en-US" xlink:label="lab_gemi_BitcoinMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Bitcoin</link:label>
    <link:label id="lab_gemi_BitcoinMember_c7ad6f04-6b23-4a03-9354-3c07424c0ce5_verboseLabel_en-US" xlink:label="lab_gemi_BitcoinMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Bitcoin</link:label>
    <link:label id="lab_gemi_BitcoinMember_label_en-US" xlink:label="lab_gemi_BitcoinMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Bitcoin [Member]</link:label>
    <link:label id="lab_gemi_BitcoinMember_documentation_en-US" xlink:label="lab_gemi_BitcoinMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Bitcoin</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_BitcoinMember" xlink:href="gemi-20250930.xsd#gemi_BitcoinMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_BitcoinMember" xlink:to="lab_gemi_BitcoinMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonPeoNeoMember_terseLabel_en-US" xlink:label="lab_ecd_NonPeoNeoMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-PEO NEO</link:label>
    <link:label id="lab_ecd_NonPeoNeoMember_label_en-US" xlink:label="lab_ecd_NonPeoNeoMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-PEO NEO [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonPeoNeoMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonPeoNeoMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonPeoNeoMember" xlink:to="lab_ecd_NonPeoNeoMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue_e80e45ec-50ca-4a88-8b7f-e4d870bf4d13_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate intrinsic value of options outstanding</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Intrinsic Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingIntrinsicValue" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_NumberOfVotesPerShareOfCommonStockHeld_bb2bb074-5b53-4e32-ba08-89d8b820c9df_terseLabel_en-US" xlink:label="lab_gemi_NumberOfVotesPerShareOfCommonStockHeld" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of votes for each share of common stock held</link:label>
    <link:label id="lab_gemi_NumberOfVotesPerShareOfCommonStockHeld_label_en-US" xlink:label="lab_gemi_NumberOfVotesPerShareOfCommonStockHeld" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Number Of Votes Per Share Of Common Stock Held</link:label>
    <link:label id="lab_gemi_NumberOfVotesPerShareOfCommonStockHeld_documentation_en-US" xlink:label="lab_gemi_NumberOfVotesPerShareOfCommonStockHeld" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Number Of Votes Per Share Of Common Stock Held</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_NumberOfVotesPerShareOfCommonStockHeld" xlink:href="gemi-20250930.xsd#gemi_NumberOfVotesPerShareOfCommonStockHeld"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_NumberOfVotesPerShareOfCommonStockHeld" xlink:to="lab_gemi_NumberOfVotesPerShareOfCommonStockHeld" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock_9ca5506c-68c3-46ea-b124-cfc2a33a3bbe_terseLabel_en-US" xlink:label="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Prepaid Expenses and Other Assets</link:label>
    <link:label id="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Deferred Costs, Capitalized, Prepaid, and Other Assets Disclosure [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock" xlink:to="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SubsidiarySaleOfStockAxis_5c15b17a-ef8c-45a1-a3a5-99284baeb524_terseLabel_en-US" xlink:label="lab_us-gaap_SubsidiarySaleOfStockAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Sale of Stock [Axis]</link:label>
    <link:label id="lab_us-gaap_SubsidiarySaleOfStockAxis_label_en-US" xlink:label="lab_us-gaap_SubsidiarySaleOfStockAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Sale of Stock [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubsidiarySaleOfStockAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubsidiarySaleOfStockAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SubsidiarySaleOfStockAxis" xlink:to="lab_us-gaap_SubsidiarySaleOfStockAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfRepurchaseAgreements_121e3e50-562d-48fa-bfd3-84c766f62386_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Repurchase Agreements</link:label>
    <link:label id="lab_us-gaap_ScheduleOfRepurchaseAgreements_label_en-US" xlink:label="lab_us-gaap_ScheduleOfRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Repurchase Agreements [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfRepurchaseAgreements" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfRepurchaseAgreements"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfRepurchaseAgreements" xlink:to="lab_us-gaap_ScheduleOfRepurchaseAgreements" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain_6566deb2-b2d1-4ef8-8d1a-5f426955fb81_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Type [Domain]</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardAwardTypeAndPlanNameDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_bd56d793-c495-412c-8e0b-9493bf1108bc_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2026</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year One</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetRollForward_99b13b8c-27d0-4cba-ae2e-600e5397450e_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetRollForward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto Asset [Roll Forward]</link:label>
    <link:label id="lab_us-gaap_CryptoAssetRollForward_label_en-US" xlink:label="lab_us-gaap_CryptoAssetRollForward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset [Roll Forward]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetRollForward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetRollForward"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetRollForward" xlink:to="lab_us-gaap_CryptoAssetRollForward" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_EqtyAwrdsAdjsMember_terseLabel_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments</link:label>
    <link:label id="lab_ecd_EqtyAwrdsAdjsMember_label_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_EqtyAwrdsAdjsMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_EqtyAwrdsAdjsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_EqtyAwrdsAdjsMember" xlink:to="lab_ecd_EqtyAwrdsAdjsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LeaseholdImprovementsMember_e9fd633c-6d86-41a9-ba5a-444d87912870_terseLabel_en-US" xlink:label="lab_us-gaap_LeaseholdImprovementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Leasehold improvements</link:label>
    <link:label id="lab_us-gaap_LeaseholdImprovementsMember_label_en-US" xlink:label="lab_us-gaap_LeaseholdImprovementsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Leasehold Improvements [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LeaseholdImprovementsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LeaseholdImprovementsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LeaseholdImprovementsMember" xlink:to="lab_us-gaap_LeaseholdImprovementsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdvertisingExpense_c7dae213-a814-47b5-9b19-f0aa46ab6710_terseLabel_en-US" xlink:label="lab_us-gaap_AdvertisingExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Advertising expense</link:label>
    <link:label id="lab_us-gaap_AdvertisingExpense_label_en-US" xlink:label="lab_us-gaap_AdvertisingExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Advertising Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdvertisingExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdvertisingExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdvertisingExpense" xlink:to="lab_us-gaap_AdvertisingExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherNoncashIncomeExpense_f8282ad6-e358-4b0c-996c-eec4f406bc38_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_OtherNoncashIncomeExpense" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Other operating activities, net</link:label>
    <link:label id="lab_us-gaap_OtherNoncashIncomeExpense_label_en-US" xlink:label="lab_us-gaap_OtherNoncashIncomeExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Noncash Income (Expense)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherNoncashIncomeExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherNoncashIncomeExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherNoncashIncomeExpense" xlink:to="lab_us-gaap_OtherNoncashIncomeExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PnsnBnftsAdjFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_PnsnBnftsAdjFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pension Benefits Adjustments, Footnote</link:label>
    <link:label id="lab_ecd_PnsnBnftsAdjFnTextBlock_label_en-US" xlink:label="lab_ecd_PnsnBnftsAdjFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pension Benefits Adjustments, Footnote [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PnsnBnftsAdjFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PnsnBnftsAdjFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PnsnBnftsAdjFnTextBlock" xlink:to="lab_ecd_PnsnBnftsAdjFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisaggregationOfRevenueLineItems_76116b9c-a293-4698-b8f5-9d21199ee748_terseLabel_en-US" xlink:label="lab_us-gaap_DisaggregationOfRevenueLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Disaggregation of Revenue [Line Items]</link:label>
    <link:label id="lab_us-gaap_DisaggregationOfRevenueLineItems_label_en-US" xlink:label="lab_us-gaap_DisaggregationOfRevenueLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disaggregation of Revenue [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisaggregationOfRevenueLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisaggregationOfRevenueLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisaggregationOfRevenueLineItems" xlink:to="lab_us-gaap_DisaggregationOfRevenueLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_OutstandingRecoveryCompAmt_terseLabel_en-US" xlink:label="lab_ecd_OutstandingRecoveryCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Amount</link:label>
    <link:label id="lab_ecd_OutstandingRecoveryCompAmt_label_en-US" xlink:label="lab_ecd_OutstandingRecoveryCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Outstanding Recovery Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_OutstandingRecoveryCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_OutstandingRecoveryCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_OutstandingRecoveryCompAmt" xlink:to="lab_ecd_OutstandingRecoveryCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding_ee2883cb-c871-4881-85ef-9fc673691dbd_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity awards issued, vested and outstanding</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding_label_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Issued, Vested And Outstanding</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding_documentation_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Issued, Vested And Outstanding</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding" xlink:href="gemi-20250930.xsd#gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsIssuedVestedAndOutstanding"/>
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    <link:label id="lab_gemi_MarchNoteBTCMember_e30de877-5ff9-4a05-8291-b47804e48063_terseLabel_en-US" xlink:label="lab_gemi_MarchNoteBTCMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">March Note (BTC)</link:label>
    <link:label id="lab_gemi_MarchNoteBTCMember_label_en-US" xlink:label="lab_gemi_MarchNoteBTCMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">March Note (BTC) [Member]</link:label>
    <link:label id="lab_gemi_MarchNoteBTCMember_documentation_en-US" xlink:label="lab_gemi_MarchNoteBTCMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">March Note (BTC)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_MarchNoteBTCMember" xlink:href="gemi-20250930.xsd#gemi_MarchNoteBTCMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_MarchNoteBTCMember" xlink:to="lab_gemi_MarchNoteBTCMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentNameDomain_4461579e-d192-4c3f-a341-4199ff78cc82_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentNameDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Instrument, Name [Domain]</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentNameDomain_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentNameDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Name [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentNameDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentNameDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentNameDomain" xlink:to="lab_us-gaap_DebtInstrumentNameDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LitigationSettlementExpense_deba951b-afdd-4299-8ad4-0396f5cb2e91_terseLabel_en-US" xlink:label="lab_us-gaap_LitigationSettlementExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Expenses from litigation settlement</link:label>
    <link:label id="lab_us-gaap_LitigationSettlementExpense_label_en-US" xlink:label="lab_us-gaap_LitigationSettlementExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Litigation Settlement, Fee Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LitigationSettlementExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LitigationSettlementExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LitigationSettlementExpense" xlink:to="lab_us-gaap_LitigationSettlementExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_ccaa58b5-49b4-4f1c-9fb2-47058d9e35ee_verboseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Expected dividend yield</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_5979a214-305b-4cc1-9692-e4410c3a4fda_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Dividend Yield</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Expected Dividend Rate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate"/>
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    <link:label id="lab_ecd_RecoveryOfErrCompDisclosureLineItems_label_en-US" xlink:label="lab_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Recovery of Erroneously Awarded Compensation Disclosure [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_RecoveryOfErrCompDisclosureLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:to="lab_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ConvertibleNotesAndRelatedPartyTermLoansMember_e2ee70f7-20f9-4eb9-99c4-ab03889e21b6_terseLabel_en-US" xlink:label="lab_gemi_ConvertibleNotesAndRelatedPartyTermLoansMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Conversion of convertible notes and related party terms loans to common stock in connection with IPO</link:label>
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    <link:label id="lab_gemi_ConvertibleNotesAndRelatedPartyTermLoansMember_documentation_en-US" xlink:label="lab_gemi_ConvertibleNotesAndRelatedPartyTermLoansMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Convertible Notes And Related Party Term Loans</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ConvertibleNotesAndRelatedPartyTermLoansMember" xlink:href="gemi-20250930.xsd#gemi_ConvertibleNotesAndRelatedPartyTermLoansMember"/>
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    <link:label id="lab_gemi_TransactionRevenueOverTheCounterMember_ba436f6a-1b16-49e7-9709-71d918a8f855_terseLabel_en-US" xlink:label="lab_gemi_TransactionRevenueOverTheCounterMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">OTC</link:label>
    <link:label id="lab_gemi_TransactionRevenueOverTheCounterMember_label_en-US" xlink:label="lab_gemi_TransactionRevenueOverTheCounterMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Transaction Revenue, Over-The-Counter [Member]</link:label>
    <link:label id="lab_gemi_TransactionRevenueOverTheCounterMember_documentation_en-US" xlink:label="lab_gemi_TransactionRevenueOverTheCounterMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Transaction Revenue, Over-The-Counter</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TransactionRevenueOverTheCounterMember" xlink:href="gemi-20250930.xsd#gemi_TransactionRevenueOverTheCounterMember"/>
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    <link:label id="lab_us-gaap_AccountsReceivableAllowanceForCreditLossTableTextBlock_2c3b4f39-2069-483f-8127-359f9b57cd16_terseLabel_en-US" xlink:label="lab_us-gaap_AccountsReceivableAllowanceForCreditLossTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Allowance for Transaction Losses</link:label>
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    <link:label id="lab_us-gaap_FairValueOptionQuantitativeDisclosuresLineItems_294e406a-aa4d-4ddf-9ea7-54563227c213_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueOptionQuantitativeDisclosuresLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, Option, Quantitative Disclosures [Line Items]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueOptionQuantitativeDisclosuresLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueOptionQuantitativeDisclosuresLineItems"/>
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    <link:label id="lab_gemi_WarrantsReceivedAsRevenue_aeb4ed80-156d-44e7-8df9-9ea28548a275_negatedTerseLabel_en-US" xlink:label="lab_gemi_WarrantsReceivedAsRevenue" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Warrants received as revenue</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_WarrantsReceivedAsRevenue" xlink:to="lab_gemi_WarrantsReceivedAsRevenue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EmployeeStockMember_74900a39-6999-44b5-99c6-ac92b634895d_terseLabel_en-US" xlink:label="lab_us-gaap_EmployeeStockMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">ESPP</link:label>
    <link:label id="lab_us-gaap_EmployeeStockMember_label_en-US" xlink:label="lab_us-gaap_EmployeeStockMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Employee Stock [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EmployeeStockMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EmployeeStockMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EmployeeStockMember" xlink:to="lab_us-gaap_EmployeeStockMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RevenuesAbstract_64e66d86-bb1a-4c19-a789-77b2ea99983b_terseLabel_en-US" xlink:label="lab_us-gaap_RevenuesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Revenue:</link:label>
    <link:label id="lab_us-gaap_RevenuesAbstract_label_en-US" xlink:label="lab_us-gaap_RevenuesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenues [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RevenuesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RevenuesAbstract"/>
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    <link:label id="lab_gemi_SeriesAPreferredUnitsMember_ae0cd204-32c2-4888-b055-f7c511426944_terseLabel_en-US" xlink:label="lab_gemi_SeriesAPreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Series A Units</link:label>
    <link:label id="lab_gemi_SeriesAPreferredUnitsMember_label_en-US" xlink:label="lab_gemi_SeriesAPreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Series A Preferred Units [Member]</link:label>
    <link:label id="lab_gemi_SeriesAPreferredUnitsMember_documentation_en-US" xlink:label="lab_gemi_SeriesAPreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Series A Preferred Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_SeriesAPreferredUnitsMember" xlink:href="gemi-20250930.xsd#gemi_SeriesAPreferredUnitsMember"/>
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    <link:label id="lab_gemi_PreferredUnitsLiquidationPreferenceValue_8b4b2f99-3fb4-406f-97b1-8cf21ce955ee_terseLabel_en-US" xlink:label="lab_gemi_PreferredUnitsLiquidationPreferenceValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Liquidation Preference</link:label>
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    <link:label id="lab_gemi_CommonUnitsMember_documentation_en-US" xlink:label="lab_gemi_CommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CommonUnitsMember" xlink:href="gemi-20250930.xsd#gemi_CommonUnitsMember"/>
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    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetary_ca652223-e275-4e6f-be86-ff931c670653_terseLabel_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetary" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash received for issuance of long term debt</link:label>
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    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetary_documentation_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetary" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Proceeds From Issuance Of Long-Term Debt, Monetary</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ProceedsFromIssuanceOfLongTermDebtMonetary" xlink:to="lab_gemi_ProceedsFromIssuanceOfLongTermDebtMonetary" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AggtPnsnAdjsSvcCstMember_terseLabel_en-US" xlink:label="lab_ecd_AggtPnsnAdjsSvcCstMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate Pension Adjustments Service Cost</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_AggtPnsnAdjsSvcCstMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AggtPnsnAdjsSvcCstMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AggtPnsnAdjsSvcCstMember" xlink:to="lab_ecd_AggtPnsnAdjsSvcCstMember" xlink:type="arc" order="1"/>
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    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsExchanged_8c4e0c31-7113-4fba-833f-f21cd1cfb0c8_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsExchanged" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Units exchanged (in shares)</link:label>
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    <link:label id="lab_us-gaap_DevelopedTechnologyRightsMember_7ca3ece3-a9db-4939-8e13-19d56cf535a0_terseLabel_en-US" xlink:label="lab_us-gaap_DevelopedTechnologyRightsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Developed technology</link:label>
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    <link:label id="lab_us-gaap_AccountsNotesAndLoansReceivableLineItems_b069688f-3791-4e35-aed0-95897ac5472b_terseLabel_en-US" xlink:label="lab_us-gaap_AccountsNotesAndLoansReceivableLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounts, Notes, Loans and Financing Receivable [Line Items]</link:label>
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    <link:label id="lab_gemi_SoftwareMember_b52f649d-aa94-4cfd-85b7-061b13a600f6_terseLabel_en-US" xlink:label="lab_gemi_SoftwareMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Software</link:label>
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    <link:label id="lab_us-gaap_FairValueDisclosureItemAmountsDomain_418dce15-31f0-4703-b68b-ec61d06e4a60_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueDisclosureItemAmountsDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value Measurement [Domain]</link:label>
    <link:label id="lab_us-gaap_FairValueDisclosureItemAmountsDomain_label_en-US" xlink:label="lab_us-gaap_FairValueDisclosureItemAmountsDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Measurement [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueDisclosureItemAmountsDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueDisclosureItemAmountsDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueDisclosureItemAmountsDomain" xlink:to="lab_us-gaap_FairValueDisclosureItemAmountsDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan14Member_15458644-7127-4b11-924d-e4f8d65009e8_terseLabel_en-US" xlink:label="lab_gemi_Loan14Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 14</link:label>
    <link:label id="lab_gemi_Loan14Member_label_en-US" xlink:label="lab_gemi_Loan14Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 14 [Member]</link:label>
    <link:label id="lab_gemi_Loan14Member_documentation_en-US" xlink:label="lab_gemi_Loan14Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 14</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan14Member" xlink:href="gemi-20250930.xsd#gemi_Loan14Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan14Member" xlink:to="lab_gemi_Loan14Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtDisclosureTextBlock_944cd3e3-c320-4a6a-b96b-355bf4b13ba3_terseLabel_en-US" xlink:label="lab_us-gaap_DebtDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Loans and Convertible Notes</link:label>
    <link:label id="lab_us-gaap_DebtDisclosureTextBlock_06b57867-5ea9-4a34-8beb-6f5160c293a0_netLabel_en-US" xlink:label="lab_us-gaap_DebtDisclosureTextBlock" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Third Party Loans</link:label>
    <link:label id="lab_us-gaap_DebtDisclosureTextBlock_f7bfcabb-dcc5-4a3e-ad91-7db2ab87dba5_verboseLabel_en-US" xlink:label="lab_us-gaap_DebtDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Funding Debt</link:label>
    <link:label id="lab_us-gaap_DebtDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_DebtDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtDisclosureTextBlock" xlink:to="lab_us-gaap_DebtDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrSecuritiesAggAvailAmt_terseLabel_en-US" xlink:label="lab_ecd_TrdArrSecuritiesAggAvailAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate Available</link:label>
    <link:label id="lab_ecd_TrdArrSecuritiesAggAvailAmt_label_en-US" xlink:label="lab_ecd_TrdArrSecuritiesAggAvailAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement, Securities Aggregate Available Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrSecuritiesAggAvailAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrSecuritiesAggAvailAmt"/>
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    <link:label id="lab_us-gaap_AccountsPayableCurrent_1a2eeffd-b942-4a88-b21c-9e91fee9d41f_terseLabel_en-US" xlink:label="lab_us-gaap_AccountsPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounts payable</link:label>
    <link:label id="lab_us-gaap_AccountsPayableCurrent_label_en-US" xlink:label="lab_us-gaap_AccountsPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts Payable, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccountsPayableCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccountsPayableCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccountsPayableCurrent" xlink:to="lab_us-gaap_AccountsPayableCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockAppreciationRightsSARSMember_terseLabel_en-US" xlink:label="lab_us-gaap_StockAppreciationRightsSARSMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock Appreciation Rights (SARs)</link:label>
    <link:label id="lab_us-gaap_StockAppreciationRightsSARSMember_label_en-US" xlink:label="lab_us-gaap_StockAppreciationRightsSARSMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Appreciation Rights (SARs) [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockAppreciationRightsSARSMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockAppreciationRightsSARSMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockAppreciationRightsSARSMember" xlink:to="lab_us-gaap_StockAppreciationRightsSARSMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted_571e550c-35da-4175-9487-059ebf96b9b6_totalLabel_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net loss attributable to common stockholders, diluted</link:label>
    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted_label_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Net Income (Loss) Available to Common Stockholders, Diluted</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xlink:to="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersDiluted" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommitmentsAndContingenciesDisclosureTextBlock_60e522e5-38f8-4fcf-98fc-711da8b850d0_terseLabel_en-US" xlink:label="lab_us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Commitments and Contingencies</link:label>
    <link:label id="lab_us-gaap_CommitmentsAndContingenciesDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Commitments and Contingencies Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommitmentsAndContingenciesDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xlink:to="lab_us-gaap_CommitmentsAndContingenciesDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationAwardTrancheTwoMember_6101f347-31ab-42a2-94e9-94c7c111691c_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationAwardTrancheTwoMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Tranche Two</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationAwardTrancheTwoMember_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationAwardTrancheTwoMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Tranche Two [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationAwardTrancheTwoMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationAwardTrancheTwoMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationAwardTrancheTwoMember" xlink:to="lab_us-gaap_ShareBasedCompensationAwardTrancheTwoMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherAssetsNoncurrentAbstract_1b297af7-3d49-419f-8961-928669feb47f_terseLabel_en-US" xlink:label="lab_us-gaap_OtherAssetsNoncurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other non-current assets</link:label>
    <link:label id="lab_us-gaap_OtherAssetsNoncurrentAbstract_label_en-US" xlink:label="lab_us-gaap_OtherAssetsNoncurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Assets, Noncurrent [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherAssetsNoncurrentAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherAssetsNoncurrentAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherAssetsNoncurrentAbstract" xlink:to="lab_us-gaap_OtherAssetsNoncurrentAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_0a7ebdc7-6e17-486a-8ed1-aa0c4ac0c308_terseLabel_en-US" xlink:label="lab_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Weighted average shares outstanding - diluted (in shares)</link:label>
    <link:label id="lab_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_d3ab91a4-883d-477e-ab37-91ff0787c7e6_verboseLabel_en-US" xlink:label="lab_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Weighted average common stock outstanding, diluted (in shares)</link:label>
    <link:label id="lab_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_label_en-US" xlink:label="lab_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Weighted Average Number of Shares Outstanding, Diluted</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:to="lab_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OfficeSpaceInSanFranciscoCaliforniaMember_f3d2449a-cfcc-4e81-a4b9-a4013e298e79_terseLabel_en-US" xlink:label="lab_gemi_OfficeSpaceInSanFranciscoCaliforniaMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Office Space In San Francisco, California</link:label>
    <link:label id="lab_gemi_OfficeSpaceInSanFranciscoCaliforniaMember_label_en-US" xlink:label="lab_gemi_OfficeSpaceInSanFranciscoCaliforniaMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Office Space In San Francisco, California [Member]</link:label>
    <link:label id="lab_gemi_OfficeSpaceInSanFranciscoCaliforniaMember_documentation_en-US" xlink:label="lab_gemi_OfficeSpaceInSanFranciscoCaliforniaMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Office Space In San Francisco, California</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OfficeSpaceInSanFranciscoCaliforniaMember" xlink:to="lab_gemi_OfficeSpaceInSanFranciscoCaliforniaMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonGaapMeasureDescriptionTextBlock_terseLabel_en-US" xlink:label="lab_ecd_NonGaapMeasureDescriptionTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-GAAP Measure Description</link:label>
    <link:label id="lab_ecd_NonGaapMeasureDescriptionTextBlock_label_en-US" xlink:label="lab_ecd_NonGaapMeasureDescriptionTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-GAAP Measure Description [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonGaapMeasureDescriptionTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonGaapMeasureDescriptionTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonGaapMeasureDescriptionTextBlock" xlink:to="lab_ecd_NonGaapMeasureDescriptionTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A6750ETHMay12024Member_c9d4b9ae-dbe2-4d66-a9d5-60a3999937c2_terseLabel_en-US" xlink:label="lab_gemi_A6750ETHMay12024Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">6,750 ETH, May 1, 2024</link:label>
    <link:label id="lab_gemi_A6750ETHMay12024Member_label_en-US" xlink:label="lab_gemi_A6750ETHMay12024Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">6,750 ETH, May 1, 2024 [Member]</link:label>
    <link:label id="lab_gemi_A6750ETHMay12024Member_documentation_en-US" xlink:label="lab_gemi_A6750ETHMay12024Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">6,750 ETH, May 1, 2024</link:label>
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    <link:label id="lab_us-gaap_DerivativeTable_5b4482f3-c0c4-48eb-8f77-5bc37c72fed6_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative [Table]</link:label>
    <link:label id="lab_us-gaap_DerivativeTable_label_en-US" xlink:label="lab_us-gaap_DerivativeTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeTable" xlink:to="lab_us-gaap_DerivativeTable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrencyUsedDuringPeriod_ce5a5f2d-6d20-4f9f-96ff-eb95dbf3cb25_terseLabel_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrencyUsedDuringPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets received for issuance of long term debt, utilized during the period (in units)</link:label>
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    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrencyUsedDuringPeriod_documentation_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrencyUsedDuringPeriod" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Proceeds From Issuance Of Long-Term Debt, Crypto Currency Used During Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrencyUsedDuringPeriod" xlink:href="gemi-20250930.xsd#gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrencyUsedDuringPeriod"/>
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    <link:label id="lab_dei_EntitySmallBusiness_8cae2631-9515-4e19-bbe3-18b3f172447e_terseLabel_en-US" xlink:label="lab_dei_EntitySmallBusiness" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Small Business</link:label>
    <link:label id="lab_dei_EntitySmallBusiness_label_en-US" xlink:label="lab_dei_EntitySmallBusiness" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Small Business</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntitySmallBusiness" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntitySmallBusiness"/>
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    <link:label id="lab_gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember_38b925b3-7965-44a9-9d30-276151e48c54_terseLabel_en-US" xlink:label="lab_gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted For Regulatory Capital Requirements</link:label>
    <link:label id="lab_gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember_label_en-US" xlink:label="lab_gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Bitcoin, Restricted For Regulatory Capital Requirements [Member]</link:label>
    <link:label id="lab_gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember_documentation_en-US" xlink:label="lab_gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Bitcoin, Restricted Under Related Party Crypto Asset Loan Agreements</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember" xlink:href="gemi-20250930.xsd#gemi_BitcoinRestrictedForRegulatoryCapitalRequirementsMember"/>
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    <link:label id="lab_us-gaap_PaymentsOfStockIssuanceCosts_0ce53af5-f2df-40fa-a8e9-bb8cdc492ec7_terseLabel_en-US" xlink:label="lab_us-gaap_PaymentsOfStockIssuanceCosts" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Offering costs</link:label>
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    <link:label id="lab_us-gaap_ConvertibleDebtFairValueDisclosures_1d2bc2ce-05ee-46cd-b63c-a083984c0044_terseLabel_en-US" xlink:label="lab_us-gaap_ConvertibleDebtFairValueDisclosures" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible notes</link:label>
    <link:label id="lab_us-gaap_ConvertibleDebtFairValueDisclosures_label_en-US" xlink:label="lab_us-gaap_ConvertibleDebtFairValueDisclosures" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Convertible Debt, Fair Value Disclosures</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConvertibleDebtFairValueDisclosures" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConvertibleDebtFairValueDisclosures"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConvertibleDebtFairValueDisclosures" xlink:to="lab_us-gaap_ConvertibleDebtFairValueDisclosures" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_TotalCollateralAssociatedWithLoan_1a0d1a13-5006-49f0-8fe4-3dacdfe11b0d_terseLabel_en-US" xlink:label="lab_gemi_TotalCollateralAssociatedWithLoan" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total collateral associated with loan</link:label>
    <link:label id="lab_gemi_TotalCollateralAssociatedWithLoan_label_en-US" xlink:label="lab_gemi_TotalCollateralAssociatedWithLoan" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Total Collateral Associated With Loan</link:label>
    <link:label id="lab_gemi_TotalCollateralAssociatedWithLoan_documentation_en-US" xlink:label="lab_gemi_TotalCollateralAssociatedWithLoan" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Total Collateral Associated With Loan</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TotalCollateralAssociatedWithLoan" xlink:href="gemi-20250930.xsd#gemi_TotalCollateralAssociatedWithLoan"/>
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    <link:label id="lab_gemi_WarehouseCreditAgreementMember_6b7f05d9-1e90-4980-b44f-0b155dc256ed_terseLabel_en-US" xlink:label="lab_gemi_WarehouseCreditAgreementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Warehouse Credit Agreement</link:label>
    <link:label id="lab_gemi_WarehouseCreditAgreementMember_label_en-US" xlink:label="lab_gemi_WarehouseCreditAgreementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Warehouse Credit Agreement [Member]</link:label>
    <link:label id="lab_gemi_WarehouseCreditAgreementMember_documentation_en-US" xlink:label="lab_gemi_WarehouseCreditAgreementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Warehouse Credit Agreement</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_WarehouseCreditAgreementMember" xlink:href="gemi-20250930.xsd#gemi_WarehouseCreditAgreementMember"/>
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    <link:label id="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating_2550f528-9f8c-4063-bb90-a5b17c4fdd5c_terseLabel_en-US" xlink:label="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Realized and unrealized gain (loss) on related party crypto loans</link:label>
    <link:label id="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating_abb947d7-66ce-402a-b1d9-be420c69de94_negatedTerseLabel_en-US" xlink:label="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Realized and unrealized gain (loss) on related party crypto loans</link:label>
    <link:label id="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating_label_en-US" xlink:label="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Loans Payable, Realized And Unrealized Gain (Loss), Nonoperating</link:label>
    <link:label id="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating_documentation_en-US" xlink:label="lab_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Loans Payable, Realized And Unrealized Gain (Loss), Nonoperating</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating" xlink:href="gemi-20250930.xsd#gemi_CryptoLoansPayableRealizedAndUnrealizedGainLossNonoperating"/>
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    <link:label id="lab_dei_DocumentType_10ba9789-d874-4bf7-a203-3d3f5f09d2dd_terseLabel_en-US" xlink:label="lab_dei_DocumentType" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Document Type</link:label>
    <link:label id="lab_dei_DocumentType_label_en-US" xlink:label="lab_dei_DocumentType" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Document Type</link:label>
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    <link:label id="lab_us-gaap_FairValueInputsLevel3Member_5ecaef9d-a1e7-4014-8ae3-6f13c1b0e925_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueInputsLevel3Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Level 3</link:label>
    <link:label id="lab_us-gaap_FairValueInputsLevel3Member_label_en-US" xlink:label="lab_us-gaap_FairValueInputsLevel3Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Inputs, Level 3 [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueInputsLevel3Member" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueInputsLevel3Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueInputsLevel3Member" xlink:to="lab_us-gaap_FairValueInputsLevel3Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EarningsPerShareTextBlock_8bc18fe9-b7db-4326-b545-34a6a37ce07d_terseLabel_en-US" xlink:label="lab_us-gaap_EarningsPerShareTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net Loss Per Share</link:label>
    <link:label id="lab_us-gaap_EarningsPerShareTextBlock_label_en-US" xlink:label="lab_us-gaap_EarningsPerShareTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EarningsPerShareTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EarningsPerShareTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EarningsPerShareTextBlock" xlink:to="lab_us-gaap_EarningsPerShareTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_OutstandingRecoveryIndName_terseLabel_en-US" xlink:label="lab_ecd_OutstandingRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Name</link:label>
    <link:label id="lab_ecd_OutstandingRecoveryIndName_label_en-US" xlink:label="lab_ecd_OutstandingRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Outstanding Recovery, Individual Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_OutstandingRecoveryIndName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_OutstandingRecoveryIndName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_OutstandingRecoveryIndName" xlink:to="lab_ecd_OutstandingRecoveryIndName" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LossContingenciesTable_bb071dc4-9f84-43ed-9c64-a7d90bc1fd78_terseLabel_en-US" xlink:label="lab_us-gaap_LossContingenciesTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loss Contingencies [Table]</link:label>
    <link:label id="lab_us-gaap_LossContingenciesTable_label_en-US" xlink:label="lab_us-gaap_LossContingenciesTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loss Contingencies [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LossContingenciesTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LossContingenciesTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LossContingenciesTable" xlink:to="lab_us-gaap_LossContingenciesTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdditionalPaidInCapital_8fd30ca3-6e14-4358-bd0a-6934c9aa5644_terseLabel_en-US" xlink:label="lab_us-gaap_AdditionalPaidInCapital" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Additional paid-in capital</link:label>
    <link:label id="lab_us-gaap_AdditionalPaidInCapital_label_en-US" xlink:label="lab_us-gaap_AdditionalPaidInCapital" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Additional Paid in Capital</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdditionalPaidInCapital" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdditionalPaidInCapital"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdditionalPaidInCapital" xlink:to="lab_us-gaap_AdditionalPaidInCapital" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_ProductOrServiceAxis_f41e868c-b851-4f05-8145-957315691281_terseLabel_en-US" xlink:label="lab_srt_ProductOrServiceAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Product and Service [Axis]</link:label>
    <link:label id="lab_srt_ProductOrServiceAxis_label_en-US" xlink:label="lab_srt_ProductOrServiceAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Product and Service [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_ProductOrServiceAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ProductOrServiceAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_ProductOrServiceAxis" xlink:to="lab_srt_ProductOrServiceAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_cd33870b-6c1e-4cee-a619-179fd465ddf9_verboseLabel_en-US" xlink:label="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Net revenue</link:label>
    <link:label id="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_6d9e95e1-b8ed-430e-aaa0-df33e7abbcaf_terseLabel_en-US" xlink:label="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total net revenue from contracts with customers</link:label>
    <link:label id="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_55e9eab7-a7ff-472f-928f-e5bc28764a31_netLabel_en-US" xlink:label="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Revenue</link:label>
    <link:label id="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_label_en-US" xlink:label="lab_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenue from Contract with Customer, Excluding Assessed Tax</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax"/>
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    <link:label id="lab_us-gaap_AssetsCurrent_e1082d7f-c681-4750-847f-870bc1edea90_totalLabel_en-US" xlink:label="lab_us-gaap_AssetsCurrent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total current assets</link:label>
    <link:label id="lab_us-gaap_AssetsCurrent_label_en-US" xlink:label="lab_us-gaap_AssetsCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Assets, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AssetsCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssetsCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AssetsCurrent" xlink:to="lab_us-gaap_AssetsCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_RangeAxis_b4757fb0-0624-46db-8f91-d71dabea2800_terseLabel_en-US" xlink:label="lab_srt_RangeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statistical Measurement [Axis]</link:label>
    <link:label id="lab_srt_RangeAxis_label_en-US" xlink:label="lab_srt_RangeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statistical Measurement [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_RangeAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_RangeAxis"/>
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    <link:label id="lab_gemi_IncreaseDecreaseInCryptoAssetDisposition_c255cb2e-a262-4428-a882-14cc8c1176a6_terseLabel_en-US" xlink:label="lab_gemi_IncreaseDecreaseInCryptoAssetDisposition" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Disposal of crypto assets</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_gemi_IncreaseDecreaseInCryptoAssetDisposition" xlink:href="gemi-20250930.xsd#gemi_IncreaseDecreaseInCryptoAssetDisposition"/>
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    <link:label id="lab_us-gaap_LossContingencyAccrualAtCarryingValue_4fbc901b-d977-4432-8c67-6988e8813313_terseLabel_en-US" xlink:label="lab_us-gaap_LossContingencyAccrualAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Estimated loss contingency accrued</link:label>
    <link:label id="lab_us-gaap_LossContingencyAccrualAtCarryingValue_label_en-US" xlink:label="lab_us-gaap_LossContingencyAccrualAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loss Contingency Accrual</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LossContingencyAccrualAtCarryingValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LossContingencyAccrualAtCarryingValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LossContingencyAccrualAtCarryingValue" xlink:to="lab_us-gaap_LossContingencyAccrualAtCarryingValue" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_BitcoinBitcoinRestrictedForCollateralRequirementsMember_43c12657-162a-4bd7-a9f0-8cc998657459_terseLabel_en-US" xlink:label="lab_gemi_BitcoinBitcoinRestrictedForCollateralRequirementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted For Collateral Requirements</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_BitcoinBitcoinRestrictedForCollateralRequirementsMember" xlink:to="lab_gemi_BitcoinBitcoinRestrictedForCollateralRequirementsMember" xlink:type="arc" order="1"/>
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    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_dfe6eee5-3ef6-4b56-8be7-3b9f4e7ee60e_verboseLabel_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Net loss attributable to common stockholders, basic</link:label>
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    <link:label id="lab_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable_a0b264ed-ac0d-497d-a019-0a8e63a18a81_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transaction [Table]</link:label>
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    <link:label id="lab_gemi_BasicCommonUnitsMember_documentation_en-US" xlink:label="lab_gemi_BasicCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Basic Common Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_BasicCommonUnitsMember" xlink:href="gemi-20250930.xsd#gemi_BasicCommonUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_BasicCommonUnitsMember" xlink:to="lab_gemi_BasicCommonUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AccountsReceivableNetCurrent_670e5a2e-d38c-4103-8b09-726a458b393c_verboseLabel_en-US" xlink:label="lab_us-gaap_AccountsReceivableNetCurrent" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Accounts receivable, net</link:label>
    <link:label id="lab_us-gaap_AccountsReceivableNetCurrent_b68b9e92-0715-4dda-bacd-ed5e13a6bcd5_totalLabel_en-US" xlink:label="lab_us-gaap_AccountsReceivableNetCurrent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total accounts receivable, net of allowance</link:label>
    <link:label id="lab_us-gaap_AccountsReceivableNetCurrent_label_en-US" xlink:label="lab_us-gaap_AccountsReceivableNetCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts Receivable, after Allowance for Credit Loss, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccountsReceivableNetCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccountsReceivableNetCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccountsReceivableNetCurrent" xlink:to="lab_us-gaap_AccountsReceivableNetCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction_c659c12c-c9b2-433c-9237-b54f3aedd46b_verboseLabel_en-US" xlink:label="lab_us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Shares issued and sold (in shares)</link:label>
    <link:label id="lab_us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction_label_en-US" xlink:label="lab_us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Sale of Stock, Number of Shares Issued in Transaction</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xlink:to="lab_us-gaap_SaleOfStockNumberOfSharesIssuedInTransaction" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TreasuryStockCommonShares_e4752602-a8cc-4ebd-92a8-48f113f661f5_terseLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonShares" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Treasury stock (in shares)</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonShares_bdfb9ded-0c86-4f31-b958-51e1063989e3_negatedPeriodStartLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonShares" xlink:role="http://www.xbrl.org/2009/role/negatedPeriodStartLabel" xlink:type="resource" xml:lang="en-US">Beginning treasury balance (in shares)</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonShares_46e9815b-f733-4337-b8b8-715f770c3dc2_negatedPeriodEndLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonShares" xlink:role="http://www.xbrl.org/2009/role/negatedPeriodEndLabel" xlink:type="resource" xml:lang="en-US">Ending treasury balance (in shares)</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonShares_label_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonShares" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Treasury Stock, Common, Shares</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TreasuryStockCommonShares" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TreasuryStockCommonShares"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TreasuryStockCommonShares" xlink:to="lab_us-gaap_TreasuryStockCommonShares" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OtherFeesReceivableNettingMember_20569a43-b15f-40ea-9945-b4ce7d2d2664_terseLabel_en-US" xlink:label="lab_gemi_OtherFeesReceivableNettingMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other fees receivable</link:label>
    <link:label id="lab_gemi_OtherFeesReceivableNettingMember_label_en-US" xlink:label="lab_gemi_OtherFeesReceivableNettingMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Fees Receivable, Netting [Member]</link:label>
    <link:label id="lab_gemi_OtherFeesReceivableNettingMember_documentation_en-US" xlink:label="lab_gemi_OtherFeesReceivableNettingMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Other Fees Receivable, Netting</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OtherFeesReceivableNettingMember" xlink:href="gemi-20250930.xsd#gemi_OtherFeesReceivableNettingMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OtherFeesReceivableNettingMember" xlink:to="lab_gemi_OtherFeesReceivableNettingMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt_3fd1c3d8-208c-4a98-9de6-9d6c614f9542_terseLabel_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value on related party loans</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt_f1025d87-9b22-4902-a11d-cf6d794b51a0_negatedTerseLabel_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value on related party loans</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt_9f812a02-bfe0-4f8e-bb99-2d7323d40197_negatedNetLabel_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:role="http://www.xbrl.org/2009/role/negatedNetLabel" xlink:type="resource" xml:lang="en-US">Unrealized loss</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt_label_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Gain (Loss) On Change In Fair Value Of Related Party Debt</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt_documentation_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Gain (Loss) On Change In Fair Value Of Related Party Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:href="gemi-20250930.xsd#gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:to="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_IncreaseDecreaseInCryptoAssetAddition_ab7e4456-7f5d-4914-8264-244bc2112ab4_negatedTerseLabel_en-US" xlink:label="lab_gemi_IncreaseDecreaseInCryptoAssetAddition" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Purchase of crypto assets</link:label>
    <link:label id="lab_gemi_IncreaseDecreaseInCryptoAssetAddition_label_en-US" xlink:label="lab_gemi_IncreaseDecreaseInCryptoAssetAddition" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) In Crypto Asset, Addition</link:label>
    <link:label id="lab_gemi_IncreaseDecreaseInCryptoAssetAddition_documentation_en-US" xlink:label="lab_gemi_IncreaseDecreaseInCryptoAssetAddition" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Increase (Decrease) In Crypto Asset, Addition</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_IncreaseDecreaseInCryptoAssetAddition" xlink:href="gemi-20250930.xsd#gemi_IncreaseDecreaseInCryptoAssetAddition"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_IncreaseDecreaseInCryptoAssetAddition" xlink:to="lab_gemi_IncreaseDecreaseInCryptoAssetAddition" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount_9ae795d9-0e91-452d-a9bf-a3fbf66edfd1_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Less: imputed interest</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, Undiscounted Excess Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_FundingDebtIncludingAccruedInterestCurrent_056d0274-a923-4bce-9ccd-600f9b3b8cf7_terseLabel_en-US" xlink:label="lab_gemi_FundingDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Funding debt</link:label>
    <link:label id="lab_gemi_FundingDebtIncludingAccruedInterestCurrent_label_en-US" xlink:label="lab_gemi_FundingDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Funding Debt, Including Accrued Interest, Current</link:label>
    <link:label id="lab_gemi_FundingDebtIncludingAccruedInterestCurrent_documentation_en-US" xlink:label="lab_gemi_FundingDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Funding Debt, Including Accrued Interest, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_FundingDebtIncludingAccruedInterestCurrent" xlink:href="gemi-20250930.xsd#gemi_FundingDebtIncludingAccruedInterestCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_FundingDebtIncludingAccruedInterestCurrent" xlink:to="lab_gemi_FundingDebtIncludingAccruedInterestCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncomeTaxesPaidNet_9092cb88-0e5b-4ec3-be25-0f914db159b7_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeTaxesPaidNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash paid during the period for income taxes</link:label>
    <link:label id="lab_us-gaap_IncomeTaxesPaidNet_label_en-US" xlink:label="lab_us-gaap_IncomeTaxesPaidNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income Taxes Paid, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeTaxesPaidNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeTaxesPaidNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncomeTaxesPaidNet" xlink:to="lab_us-gaap_IncomeTaxesPaidNet" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TemporaryEquitySharesIssued_94056f93-3fe5-46d9-9297-4c16dc3dc83f_terseLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesIssued" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible preferred units, shares issued (in shares)</link:label>
    <link:label id="lab_us-gaap_TemporaryEquitySharesIssued_label_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesIssued" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Temporary Equity, Shares Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TemporaryEquitySharesIssued" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TemporaryEquitySharesIssued"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TemporaryEquitySharesIssued" xlink:to="lab_us-gaap_TemporaryEquitySharesIssued" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo_02212d50-45ad-47bc-95ef-16ce6299f614_terseLabel_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2027</link:label>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo_label_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Finite-Lived Intangible Asset, Expected Amortization, Year Two</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo" xlink:to="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember_eb6e1a19-f423-4964-8020-9797f21a4cad_terseLabel_en-US" xlink:label="lab_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common Units</link:label>
    <link:label id="lab_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember_label_en-US" xlink:label="lab_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Basic Common Units And Capital Common Units [Member]</link:label>
    <link:label id="lab_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember_documentation_en-US" xlink:label="lab_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Basic Common Units And Capital Common Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember" xlink:href="gemi-20250930.xsd#gemi_BasicCommonUnitsAndCapitalCommonUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember" xlink:to="lab_gemi_BasicCommonUnitsAndCapitalCommonUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetFairValue_9170663c-53f1-4529-967e-f417fab91df8_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets held:</link:label>
    <link:label id="lab_us-gaap_CryptoAssetFairValue_5b99ab96-d52f-4c6d-880e-99cfb06233c1_verboseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Fair Value</link:label>
    <link:label id="lab_us-gaap_CryptoAssetFairValue_d325840d-4bbc-47bb-bda8-e476b3ec6b19_periodStartLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValue" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Crypto assets held beginning balance</link:label>
    <link:label id="lab_us-gaap_CryptoAssetFairValue_440ad18e-d31a-41ab-8b25-90c16a8c6687_periodEndLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValue" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Crypto assets held, ending balance</link:label>
    <link:label id="lab_us-gaap_CryptoAssetFairValue_label_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetFairValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetFairValue" xlink:to="lab_us-gaap_CryptoAssetFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice_887b10e8-d4bd-48b2-9582-6c1956927aa3_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Granted (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Grants in Period, Weighted Average Exercise Price</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageExercisePrice" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EmployeeRelatedLiabilitiesCurrent_62e83f5e-731f-4528-8fcd-7ed268920655_verboseLabel_en-US" xlink:label="lab_us-gaap_EmployeeRelatedLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Accrued salaries and benefits</link:label>
    <link:label id="lab_us-gaap_EmployeeRelatedLiabilitiesCurrent_label_en-US" xlink:label="lab_us-gaap_EmployeeRelatedLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Employee-related Liabilities, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EmployeeRelatedLiabilitiesCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EmployeeRelatedLiabilitiesCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EmployeeRelatedLiabilitiesCurrent" xlink:to="lab_us-gaap_EmployeeRelatedLiabilitiesCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock_74d3c6a4-df5c-4cf9-a6b3-4a7decb782b0_terseLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Software, Property and Equipment, Net</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_8374cac7-814b-44c3-bced-3b88d8f5d8f4_verboseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Total present value of lease payments</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_102ea685-c46a-4dde-a8e8-da43a083731f_totalLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total undiscounted lease payments</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity_7cd870c0-aa92-4390-b046-dd4262c207b4_terseLabel_en-US" xlink:label="lab_us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Line of credit facility, current borrowing capacity</link:label>
    <link:label id="lab_us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity_label_en-US" xlink:label="lab_us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Line of Credit Facility, Current Borrowing Capacity</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity" xlink:to="lab_us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrExpirationDate_terseLabel_en-US" xlink:label="lab_ecd_TrdArrExpirationDate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Expiration Date</link:label>
    <link:label id="lab_ecd_TrdArrExpirationDate_label_en-US" xlink:label="lab_ecd_TrdArrExpirationDate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement Expiration Date</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrExpirationDate" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrExpirationDate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TrdArrExpirationDate" xlink:to="lab_ecd_TrdArrExpirationDate" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_TotalCollateralAssociatedWithRepurchaseAgreements_69a2f471-2392-4554-adfe-509117807123_terseLabel_en-US" xlink:label="lab_gemi_TotalCollateralAssociatedWithRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total collateral associated with repurchase agreements</link:label>
    <link:label id="lab_gemi_TotalCollateralAssociatedWithRepurchaseAgreements_label_en-US" xlink:label="lab_gemi_TotalCollateralAssociatedWithRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Total Collateral Associated With Repurchase Agreements</link:label>
    <link:label id="lab_gemi_TotalCollateralAssociatedWithRepurchaseAgreements_documentation_en-US" xlink:label="lab_gemi_TotalCollateralAssociatedWithRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Total Collateral Associated With Repurchase Agreements</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TotalCollateralAssociatedWithRepurchaseAgreements" xlink:href="gemi-20250930.xsd#gemi_TotalCollateralAssociatedWithRepurchaseAgreements"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_TotalCollateralAssociatedWithRepurchaseAgreements" xlink:to="lab_gemi_TotalCollateralAssociatedWithRepurchaseAgreements" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOtherOperatingLiabilities_6df8d318-5130-48bb-b579-781684d7cc60_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOtherOperatingLiabilities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other liabilities</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOtherOperatingLiabilities_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOtherOperatingLiabilities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Other Operating Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInOtherOperatingLiabilities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInOtherOperatingLiabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInOtherOperatingLiabilities" xlink:to="lab_us-gaap_IncreaseDecreaseInOtherOperatingLiabilities" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentNet_061c64e2-be77-4c4e-913d-0288ce4a37d3_verboseLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentNet" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Software, property and equipment, net</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentNet_6cfc87b1-adde-43bd-a894-39853a59cd27_totalLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentNet" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Software, property and equipment, net</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentNet_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentNet" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentNet" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CompActuallyPaidVsNetIncomeTextBlock_terseLabel_en-US" xlink:label="lab_ecd_CompActuallyPaidVsNetIncomeTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Net Income</link:label>
    <link:label id="lab_ecd_CompActuallyPaidVsNetIncomeTextBlock_label_en-US" xlink:label="lab_ecd_CompActuallyPaidVsNetIncomeTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Net Income [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CompActuallyPaidVsNetIncomeTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CompActuallyPaidVsNetIncomeTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CompActuallyPaidVsNetIncomeTextBlock" xlink:to="lab_ecd_CompActuallyPaidVsNetIncomeTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInAccountsReceivable_8dc73427-c927-4da0-adf9-c7cc9093500d_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInAccountsReceivable" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Accounts receivable</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInAccountsReceivable_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInAccountsReceivable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Accounts Receivable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInAccountsReceivable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInAccountsReceivable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInAccountsReceivable" xlink:to="lab_us-gaap_IncreaseDecreaseInAccountsReceivable" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityCurrentReportingStatus_2a85af97-75b8-4cc6-8cc9-a67ccfee7e82_terseLabel_en-US" xlink:label="lab_dei_EntityCurrentReportingStatus" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Current Reporting Status</link:label>
    <link:label id="lab_dei_EntityCurrentReportingStatus_label_en-US" xlink:label="lab_dei_EntityCurrentReportingStatus" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Current Reporting Status</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityCurrentReportingStatus" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityCurrentReportingStatus"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityCurrentReportingStatus" xlink:to="lab_dei_EntityCurrentReportingStatus" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReceivablesAbstract_label_en-US" xlink:label="lab_us-gaap_ReceivablesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Receivables [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReceivablesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReceivablesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReceivablesAbstract" xlink:to="lab_us-gaap_ReceivablesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativeLineItems_1165000f-d478-4835-9304-85aa81096be2_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative [Line Items]</link:label>
    <link:label id="lab_us-gaap_DerivativeLineItems_label_en-US" xlink:label="lab_us-gaap_DerivativeLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeLineItems" xlink:to="lab_us-gaap_DerivativeLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardsCloseToMnpiDiscTable_terseLabel_en-US" xlink:label="lab_ecd_AwardsCloseToMnpiDiscTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Awards Close in Time to MNPI Disclosures</link:label>
    <link:label id="lab_ecd_AwardsCloseToMnpiDiscTable_label_en-US" xlink:label="lab_ecd_AwardsCloseToMnpiDiscTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Awards Close in Time to MNPI Disclosures [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardsCloseToMnpiDiscTable" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardsCloseToMnpiDiscTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AwardsCloseToMnpiDiscTable" xlink:to="lab_ecd_AwardsCloseToMnpiDiscTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ProceedsFromConvertibleDebt_87fed924-e100-4b39-a7e3-c1b52cd0b351_verboseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from related party convertible notes</link:label>
    <link:label id="lab_us-gaap_ProceedsFromConvertibleDebt_1a356ede-66f4-4a72-89ec-76384c624a59_terseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amount Drawn</link:label>
    <link:label id="lab_us-gaap_ProceedsFromConvertibleDebt_label_en-US" xlink:label="lab_us-gaap_ProceedsFromConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds from Convertible Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromConvertibleDebt" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromConvertibleDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ProceedsFromConvertibleDebt" xlink:to="lab_us-gaap_ProceedsFromConvertibleDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FinancialInstrumentAxis_b7fbda31-c394-4114-b3ab-a1d1fc9de4e1_terseLabel_en-US" xlink:label="lab_us-gaap_FinancialInstrumentAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Financial Instrument [Axis]</link:label>
    <link:label id="lab_us-gaap_FinancialInstrumentAxis_label_en-US" xlink:label="lab_us-gaap_FinancialInstrumentAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Financial Instrument [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancialInstrumentAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancialInstrumentAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FinancialInstrumentAxis" xlink:to="lab_us-gaap_FinancialInstrumentAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementOfFinancialPositionAbstract_label_en-US" xlink:label="lab_us-gaap_StatementOfFinancialPositionAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Financial Position [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfFinancialPositionAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfFinancialPositionAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfFinancialPositionAbstract" xlink:to="lab_us-gaap_StatementOfFinancialPositionAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ExecutiveCategoryAxis_terseLabel_en-US" xlink:label="lab_ecd_ExecutiveCategoryAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Executive Category:</link:label>
    <link:label id="lab_ecd_ExecutiveCategoryAxis_label_en-US" xlink:label="lab_ecd_ExecutiveCategoryAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Executive Category [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ExecutiveCategoryAxis" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ExecutiveCategoryAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ExecutiveCategoryAxis" xlink:to="lab_ecd_ExecutiveCategoryAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod_bae3816c-3989-4fac-ae08-292dd836fdfd_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Issuance of Common Units due to equity awards granted, net of forfeitures (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Grants in Period, Net of Forfeitures</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetActivityTable_26991516-bc8f-4930-8960-4b824033edd8_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetActivityTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto Asset, Activity [Table]</link:label>
    <link:label id="lab_us-gaap_CryptoAssetActivityTable_label_en-US" xlink:label="lab_us-gaap_CryptoAssetActivityTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Activity [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetActivityTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetActivityTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetActivityTable" xlink:to="lab_us-gaap_CryptoAssetActivityTable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_SeriesA2PreferredUnitsMember_35136ff8-2cf3-41a4-b330-a1775fd6c12f_terseLabel_en-US" xlink:label="lab_gemi_SeriesA2PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Series A-2</link:label>
    <link:label id="lab_gemi_SeriesA2PreferredUnitsMember_label_en-US" xlink:label="lab_gemi_SeriesA2PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Series A-2 Preferred Units [Member]</link:label>
    <link:label id="lab_gemi_SeriesA2PreferredUnitsMember_documentation_en-US" xlink:label="lab_gemi_SeriesA2PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Series A-2 Preferred Units</link:label>
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    <link:label id="lab_us-gaap_ScheduleOfEarningsPerShareBasicByCommonClassTable_7973acd8-f548-4ede-ae8e-13517ce6fd9a_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfEarningsPerShareBasicByCommonClassTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Basic, by Common Class, Including Two-Class Method [Table]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xlink:to="lab_us-gaap_ScheduleOfFairValueAssetsAndLiabilitiesMeasuredOnRecurringBasisTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EquitySecuritiesFvNi_a058060d-7c28-4077-9ec4-6398612be2fd_terseLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNi" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity securities</link:label>
    <link:label id="lab_us-gaap_EquitySecuritiesFvNi_label_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNi" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Securities, FV-NI, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquitySecuritiesFvNi" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquitySecuritiesFvNi"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EquitySecuritiesFvNi" xlink:to="lab_us-gaap_EquitySecuritiesFvNi" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_CryptoAssetOtherMember_f3d5a95e-759d-4792-a4f8-2634eefd550b_terseLabel_en-US" xlink:label="lab_srt_CryptoAssetOtherMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other crypto assets</link:label>
    <link:label id="lab_srt_CryptoAssetOtherMember_63f2605b-2a14-4c48-8600-838ae0972f54_verboseLabel_en-US" xlink:label="lab_srt_CryptoAssetOtherMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Other</link:label>
    <link:label id="lab_srt_CryptoAssetOtherMember_label_en-US" xlink:label="lab_srt_CryptoAssetOtherMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Other [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_CryptoAssetOtherMember" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_CryptoAssetOtherMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_CryptoAssetOtherMember" xlink:to="lab_srt_CryptoAssetOtherMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount_bc1edbab-9d8f-4e56-a0c6-2bf75689c6a7_terseLabel_en-US" xlink:label="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Potentially dilutive units not included in calculation of diluted units outstanding (in shares)</link:label>
    <link:label id="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount_label_en-US" xlink:label="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Antidilutive Securities Excluded from Computation of Earnings Per Share, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xlink:to="lab_us-gaap_AntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_ea885070-8c6e-4719-a728-c80b0bc54d13_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, Liabilities Measured on Recurring Basis, Unobservable Input Reconciliation, Calculation [Roll Forward]</link:label>
    <link:label id="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_label_en-US" xlink:label="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Liabilities Measured on Recurring Basis, Unobservable Input Reconciliation, Calculation [Roll Forward]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:to="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_LossContingencyStayOfCasePeriod_1518fddb-3f22-4e75-a5f3-3bff80ae78b4_terseLabel_en-US" xlink:label="lab_gemi_LossContingencyStayOfCasePeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Court ordered stay of case period</link:label>
    <link:label id="lab_gemi_LossContingencyStayOfCasePeriod_label_en-US" xlink:label="lab_gemi_LossContingencyStayOfCasePeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loss Contingency, Stay Of Case Period</link:label>
    <link:label id="lab_gemi_LossContingencyStayOfCasePeriod_documentation_en-US" xlink:label="lab_gemi_LossContingencyStayOfCasePeriod" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loss Contingency, Stay Of Case Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_LossContingencyStayOfCasePeriod" xlink:href="gemi-20250930.xsd#gemi_LossContingencyStayOfCasePeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_LossContingencyStayOfCasePeriod" xlink:to="lab_gemi_LossContingencyStayOfCasePeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueByBalanceSheetGroupingTable_4dfd140d-f828-4198-a23c-51e2f9ab45e8_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueByBalanceSheetGroupingTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, by Balance Sheet Grouping [Table]</link:label>
    <link:label id="lab_us-gaap_FairValueByBalanceSheetGroupingTable_label_en-US" xlink:label="lab_us-gaap_FairValueByBalanceSheetGroupingTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, by Balance Sheet Grouping [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueByBalanceSheetGroupingTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueByBalanceSheetGroupingTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueByBalanceSheetGroupingTable" xlink:to="lab_us-gaap_FairValueByBalanceSheetGroupingTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherLiabilitiesCurrentAbstract_3895caf2-e5d1-4cbe-b527-c95a59aeede9_terseLabel_en-US" xlink:label="lab_us-gaap_OtherLiabilitiesCurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other current liabilities:</link:label>
    <link:label id="lab_us-gaap_OtherLiabilitiesCurrentAbstract_label_en-US" xlink:label="lab_us-gaap_OtherLiabilitiesCurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Liabilities, Current [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherLiabilitiesCurrentAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherLiabilitiesCurrentAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherLiabilitiesCurrentAbstract" xlink:to="lab_us-gaap_OtherLiabilitiesCurrentAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeasesTextBlock_d901c12f-96fd-4060-95ba-3663ab88e3fe_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeasesTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Leases</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeasesTextBlock_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeasesTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Leases [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeasesTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeasesTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeasesTextBlock" xlink:to="lab_us-gaap_LesseeOperatingLeasesTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInFinancingActivities_2365503f-8591-4d1b-92d2-d875dd6432b9_totalLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net cash provided by financing activities</link:label>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInFinancingActivities_label_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Financing Activity, Including Discontinued Operation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:to="lab_us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_MeasureAxis_terseLabel_en-US" xlink:label="lab_ecd_MeasureAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Measure:</link:label>
    <link:label id="lab_ecd_MeasureAxis_label_en-US" xlink:label="lab_ecd_MeasureAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measure [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_MeasureAxis" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_MeasureAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_MeasureAxis" xlink:to="lab_ecd_MeasureAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RetailReceivableMember_fbb90922-a4e8-43cb-bfcb-095efe17b3a7_terseLabel_en-US" xlink:label="lab_gemi_RetailReceivableMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Retail receivables</link:label>
    <link:label id="lab_gemi_RetailReceivableMember_label_en-US" xlink:label="lab_gemi_RetailReceivableMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Retail Receivable [Member]</link:label>
    <link:label id="lab_gemi_RetailReceivableMember_documentation_en-US" xlink:label="lab_gemi_RetailReceivableMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Retail Receivable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RetailReceivableMember" xlink:href="gemi-20250930.xsd#gemi_RetailReceivableMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RetailReceivableMember" xlink:to="lab_gemi_RetailReceivableMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoLegalSettlementAgreementContributions_82370d8c-87d1-4261-9d54-c5710d9460bf_terseLabel_en-US" xlink:label="lab_gemi_CryptoLegalSettlementAgreementContributions" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Bitcoin contributed for settlement agreement</link:label>
    <link:label id="lab_gemi_CryptoLegalSettlementAgreementContributions_label_en-US" xlink:label="lab_gemi_CryptoLegalSettlementAgreementContributions" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto, Legal Settlement Agreement, Contributions</link:label>
    <link:label id="lab_gemi_CryptoLegalSettlementAgreementContributions_documentation_en-US" xlink:label="lab_gemi_CryptoLegalSettlementAgreementContributions" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto, Legal Settlement Agreement, Contributions</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoLegalSettlementAgreementContributions" xlink:href="gemi-20250930.xsd#gemi_CryptoLegalSettlementAgreementContributions"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoLegalSettlementAgreementContributions" xlink:to="lab_gemi_CryptoLegalSettlementAgreementContributions" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentTypeDomain_a074d7e1-091d-4bba-8803-0552444447d0_terseLabel_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Long-Lived Tangible Asset [Domain]</link:label>
    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentTypeDomain_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Lived Tangible Asset [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentTypeDomain" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures_fa4b52e3-49a0-4fc2-8f30-adbd7397ac6c_terseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Issuance of common stock upon settlement of RSUs, net of shares withheld</link:label>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures_label_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Issued During Period, Value, Restricted Stock Award, Net of Forfeitures</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures" xlink:to="lab_us-gaap_StockIssuedDuringPeriodValueRestrictedStockAwardNetOfForfeitures" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetsReceivedInKind_629e4988-2f28-4f25-9828-6c5ab16ceee7_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetsReceivedInKind" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets received in kind</link:label>
    <link:label id="lab_gemi_CryptoAssetsReceivedInKind_label_en-US" xlink:label="lab_gemi_CryptoAssetsReceivedInKind" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Assets, Received In Kind</link:label>
    <link:label id="lab_gemi_CryptoAssetsReceivedInKind_documentation_en-US" xlink:label="lab_gemi_CryptoAssetsReceivedInKind" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Assets, Received In Kind</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetsReceivedInKind" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetsReceivedInKind"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetsReceivedInKind" xlink:to="lab_gemi_CryptoAssetsReceivedInKind" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain_822ffb3b-f233-4aaa-8bc9-95ef7203b869_terseLabel_en-US" xlink:label="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement of Financial Position Location, Activity, Accrual [Domain]</link:label>
    <link:label id="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain_label_en-US" xlink:label="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Financial Position Location, Activity, Accrual [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain" xlink:to="lab_us-gaap_StatementOfFinancialPositionLocationActivityAccrualDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A1275BTCJuly242025Member_34a4d1f5-d854-4aad-bbed-5660d7c173fc_terseLabel_en-US" xlink:label="lab_gemi_A1275BTCJuly242025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">1, 275 BTC, July 24, 2025</link:label>
    <link:label id="lab_gemi_A1275BTCJuly242025Member_label_en-US" xlink:label="lab_gemi_A1275BTCJuly242025Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">1, 275 BTC, July 24, 2025 [Member]</link:label>
    <link:label id="lab_gemi_A1275BTCJuly242025Member_documentation_en-US" xlink:label="lab_gemi_A1275BTCJuly242025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">1, 275 BTC, July 24, 2025</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A1275BTCJuly242025Member" xlink:href="gemi-20250930.xsd#gemi_A1275BTCJuly242025Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A1275BTCJuly242025Member" xlink:to="lab_gemi_A1275BTCJuly242025Member" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PvpTableTextBlock_terseLabel_en-US" xlink:label="lab_ecd_PvpTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pay vs Performance Disclosure, Table</link:label>
    <link:label id="lab_ecd_PvpTableTextBlock_label_en-US" xlink:label="lab_ecd_PvpTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pay vs Performance [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PvpTableTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PvpTableTextBlock"/>
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    <link:label id="lab_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Forgone Recovery due to Violation of Home Country Law, Amount</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt"/>
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    <link:label id="lab_dei_EntityTaxIdentificationNumber_ab51b663-8044-4915-bb21-1dc89ee88375_terseLabel_en-US" xlink:label="lab_dei_EntityTaxIdentificationNumber" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Tax Identification Number</link:label>
    <link:label id="lab_dei_EntityTaxIdentificationNumber_label_en-US" xlink:label="lab_dei_EntityTaxIdentificationNumber" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Tax Identification Number</link:label>
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    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_ab6aa10c-1ae4-4858-b7cd-484912e3f77b_periodStartLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Beginning balance (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_14108254-df4b-4b13-96ee-ba89b42445b7_periodEndLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Ending balance (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Exercise Price</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePrice"/>
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    <link:label id="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable_422ed45f-a18e-4ed2-8b75-436b075abdbe_terseLabel_en-US" xlink:label="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Organization, Consolidation And Presentation Of Financial Statements [Table]</link:label>
    <link:label id="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable_label_en-US" xlink:label="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Organization, Consolidation And Presentation Of Financial Statements [Table]</link:label>
    <link:label id="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable_documentation_en-US" xlink:label="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Organization, Consolidation And Presentation Of Financial Statements</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable" xlink:href="gemi-20250930.xsd#gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable" xlink:to="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsTable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_FoundersMember_594136f4-cae1-4804-93f2-1d9ed6ebdf08_terseLabel_en-US" xlink:label="lab_gemi_FoundersMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Founders</link:label>
    <link:label id="lab_gemi_FoundersMember_label_en-US" xlink:label="lab_gemi_FoundersMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Founders [Member]</link:label>
    <link:label id="lab_gemi_FoundersMember_documentation_en-US" xlink:label="lab_gemi_FoundersMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Founders</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_FoundersMember" xlink:href="gemi-20250930.xsd#gemi_FoundersMember"/>
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    <link:label id="lab_us-gaap_CommonStockSharesIssued_3b8e744b-e30c-49d8-beab-4ff42139dc24_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockSharesIssued" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common stock, shares issued (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonStockSharesIssued_label_en-US" xlink:label="lab_us-gaap_CommonStockSharesIssued" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Shares, Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockSharesIssued" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockSharesIssued"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockSharesIssued" xlink:to="lab_us-gaap_CommonStockSharesIssued" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_SeptemberNoteUSDMember_7a1860bf-0f90-4bbe-beb4-b468942b8f9a_terseLabel_en-US" xlink:label="lab_gemi_SeptemberNoteUSDMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">September Note (USD)</link:label>
    <link:label id="lab_gemi_SeptemberNoteUSDMember_label_en-US" xlink:label="lab_gemi_SeptemberNoteUSDMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">September Note (USD) [Member]</link:label>
    <link:label id="lab_gemi_SeptemberNoteUSDMember_documentation_en-US" xlink:label="lab_gemi_SeptemberNoteUSDMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">September Note (USD)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_SeptemberNoteUSDMember" xlink:href="gemi-20250930.xsd#gemi_SeptemberNoteUSDMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_SeptemberNoteUSDMember" xlink:to="lab_gemi_SeptemberNoteUSDMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1_dc55cbd2-2b96-489e-88b3-2e347a92b321_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Weighted-average remaining lease term (in years)</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Weighted Average Remaining Lease Term</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1"/>
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    <link:label id="lab_ecd_TradingArrAxis_terseLabel_en-US" xlink:label="lab_ecd_TradingArrAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Trading Arrangement:</link:label>
    <link:label id="lab_ecd_TradingArrAxis_label_en-US" xlink:label="lab_ecd_TradingArrAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TradingArrAxis" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TradingArrAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TradingArrAxis" xlink:to="lab_ecd_TradingArrAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativeAssetsCurrent_27dac9e7-6b09-4fbd-97f5-afbbd15d2db8_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeAssetsCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative assets</link:label>
    <link:label id="lab_us-gaap_DerivativeAssetsCurrent_label_en-US" xlink:label="lab_us-gaap_DerivativeAssetsCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Asset, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeAssetsCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeAssetsCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeAssetsCurrent" xlink:to="lab_us-gaap_DerivativeAssetsCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TotalShareholderRtnAmt_terseLabel_en-US" xlink:label="lab_ecd_TotalShareholderRtnAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total Shareholder Return Amount</link:label>
    <link:label id="lab_ecd_TotalShareholderRtnAmt_label_en-US" xlink:label="lab_ecd_TotalShareholderRtnAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Total Shareholder Return Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TotalShareholderRtnAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TotalShareholderRtnAmt"/>
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    <link:label id="lab_us-gaap_LossContingenciesLineItems_9d350e5d-0b30-4487-a1ac-e32cc8f0b167_terseLabel_en-US" xlink:label="lab_us-gaap_LossContingenciesLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loss Contingencies [Line Items]</link:label>
    <link:label id="lab_us-gaap_LossContingenciesLineItems_label_en-US" xlink:label="lab_us-gaap_LossContingenciesLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loss Contingencies [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LossContingenciesLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LossContingenciesLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LossContingenciesLineItems" xlink:to="lab_us-gaap_LossContingenciesLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityGainLossIncludedInEarnings_fbd684de-11bd-49e4-8cd5-7eca12d5ac6c_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityGainLossIncludedInEarnings" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Unrealized loss attributable to change in fair value</link:label>
    <link:label id="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityGainLossIncludedInEarnings_label_en-US" xlink:label="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityGainLossIncludedInEarnings" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Liability, Gain (Loss) Included in Earnings</link:label>
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    <link:label id="lab_dei_SecurityExchangeName_7c7acf5d-eba1-4b37-b9b2-93bf89d75bba_terseLabel_en-US" xlink:label="lab_dei_SecurityExchangeName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Security Exchange Name</link:label>
    <link:label id="lab_dei_SecurityExchangeName_label_en-US" xlink:label="lab_dei_SecurityExchangeName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Security Exchange Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_SecurityExchangeName" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_SecurityExchangeName"/>
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    <link:label id="lab_gemi_OfficeSpaceInLondonUnitedKingdomMember_dc1f5a21-50c1-4d15-92e8-6f01b4ce1cc3_terseLabel_en-US" xlink:label="lab_gemi_OfficeSpaceInLondonUnitedKingdomMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Office Space In London, United Kingdom</link:label>
    <link:label id="lab_gemi_OfficeSpaceInLondonUnitedKingdomMember_label_en-US" xlink:label="lab_gemi_OfficeSpaceInLondonUnitedKingdomMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Office Space In London, United Kingdom [Member]</link:label>
    <link:label id="lab_gemi_OfficeSpaceInLondonUnitedKingdomMember_documentation_en-US" xlink:label="lab_gemi_OfficeSpaceInLondonUnitedKingdomMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Office Space In London, United Kingdom</link:label>
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    <link:label id="lab_us-gaap_LiabilitiesNoncurrent_1fc39048-4091-495e-93fb-161d03101daf_totalLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesNoncurrent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total non-current liabilities</link:label>
    <link:label id="lab_us-gaap_LiabilitiesNoncurrent_label_en-US" xlink:label="lab_us-gaap_LiabilitiesNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liabilities, Noncurrent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesNoncurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesNoncurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilitiesNoncurrent" xlink:to="lab_us-gaap_LiabilitiesNoncurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A5000BTCDecember292022Member_ab001882-2411-4541-af77-34fbe6bd7953_terseLabel_en-US" xlink:label="lab_gemi_A5000BTCDecember292022Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">5,000 BTC December 29, 2022</link:label>
    <link:label id="lab_gemi_A5000BTCDecember292022Member_label_en-US" xlink:label="lab_gemi_A5000BTCDecember292022Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">5,000 BTC December 29, 2022 [Member]</link:label>
    <link:label id="lab_gemi_A5000BTCDecember292022Member_documentation_en-US" xlink:label="lab_gemi_A5000BTCDecember292022Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">5,000 BTC December 29, 2022</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A5000BTCDecember292022Member" xlink:href="gemi-20250930.xsd#gemi_A5000BTCDecember292022Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A5000BTCDecember292022Member" xlink:to="lab_gemi_A5000BTCDecember292022Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity_716d69bc-5f8c-402c-b788-299221cf324d_terseLabel_en-US" xlink:label="lab_us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Line of credit facility, maximum borrowing capacity</link:label>
    <link:label id="lab_us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity_label_en-US" xlink:label="lab_us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Line of Credit Facility, Maximum Borrowing Capacity</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity" xlink:to="lab_us-gaap_LineOfCreditFacilityMaximumBorrowingCapacity" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CatchUpCommonUnitsMember_8cd53473-89af-41aa-9c5b-3891a38c7b0f_terseLabel_en-US" xlink:label="lab_gemi_CatchUpCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Catch Up Common Units</link:label>
    <link:label id="lab_gemi_CatchUpCommonUnitsMember_label_en-US" xlink:label="lab_gemi_CatchUpCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Catch Up Common Units [Member]</link:label>
    <link:label id="lab_gemi_CatchUpCommonUnitsMember_documentation_en-US" xlink:label="lab_gemi_CatchUpCommonUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Catch Up Common Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CatchUpCommonUnitsMember" xlink:href="gemi-20250930.xsd#gemi_CatchUpCommonUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CatchUpCommonUnitsMember" xlink:to="lab_gemi_CatchUpCommonUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ProceedsFromIssuanceInitialPublicOffering_8ea7ff75-f1ff-48fb-9ca8-d21bcf3dd5fc_terseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromIssuanceInitialPublicOffering" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from issuance of common stock in connection with initial public offering, net of underwriting discounts and commissions and offering costs</link:label>
    <link:label id="lab_us-gaap_ProceedsFromIssuanceInitialPublicOffering_label_en-US" xlink:label="lab_us-gaap_ProceedsFromIssuanceInitialPublicOffering" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds from Issuance Initial Public Offering</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromIssuanceInitialPublicOffering" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromIssuanceInitialPublicOffering"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ProceedsFromIssuanceInitialPublicOffering" xlink:to="lab_us-gaap_ProceedsFromIssuanceInitialPublicOffering" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PrepaidExpenseCurrent_ab9408b4-d2e2-443b-9d72-eb1cb3d1c72b_terseLabel_en-US" xlink:label="lab_us-gaap_PrepaidExpenseCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Prepaid expenses</link:label>
    <link:label id="lab_us-gaap_PrepaidExpenseCurrent_label_en-US" xlink:label="lab_us-gaap_PrepaidExpenseCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Prepaid Expense, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PrepaidExpenseCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PrepaidExpenseCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PrepaidExpenseCurrent" xlink:to="lab_us-gaap_PrepaidExpenseCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingExpensesAbstract_701a8cd5-9e07-4696-a9de-fea3aa62abbd_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingExpensesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating expenses:</link:label>
    <link:label id="lab_us-gaap_OperatingExpensesAbstract_label_en-US" xlink:label="lab_us-gaap_OperatingExpensesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Expenses [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingExpensesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingExpensesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingExpensesAbstract" xlink:to="lab_us-gaap_OperatingExpensesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherLiabilitiesCurrent_4a85a18b-93a5-43d5-bf35-2154316db56a_terseLabel_en-US" xlink:label="lab_us-gaap_OtherLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other current liabilities</link:label>
    <link:label id="lab_us-gaap_OtherLiabilitiesCurrent_b034ba09-917a-45c3-bd8e-7eaee00e137a_totalLabel_en-US" xlink:label="lab_us-gaap_OtherLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total other current liabilities</link:label>
    <link:label id="lab_us-gaap_OtherLiabilitiesCurrent_label_en-US" xlink:label="lab_us-gaap_OtherLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Liabilities, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherLiabilitiesCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherLiabilitiesCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherLiabilitiesCurrent" xlink:to="lab_us-gaap_OtherLiabilitiesCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PhantomShareUnitsPSUsMember_a3a14fef-d368-4911-91da-c642dfcf07d4_terseLabel_en-US" xlink:label="lab_us-gaap_PhantomShareUnitsPSUsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Phantom Units</link:label>
    <link:label id="lab_us-gaap_PhantomShareUnitsPSUsMember_label_en-US" xlink:label="lab_us-gaap_PhantomShareUnitsPSUsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Phantom Share Units (PSUs) [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PhantomShareUnitsPSUsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PhantomShareUnitsPSUsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PhantomShareUnitsPSUsMember" xlink:to="lab_us-gaap_PhantomShareUnitsPSUsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherSundryLiabilitiesCurrent_dae3ca81-a9e7-4aa5-b3d9-6e33b069a3db_terseLabel_en-US" xlink:label="lab_us-gaap_OtherSundryLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other current liabilities</link:label>
    <link:label id="lab_us-gaap_OtherSundryLiabilitiesCurrent_label_en-US" xlink:label="lab_us-gaap_OtherSundryLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Sundry Liabilities, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherSundryLiabilitiesCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherSundryLiabilitiesCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherSundryLiabilitiesCurrent" xlink:to="lab_us-gaap_OtherSundryLiabilitiesCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInTemporaryEquityRollForward_eccbf447-5757-4747-a707-ec9f83714594_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Temporary Equity [Roll Forward]</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInTemporaryEquityRollForward_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Temporary Equity [Roll Forward]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInTemporaryEquityRollForward"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xlink:to="lab_us-gaap_IncreaseDecreaseInTemporaryEquityRollForward" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementLineItems_f47151be-b2ca-467f-8923-d73f531ca3af_terseLabel_en-US" xlink:label="lab_us-gaap_StatementLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement [Line Items]</link:label>
    <link:label id="lab_us-gaap_StatementLineItems_label_en-US" xlink:label="lab_us-gaap_StatementLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementLineItems" xlink:to="lab_us-gaap_StatementLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_Rule10b51ArrAdoptedFlag_terseLabel_en-US" xlink:label="lab_ecd_Rule10b51ArrAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Rule 10b5-1 Arrangement Adopted</link:label>
    <link:label id="lab_ecd_Rule10b51ArrAdoptedFlag_label_en-US" xlink:label="lab_ecd_Rule10b51ArrAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Rule 10b5-1 Arrangement Adopted [Flag]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_Rule10b51ArrAdoptedFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_Rule10b51ArrAdoptedFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_Rule10b51ArrAdoptedFlag" xlink:to="lab_ecd_Rule10b51ArrAdoptedFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue_7fcb8452-594c-4838-a30b-a046505b4976_terseLabel_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue_label_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash and Cash Equivalent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashAndCashEquivalentsAtCarryingValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:to="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ConvertiblePreferredUnitsMember_483e3280-dd6e-44f1-9fa7-6453315e4d32_verboseLabel_en-US" xlink:label="lab_gemi_ConvertiblePreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Conversion of convertible preferred units to common stock in connection with IPO</link:label>
    <link:label id="lab_gemi_ConvertiblePreferredUnitsMember_ce3d56b3-9520-4d35-a714-6181d2a5f5f3_terseLabel_en-US" xlink:label="lab_gemi_ConvertiblePreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible Preferred Units</link:label>
    <link:label id="lab_gemi_ConvertiblePreferredUnitsMember_label_en-US" xlink:label="lab_gemi_ConvertiblePreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Convertible Preferred Units [Member]</link:label>
    <link:label id="lab_gemi_ConvertiblePreferredUnitsMember_documentation_en-US" xlink:label="lab_gemi_ConvertiblePreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Convertible Preferred Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ConvertiblePreferredUnitsMember" xlink:href="gemi-20250930.xsd#gemi_ConvertiblePreferredUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ConvertiblePreferredUnitsMember" xlink:to="lab_gemi_ConvertiblePreferredUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SubsequentEventLineItems_5389f564-f13c-4292-ac22-f0c6afc6249c_terseLabel_en-US" xlink:label="lab_us-gaap_SubsequentEventLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Subsequent Event [Line Items]</link:label>
    <link:label id="lab_us-gaap_SubsequentEventLineItems_label_en-US" xlink:label="lab_us-gaap_SubsequentEventLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Subsequent Event [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubsequentEventLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubsequentEventLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SubsequentEventLineItems" xlink:to="lab_us-gaap_SubsequentEventLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A30000ETHDecember292022Member_a032a7ee-826a-482b-9707-ff7eab956e4a_terseLabel_en-US" xlink:label="lab_gemi_A30000ETHDecember292022Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">30,000 ETH, December 29, 2022</link:label>
    <link:label id="lab_gemi_A30000ETHDecember292022Member_label_en-US" xlink:label="lab_gemi_A30000ETHDecember292022Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">30,000 ETH, December 29, 2022 [Member]</link:label>
    <link:label id="lab_gemi_A30000ETHDecember292022Member_documentation_en-US" xlink:label="lab_gemi_A30000ETHDecember292022Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">30,000 ETH, December 29, 2022</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A30000ETHDecember292022Member" xlink:href="gemi-20250930.xsd#gemi_A30000ETHDecember292022Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A30000ETHDecember292022Member" xlink:to="lab_gemi_A30000ETHDecember292022Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonStockMember_551c5955-5a4b-4e67-ad29-64b5a55e76b1_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common Stock</link:label>
    <link:label id="lab_us-gaap_CommonStockMember_label_en-US" xlink:label="lab_us-gaap_CommonStockMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockMember" xlink:to="lab_us-gaap_CommonStockMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod_490754bf-bc59-4043-b271-268e3207f95b_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Vested (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Vested in Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_SeriesA6PreferredUnitsMember_d99d1e13-3d99-4246-9d8e-f8f04435cc92_terseLabel_en-US" xlink:label="lab_gemi_SeriesA6PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Series A-6</link:label>
    <link:label id="lab_gemi_SeriesA6PreferredUnitsMember_label_en-US" xlink:label="lab_gemi_SeriesA6PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Series A-6 Preferred Units [Member]</link:label>
    <link:label id="lab_gemi_SeriesA6PreferredUnitsMember_documentation_en-US" xlink:label="lab_gemi_SeriesA6PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Series A-6 Preferred Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_SeriesA6PreferredUnitsMember" xlink:href="gemi-20250930.xsd#gemi_SeriesA6PreferredUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_SeriesA6PreferredUnitsMember" xlink:to="lab_gemi_SeriesA6PreferredUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RepurchaseAgreementsInterestRate_1dd440b9-fac0-471d-acf4-c4b91d83c44c_terseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsInterestRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest Rate</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsInterestRate_137b0d9e-5e23-4e4a-b9ab-7fd18ba15f42_verboseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsInterestRate" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Interest Rate</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsInterestRate_label_en-US" xlink:label="lab_gemi_RepurchaseAgreementsInterestRate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Interest Rate</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsInterestRate_documentation_en-US" xlink:label="lab_gemi_RepurchaseAgreementsInterestRate" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Interest Rate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RepurchaseAgreementsInterestRate" xlink:href="gemi-20250930.xsd#gemi_RepurchaseAgreementsInterestRate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RepurchaseAgreementsInterestRate" xlink:to="lab_gemi_RepurchaseAgreementsInterestRate" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar_a2d4a252-40e1-4a72-a098-a8f5d125edd0_terseLabel_en-US" xlink:label="lab_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Customer custodial funds due to customers related to GUSD</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar_label_en-US" xlink:label="lab_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds Due To Customers, Gemini Dollar</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar_documentation_en-US" xlink:label="lab_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds Due To Customers, Gemini Dollar</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar" xlink:href="gemi-20250930.xsd#gemi_CustomerCustodialFundsDueToCustomersGeminiDollar"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar" xlink:to="lab_gemi_CustomerCustodialFundsDueToCustomersGeminiDollar" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SignificantAccountingPoliciesTextBlock_1d4f7287-3bb4-42d7-97e0-f4d98f7e5f19_terseLabel_en-US" xlink:label="lab_us-gaap_SignificantAccountingPoliciesTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Summary of Significant Accounting Policies</link:label>
    <link:label id="lab_us-gaap_SignificantAccountingPoliciesTextBlock_label_en-US" xlink:label="lab_us-gaap_SignificantAccountingPoliciesTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Significant Accounting Policies [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SignificantAccountingPoliciesTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SignificantAccountingPoliciesTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SignificantAccountingPoliciesTextBlock" xlink:to="lab_us-gaap_SignificantAccountingPoliciesTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_RangeMember_63f667e1-4bf2-477b-87d3-20a0f66d546e_terseLabel_en-US" xlink:label="lab_srt_RangeMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statistical Measurement [Domain]</link:label>
    <link:label id="lab_srt_RangeMember_label_en-US" xlink:label="lab_srt_RangeMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statistical Measurement [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_RangeMember" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_RangeMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_RangeMember" xlink:to="lab_srt_RangeMember" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_CryptoAssetDomain_d76a740e-15b0-4eee-aea9-88fd7f8ccb71_terseLabel_en-US" xlink:label="lab_srt_CryptoAssetDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto Asset [Domain]</link:label>
    <link:label id="lab_srt_CryptoAssetDomain_label_en-US" xlink:label="lab_srt_CryptoAssetDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_CryptoAssetDomain" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_CryptoAssetDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_CryptoAssetDomain" xlink:to="lab_srt_CryptoAssetDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ContractWithCustomerAssetVested_83f227df-ddf5-4936-9981-915d1f21d2e8_terseLabel_en-US" xlink:label="lab_gemi_ContractWithCustomerAssetVested" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Contract asset vested</link:label>
    <link:label id="lab_gemi_ContractWithCustomerAssetVested_label_en-US" xlink:label="lab_gemi_ContractWithCustomerAssetVested" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Contract With Customer, Asset, Vested</link:label>
    <link:label id="lab_gemi_ContractWithCustomerAssetVested_documentation_en-US" xlink:label="lab_gemi_ContractWithCustomerAssetVested" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Contract With Customer, Asset, Vested</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ContractWithCustomerAssetVested" xlink:href="gemi-20250930.xsd#gemi_ContractWithCustomerAssetVested"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ContractWithCustomerAssetVested" xlink:to="lab_gemi_ContractWithCustomerAssetVested" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RestrictedCashCurrent_8447ab30-ce90-49ba-9cbd-15656ce1bac0_terseLabel_en-US" xlink:label="lab_us-gaap_RestrictedCashCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_RestrictedCashCurrent_label_en-US" xlink:label="lab_us-gaap_RestrictedCashCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Cash, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RestrictedCashCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RestrictedCashCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RestrictedCashCurrent" xlink:to="lab_us-gaap_RestrictedCashCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear_73414a00-72e0-4927-94a9-e8e4db5b0bfc_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Remaining 2025</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Remainder of Fiscal Year</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoLegalSettlementContributions_838a1e92-4381-4811-8128-2f07257c57f1_terseLabel_en-US" xlink:label="lab_gemi_CryptoLegalSettlementContributions" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Bitcoin contributed in compliance with Consent Order</link:label>
    <link:label id="lab_gemi_CryptoLegalSettlementContributions_label_en-US" xlink:label="lab_gemi_CryptoLegalSettlementContributions" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto, Legal Settlement, Contributions</link:label>
    <link:label id="lab_gemi_CryptoLegalSettlementContributions_documentation_en-US" xlink:label="lab_gemi_CryptoLegalSettlementContributions" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto, Legal Settlement, Contributions</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoLegalSettlementContributions" xlink:href="gemi-20250930.xsd#gemi_CryptoLegalSettlementContributions"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoLegalSettlementContributions" xlink:to="lab_gemi_CryptoLegalSettlementContributions" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LeasesAbstract_label_en-US" xlink:label="lab_us-gaap_LeasesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Leases [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LeasesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LeasesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LeasesAbstract" xlink:to="lab_us-gaap_LeasesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EarningsPerShareBasicAbstract_1a23cbca-13a9-4c25-87f3-9141d17fe837_terseLabel_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasicAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net loss per share attributable to common stockholders:</link:label>
    <link:label id="lab_us-gaap_EarningsPerShareBasicAbstract_label_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasicAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Basic [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EarningsPerShareBasicAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EarningsPerShareBasicAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EarningsPerShareBasicAbstract" xlink:to="lab_us-gaap_EarningsPerShareBasicAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetFairValueCurrent_e8bc0393-7e8c-4afb-831d-3c6f2f0a56c7_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValueCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets held</link:label>
    <link:label id="lab_us-gaap_CryptoAssetFairValueCurrent_label_en-US" xlink:label="lab_us-gaap_CryptoAssetFairValueCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Fair Value, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetFairValueCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetFairValueCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetFairValueCurrent" xlink:to="lab_us-gaap_CryptoAssetFairValueCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetPaymentsForExpenses_e9fe416b-537f-4581-9f2e-64d388288013_negatedTerseLabel_en-US" xlink:label="lab_gemi_CryptoAssetPaymentsForExpenses" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset payments for expenses</link:label>
    <link:label id="lab_gemi_CryptoAssetPaymentsForExpenses_label_en-US" xlink:label="lab_gemi_CryptoAssetPaymentsForExpenses" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Payments For Expenses</link:label>
    <link:label id="lab_gemi_CryptoAssetPaymentsForExpenses_documentation_en-US" xlink:label="lab_gemi_CryptoAssetPaymentsForExpenses" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Payments For Expenses</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetPaymentsForExpenses" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetPaymentsForExpenses"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetPaymentsForExpenses" xlink:to="lab_gemi_CryptoAssetPaymentsForExpenses" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RelatedPartyDebtFairValueDisclosure_2299141a-ebc1-473d-a34b-264583b7750a_terseLabel_en-US" xlink:label="lab_gemi_RelatedPartyDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party loans:</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtFairValueDisclosure_label_en-US" xlink:label="lab_gemi_RelatedPartyDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Debt, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_RelatedPartyDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Debt, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RelatedPartyDebtFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_RelatedPartyDebtFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RelatedPartyDebtFairValueDisclosure" xlink:to="lab_gemi_RelatedPartyDebtFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_WarrantLiabilityMember_2e1fbf58-e687-47a7-b69c-4ebbe08c6292_terseLabel_en-US" xlink:label="lab_gemi_WarrantLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Warrant Liability</link:label>
    <link:label id="lab_gemi_WarrantLiabilityMember_label_en-US" xlink:label="lab_gemi_WarrantLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Warrant Liability [Member]</link:label>
    <link:label id="lab_gemi_WarrantLiabilityMember_documentation_en-US" xlink:label="lab_gemi_WarrantLiabilityMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Warrant Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_WarrantLiabilityMember" xlink:href="gemi-20250930.xsd#gemi_WarrantLiabilityMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_WarrantLiabilityMember" xlink:to="lab_gemi_WarrantLiabilityMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation_be590ec9-db2e-4826-a753-dbc4fcbfaa6f_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Shares withheld for employee tax withholdings</link:label>
    <link:label id="lab_us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation_label_en-US" xlink:label="lab_us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Decrease for Tax Withholding Obligation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation" xlink:to="lab_us-gaap_AdjustmentsRelatedToTaxWithholdingForShareBasedCompensation" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ComputerAndOtherEquipmentMember_a4d1ce22-a2d2-471a-8d6c-3792611e480b_terseLabel_en-US" xlink:label="lab_gemi_ComputerAndOtherEquipmentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Computers and other equipment</link:label>
    <link:label id="lab_gemi_ComputerAndOtherEquipmentMember_label_en-US" xlink:label="lab_gemi_ComputerAndOtherEquipmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Computer And Other Equipment [Member]</link:label>
    <link:label id="lab_gemi_ComputerAndOtherEquipmentMember_documentation_en-US" xlink:label="lab_gemi_ComputerAndOtherEquipmentMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Computer And Other Equipment</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ComputerAndOtherEquipmentMember" xlink:href="gemi-20250930.xsd#gemi_ComputerAndOtherEquipmentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ComputerAndOtherEquipmentMember" xlink:to="lab_gemi_ComputerAndOtherEquipmentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TradingArrByIndTable_terseLabel_en-US" xlink:label="lab_ecd_TradingArrByIndTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Trading Arrangements, by Individual</link:label>
    <link:label id="lab_ecd_TradingArrByIndTable_label_en-US" xlink:label="lab_ecd_TradingArrByIndTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangements, by Individual [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TradingArrByIndTable" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TradingArrByIndTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TradingArrByIndTable" xlink:to="lab_ecd_TradingArrByIndTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EarningsPerShareBasic_004e9cf5-a93c-4ab2-b31d-c150f06ad038_terseLabel_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasic" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Basic (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_EarningsPerShareBasic_label_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasic" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Basic</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EarningsPerShareBasic" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EarningsPerShareBasic"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EarningsPerShareBasic" xlink:to="lab_us-gaap_EarningsPerShareBasic" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities_093689e1-81e4-4399-89b7-ff49d6aa9db0_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounts payable and accrued expenses</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Accounts Payable and Accrued Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities" xlink:to="lab_us-gaap_IncreaseDecreaseInAccountsPayableAndAccruedLiabilities" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LiabilitiesFairValueDisclosureAbstract_854af77d-da48-4131-bbf6-d74cbafa25a8_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesFairValueDisclosureAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Liabilities</link:label>
    <link:label id="lab_us-gaap_LiabilitiesFairValueDisclosureAbstract_9e1ae491-f8ad-4601-89a4-7b617e59996d_verboseLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesFairValueDisclosureAbstract" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Financial Liabilities:</link:label>
    <link:label id="lab_us-gaap_LiabilitiesFairValueDisclosureAbstract_label_en-US" xlink:label="lab_us-gaap_LiabilitiesFairValueDisclosureAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liabilities, Fair Value Disclosure [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesFairValueDisclosureAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesFairValueDisclosureAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilitiesFairValueDisclosureAbstract" xlink:to="lab_us-gaap_LiabilitiesFairValueDisclosureAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock_0fdb4751-19f7-4165-b1ec-b17f4b57530a_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Derivative Assets at Fair Value</link:label>
    <link:label id="lab_us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Derivative Assets at Fair Value [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock" xlink:to="lab_us-gaap_ScheduleOfDerivativeAssetsAtFairValueTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_7eba1dbd-9421-49f1-8e2a-fd9da87538b3_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Share available for future issuance (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonStockCapitalSharesReservedForFutureIssuance_label_en-US" xlink:label="lab_us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Capital Shares Reserved for Future Issuance</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockCapitalSharesReservedForFutureIssuance"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xlink:to="lab_us-gaap_CommonStockCapitalSharesReservedForFutureIssuance" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RepurchaseAgreementsInterestPayable_60000669-3138-41dc-97b3-37ee0dc13b2c_terseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsInterestPayable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest Payable</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsInterestPayable_label_en-US" xlink:label="lab_gemi_RepurchaseAgreementsInterestPayable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Interest Payable</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsInterestPayable_documentation_en-US" xlink:label="lab_gemi_RepurchaseAgreementsInterestPayable" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Interest Payable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RepurchaseAgreementsInterestPayable" xlink:href="gemi-20250930.xsd#gemi_RepurchaseAgreementsInterestPayable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RepurchaseAgreementsInterestPayable" xlink:to="lab_gemi_RepurchaseAgreementsInterestPayable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetPurchaseNumberOfUnits_57a532c0-2c98-4ec2-b01f-45ef3f9985d8_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetPurchaseNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of crypto assets acquired (in units)</link:label>
    <link:label id="lab_gemi_CryptoAssetPurchaseNumberOfUnits_label_en-US" xlink:label="lab_gemi_CryptoAssetPurchaseNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Purchase, Number Of Units</link:label>
    <link:label id="lab_gemi_CryptoAssetPurchaseNumberOfUnits_documentation_en-US" xlink:label="lab_gemi_CryptoAssetPurchaseNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset, Purchase, Number Of Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetPurchaseNumberOfUnits" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetPurchaseNumberOfUnits"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetPurchaseNumberOfUnits" xlink:to="lab_gemi_CryptoAssetPurchaseNumberOfUnits" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_MeasurementInputDiscountRateMember_2737f9a9-cda6-4a3b-aa78-1b0ee3732c9a_terseLabel_en-US" xlink:label="lab_us-gaap_MeasurementInputDiscountRateMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Measurement Input, Discount Rate</link:label>
    <link:label id="lab_us-gaap_MeasurementInputDiscountRateMember_label_en-US" xlink:label="lab_us-gaap_MeasurementInputDiscountRateMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measurement Input, Discount Rate [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_MeasurementInputDiscountRateMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_MeasurementInputDiscountRateMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_MeasurementInputDiscountRateMember" xlink:to="lab_us-gaap_MeasurementInputDiscountRateMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_InterestExpenseThirdPartyDebt_b12660d7-c8cb-4a1c-a2c3-1ae27711c2ce_negatedTerseLabel_en-US" xlink:label="lab_gemi_InterestExpenseThirdPartyDebt" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Interest expense on third party loans</link:label>
    <link:label id="lab_gemi_InterestExpenseThirdPartyDebt_label_en-US" xlink:label="lab_gemi_InterestExpenseThirdPartyDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Expense, Third Party Debt</link:label>
    <link:label id="lab_gemi_InterestExpenseThirdPartyDebt_documentation_en-US" xlink:label="lab_gemi_InterestExpenseThirdPartyDebt" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Interest Expense, Debt, Third Party Loans</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_InterestExpenseThirdPartyDebt" xlink:href="gemi-20250930.xsd#gemi_InterestExpenseThirdPartyDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_InterestExpenseThirdPartyDebt" xlink:to="lab_gemi_InterestExpenseThirdPartyDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShortTermBorrowings_e43c18b4-f3de-4cfd-9eaf-1befcdec437c_periodStartLabel_en-US" xlink:label="lab_us-gaap_ShortTermBorrowings" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Principal Outstanding as of 12.31.24</link:label>
    <link:label id="lab_us-gaap_ShortTermBorrowings_612fc3d4-f775-48e0-8082-0cd9c804cdfe_periodEndLabel_en-US" xlink:label="lab_us-gaap_ShortTermBorrowings" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Principal Outstanding as of 9.30.25</link:label>
    <link:label id="lab_us-gaap_ShortTermBorrowings_label_en-US" xlink:label="lab_us-gaap_ShortTermBorrowings" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Short-Term Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShortTermBorrowings" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShortTermBorrowings"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShortTermBorrowings" xlink:to="lab_us-gaap_ShortTermBorrowings" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock_88c23dfb-201d-4a28-88da-977ce2802be9_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Derivative Liabilities at Fair Value</link:label>
    <link:label id="lab_us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Derivative Liabilities at Fair Value [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock" xlink:to="lab_us-gaap_ScheduleOfDerivativeLiabilitiesAtFairValueTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_MeasurementInputTypeDomain_24b9a61f-e02a-481b-a7da-8afe8f7d09f6_terseLabel_en-US" xlink:label="lab_us-gaap_MeasurementInputTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Measurement Input Type [Domain]</link:label>
    <link:label id="lab_us-gaap_MeasurementInputTypeDomain_label_en-US" xlink:label="lab_us-gaap_MeasurementInputTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measurement Input Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_MeasurementInputTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_MeasurementInputTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_MeasurementInputTypeDomain" xlink:to="lab_us-gaap_MeasurementInputTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetsPledgedAsCollateral_58c9a209-2119-46d4-98b6-0996fdf8c87e_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetsPledgedAsCollateral" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets pledged as collateral</link:label>
    <link:label id="lab_gemi_CryptoAssetsPledgedAsCollateral_label_en-US" xlink:label="lab_gemi_CryptoAssetsPledgedAsCollateral" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Assets Pledged As Collateral</link:label>
    <link:label id="lab_gemi_CryptoAssetsPledgedAsCollateral_documentation_en-US" xlink:label="lab_gemi_CryptoAssetsPledgedAsCollateral" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Assets Pledged As Collateral</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetsPledgedAsCollateral" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetsPledgedAsCollateral"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetsPledgedAsCollateral" xlink:to="lab_gemi_CryptoAssetsPledgedAsCollateral" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes_a7d2bddc-66ec-4a12-a852-d48e706e6087_terseLabel_en-US" xlink:label="lab_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-qualified financing event, maximum amount of proceeds received to trigger the option to convert the outstanding notes</link:label>
    <link:label id="lab_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes_label_en-US" xlink:label="lab_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Conversion, Non-Qualifying Financing Event, Maximum Amount Of Proceeds Received To Trigger Option To Convert Notes</link:label>
    <link:label id="lab_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes_documentation_en-US" xlink:label="lab_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Conversion, Non-Qualifying Financing Event, Option To Convert Notes, Percentage Of The Lowest Price Paid</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes" xlink:href="gemi-20250930.xsd#gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes" xlink:to="lab_gemi_DebtConversionNonQualifyingFinancingEventMaximumAmountOfProceedsReceivedToTriggerOptionToConvertNotes" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardTmgMnpiDiscTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AwardTmgMnpiDiscTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Timing MNPI Disclosure</link:label>
    <link:label id="lab_ecd_AwardTmgMnpiDiscTextBlock_label_en-US" xlink:label="lab_ecd_AwardTmgMnpiDiscTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Timing MNPI Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardTmgMnpiDiscTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardTmgMnpiDiscTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AwardTmgMnpiDiscTextBlock" xlink:to="lab_ecd_AwardTmgMnpiDiscTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis_3c22bc54-0ba6-4da4-997a-656ce3995cf9_terseLabel_en-US" xlink:label="lab_us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Category of Item Purchased [Axis]</link:label>
    <link:label id="lab_us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis_label_en-US" xlink:label="lab_us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Category of Item Purchased [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis" xlink:to="lab_us-gaap_LongTermPurchaseCommitmentByCategoryOfItemPurchasedAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_IncentiveUnitsMember_b8efd282-11e4-4f50-9dde-f1bfa761289f_terseLabel_en-US" xlink:label="lab_gemi_IncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Incentive Units</link:label>
    <link:label id="lab_gemi_IncentiveUnitsMember_label_en-US" xlink:label="lab_gemi_IncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Incentive Units [Member]</link:label>
    <link:label id="lab_gemi_IncentiveUnitsMember_documentation_en-US" xlink:label="lab_gemi_IncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Incentive Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_IncentiveUnitsMember" xlink:href="gemi-20250930.xsd#gemi_IncentiveUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_IncentiveUnitsMember" xlink:to="lab_gemi_IncentiveUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable_ecc0984c-d8e9-4bd7-b98e-86f368bf1aaf_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounts and Financing Receivables [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts and Financing Receivables [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable" xlink:to="lab_us-gaap_ScheduleOfAccountsNotesLoansAndFinancingReceivableTable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_SeriesA3PreferredUnitsMember_c15fdaee-133a-41c9-b8c5-e0ec9ae08361_terseLabel_en-US" xlink:label="lab_gemi_SeriesA3PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Series A-3</link:label>
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    <link:label id="lab_us-gaap_AllocatedShareBasedCompensationExpense_2202cd05-6901-466f-80e2-a1ffb9a020cc_terseLabel_en-US" xlink:label="lab_us-gaap_AllocatedShareBasedCompensationExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Share-based compensation expense</link:label>
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    <link:label id="lab_us-gaap_PatentsMember_1f68eaac-aaa1-4fbc-b3bc-033c811f30d0_terseLabel_en-US" xlink:label="lab_us-gaap_PatentsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Patents</link:label>
    <link:label id="lab_us-gaap_PatentsMember_label_en-US" xlink:label="lab_us-gaap_PatentsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Patents [Member]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesAndStockholdersEquity" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesAndStockholdersEquity"/>
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    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward_733566f9-d512-4a37-bfce-001127f1649a_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingRollForward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of Stock Options</link:label>
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    <link:label id="lab_gemi_TermLoan2025Member_18b57079-79a4-4486-a20c-1fef8db6b868_terseLabel_en-US" xlink:label="lab_gemi_TermLoan2025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Term Loan 2025</link:label>
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    <link:label id="lab_us-gaap_CreditCardMember_2ba6c14a-f743-4dd3-8cce-c67c07a7fe3c_terseLabel_en-US" xlink:label="lab_us-gaap_CreditCardMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit card revenue</link:label>
    <link:label id="lab_us-gaap_CreditCardMember_label_en-US" xlink:label="lab_us-gaap_CreditCardMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Card [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CreditCardMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CreditCardMember"/>
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    <link:label id="lab_gemi_DebtConversionNoteHolderOptionConversionPriceAsAPercentageOfLowestPricePaid_e26c84c0-b336-4579-9d6a-11987e41ba45_terseLabel_en-US" xlink:label="lab_gemi_DebtConversionNoteHolderOptionConversionPriceAsAPercentageOfLowestPricePaid" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Note holder option, price of conversion as a percentage of the lowest original issue price</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtConversionNoteHolderOptionConversionPriceAsAPercentageOfLowestPricePaid" xlink:to="lab_gemi_DebtConversionNoteHolderOptionConversionPriceAsAPercentageOfLowestPricePaid" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsLineItems_be0757d3-3052-4cb6-a7af-c352970917bd_terseLabel_en-US" xlink:label="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Organization, Consolidation And Presentation Of Financial Statements [Line Items]</link:label>
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    <link:label id="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsLineItems_documentation_en-US" xlink:label="lab_gemi_OrganizationConsolidationAndPresentationOfFinancialStatementsLineItems" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Organization, Consolidation And Presentation Of Financial Statements</link:label>
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    <link:label id="lab_gemi_ReceivableCryptoAssetPledgedRealizedGainOperating_f64942d4-0ffb-4e5d-a58b-4fe403d98b65_terseLabel_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetPledgedRealizedGainOperating" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Realized gain on receivable, crypto assets pledged</link:label>
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    <link:label id="lab_us-gaap_AccountingPoliciesAbstract_label_en-US" xlink:label="lab_us-gaap_AccountingPoliciesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounting Policies [Abstract]</link:label>
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    <link:label id="lab_us-gaap_PreferredUnitsPreferredPartnersCapitalAccounts_label_en-US" xlink:label="lab_us-gaap_PreferredUnitsPreferredPartnersCapitalAccounts" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Units, Preferred Partners' Capital Accounts</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredUnitsPreferredPartnersCapitalAccounts" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredUnitsPreferredPartnersCapitalAccounts"/>
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    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock_9e48156a-6f5f-4983-afca-94c82fb986c6_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Maturities of Operating Lease Liabilities</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Maturity [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityMaturityTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TreasuryStockCommonMember_6fe43d34-e6ba-43e1-be66-8c741ea85ace_terseLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Treasury Stock</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonMember_label_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Treasury Stock, Common [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TreasuryStockCommonMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TreasuryStockCommonMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TreasuryStockCommonMember" xlink:to="lab_us-gaap_TreasuryStockCommonMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PayVsPerformanceDisclosureLineItems_label_en-US" xlink:label="lab_ecd_PayVsPerformanceDisclosureLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pay vs Performance Disclosure [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PayVsPerformanceDisclosureLineItems" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PayVsPerformanceDisclosureLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PayVsPerformanceDisclosureLineItems" xlink:to="lab_ecd_PayVsPerformanceDisclosureLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ClassOfWarrantOrRightAxis_0cf1905a-7dc9-43c1-b045-8dfed5add5f4_terseLabel_en-US" xlink:label="lab_us-gaap_ClassOfWarrantOrRightAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Class of Warrant or Right [Axis]</link:label>
    <link:label id="lab_us-gaap_ClassOfWarrantOrRightAxis_label_en-US" xlink:label="lab_us-gaap_ClassOfWarrantOrRightAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class of Warrant or Right [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ClassOfWarrantOrRightAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ClassOfWarrantOrRightAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ClassOfWarrantOrRightAxis" xlink:to="lab_us-gaap_ClassOfWarrantOrRightAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseLiability_c6bfc691-ad66-4897-8b8c-1a14d110721d_netLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiability" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Lease liability</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseLiability_2d3a6c68-5843-43ae-96e9-96f13647d2ee_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiability" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total lease liabilities</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseLiability_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiability" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseLiability" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseLiability"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseLiability" xlink:to="lab_us-gaap_OperatingLeaseLiability" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent_0fdb1a53-329d-46fd-8184-ee4784a1b90c_terseLabel_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Foreign currency translation, net of tax</link:label>
    <link:label id="lab_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent_label_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Comprehensive Income (Loss), Foreign Currency Transaction and Translation Adjustment, Net of Tax, Portion Attributable to Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent" xlink:to="lab_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentNetOfTaxPortionAttributableToParent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_LesseeOperatingLeaseArea_f8af5b90-ce2e-4243-87e7-70d552b873bb_terseLabel_en-US" xlink:label="lab_gemi_LesseeOperatingLeaseArea" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Area of operating lease (in sq ft)</link:label>
    <link:label id="lab_gemi_LesseeOperatingLeaseArea_label_en-US" xlink:label="lab_gemi_LesseeOperatingLeaseArea" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Area</link:label>
    <link:label id="lab_gemi_LesseeOperatingLeaseArea_documentation_en-US" xlink:label="lab_gemi_LesseeOperatingLeaseArea" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Area</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_LesseeOperatingLeaseArea" xlink:href="gemi-20250930.xsd#gemi_LesseeOperatingLeaseArea"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_LesseeOperatingLeaseArea" xlink:to="lab_gemi_LesseeOperatingLeaseArea" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConvertibleDebtNoncurrent_7ab643a9-bcae-4f26-bf11-98e18c9b97f9_terseLabel_en-US" xlink:label="lab_us-gaap_ConvertibleDebtNoncurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party convertible notes</link:label>
    <link:label id="lab_us-gaap_ConvertibleDebtNoncurrent_label_en-US" xlink:label="lab_us-gaap_ConvertibleDebtNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Convertible Debt, Noncurrent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConvertibleDebtNoncurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConvertibleDebtNoncurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConvertibleDebtNoncurrent" xlink:to="lab_us-gaap_ConvertibleDebtNoncurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ProceedsFromCreditCardReceivables_2fc0a71d-81b5-4145-981a-c581f94c9d36_terseLabel_en-US" xlink:label="lab_gemi_ProceedsFromCreditCardReceivables" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from repayments of credit card receivables</link:label>
    <link:label id="lab_gemi_ProceedsFromCreditCardReceivables_label_en-US" xlink:label="lab_gemi_ProceedsFromCreditCardReceivables" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds From Credit Card Receivables</link:label>
    <link:label id="lab_gemi_ProceedsFromCreditCardReceivables_documentation_en-US" xlink:label="lab_gemi_ProceedsFromCreditCardReceivables" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Proceeds From Credit Card Receivables</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ProceedsFromCreditCardReceivables" xlink:href="gemi-20250930.xsd#gemi_ProceedsFromCreditCardReceivables"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ProceedsFromCreditCardReceivables" xlink:to="lab_gemi_ProceedsFromCreditCardReceivables" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementOfCashFlowsAbstract_label_en-US" xlink:label="lab_us-gaap_StatementOfCashFlowsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Cash Flows [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfCashFlowsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfCashFlowsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfCashFlowsAbstract" xlink:to="lab_us-gaap_StatementOfCashFlowsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CoSelectedMeasureAmt_terseLabel_en-US" xlink:label="lab_ecd_CoSelectedMeasureAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Company Selected Measure Amount</link:label>
    <link:label id="lab_ecd_CoSelectedMeasureAmt_label_en-US" xlink:label="lab_ecd_CoSelectedMeasureAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Company Selected Measure Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CoSelectedMeasureAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CoSelectedMeasureAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CoSelectedMeasureAmt" xlink:to="lab_ecd_CoSelectedMeasureAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_db161b15-e8f1-41a0-b805-151ae7e374b5_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2028</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year Three</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ComprehensiveIncomeNetOfTax_337a32c9-4232-4649-ae8b-bd4409b03f96_totalLabel_en-US" xlink:label="lab_us-gaap_ComprehensiveIncomeNetOfTax" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Comprehensive loss</link:label>
    <link:label id="lab_us-gaap_ComprehensiveIncomeNetOfTax_label_en-US" xlink:label="lab_us-gaap_ComprehensiveIncomeNetOfTax" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Comprehensive Income (Loss), Net of Tax, Attributable to Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ComprehensiveIncomeNetOfTax" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ComprehensiveIncomeNetOfTax"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ComprehensiveIncomeNetOfTax" xlink:to="lab_us-gaap_ComprehensiveIncomeNetOfTax" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RepurchaseAgreementCounterpartyLineItems_7d7c73d8-7b30-464b-a911-0397b72e7257_terseLabel_en-US" xlink:label="lab_us-gaap_RepurchaseAgreementCounterpartyLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Repurchase Agreement Counterparty [Line Items]</link:label>
    <link:label id="lab_us-gaap_RepurchaseAgreementCounterpartyLineItems_label_en-US" xlink:label="lab_us-gaap_RepurchaseAgreementCounterpartyLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreement Counterparty [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RepurchaseAgreementCounterpartyLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RepurchaseAgreementCounterpartyLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RepurchaseAgreementCounterpartyLineItems" xlink:to="lab_us-gaap_RepurchaseAgreementCounterpartyLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice_fae286c1-9341-4eba-ac9e-35dcc1ba3c8c_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Expirations (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangements by Share-Based Payment Award, Options, Expirations in Period, Weighted Average Exercise Price</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementsByShareBasedPaymentAwardOptionsExpirationsInPeriodWeightedAverageExercisePrice" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_MeasurementInputTypeAxis_c2cafa81-ff01-4ef9-90b2-41e77439cbb7_terseLabel_en-US" xlink:label="lab_us-gaap_MeasurementInputTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Measurement Input Type [Axis]</link:label>
    <link:label id="lab_us-gaap_MeasurementInputTypeAxis_label_en-US" xlink:label="lab_us-gaap_MeasurementInputTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measurement Input Type [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_MeasurementInputTypeAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_MeasurementInputTypeAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_MeasurementInputTypeAxis" xlink:to="lab_us-gaap_MeasurementInputTypeAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_MeasureName_terseLabel_en-US" xlink:label="lab_ecd_MeasureName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Name</link:label>
    <link:label id="lab_ecd_MeasureName_label_en-US" xlink:label="lab_ecd_MeasureName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measure Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_MeasureName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_MeasureName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_MeasureName" xlink:to="lab_ecd_MeasureName" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SecuredDebtMember_99648241-bfe6-463e-8c77-e7cbd2e16200_terseLabel_en-US" xlink:label="lab_us-gaap_SecuredDebtMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Secured Debt</link:label>
    <link:label id="lab_us-gaap_SecuredDebtMember_label_en-US" xlink:label="lab_us-gaap_SecuredDebtMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Secured Debt [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SecuredDebtMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SecuredDebtMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SecuredDebtMember" xlink:to="lab_us-gaap_SecuredDebtMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock_cee78871-f87f-40f9-ac75-4107a96c6e4b_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Roll-Forward of Level 3 Investments Measured on Recurring Basis</link:label>
    <link:label id="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock_label_en-US" xlink:label="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Liabilities Measured on Recurring Basis, Unobservable Input Reconciliation [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:to="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ClassOfStockLineItems_8b14b368-bfff-4736-a0a3-ffaaff88d27f_terseLabel_en-US" xlink:label="lab_us-gaap_ClassOfStockLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Class of Stock [Line Items]</link:label>
    <link:label id="lab_us-gaap_ClassOfStockLineItems_label_en-US" xlink:label="lab_us-gaap_ClassOfStockLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class of Stock [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ClassOfStockLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ClassOfStockLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ClassOfStockLineItems" xlink:to="lab_us-gaap_ClassOfStockLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DeferredIncomeTaxLiabilitiesNet_2b45794a-f84d-44ed-a62b-c2934c66eb7d_terseLabel_en-US" xlink:label="lab_us-gaap_DeferredIncomeTaxLiabilitiesNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Deferred tax liabilities</link:label>
    <link:label id="lab_us-gaap_DeferredIncomeTaxLiabilitiesNet_label_en-US" xlink:label="lab_us-gaap_DeferredIncomeTaxLiabilitiesNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Deferred Income Tax Liabilities, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredIncomeTaxLiabilitiesNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredIncomeTaxLiabilitiesNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DeferredIncomeTaxLiabilitiesNet" xlink:to="lab_us-gaap_DeferredIncomeTaxLiabilitiesNet" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered_46425ac5-4ab6-45fd-a9f6-8862de164eae_terseLabel_en-US" xlink:label="lab_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in control, conversion price as a percentage of the lowest price per unit offered in the change of control event</link:label>
    <link:label id="lab_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered_label_en-US" xlink:label="lab_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Conversion, Change In Control, Price Per Unit As A Percentage Of Lowest Price Offered</link:label>
    <link:label id="lab_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered_documentation_en-US" xlink:label="lab_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Conversion, Change In Control, Price Per Unit As A Percentage Of Lowest Price Offered</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered" xlink:href="gemi-20250930.xsd#gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered" xlink:to="lab_gemi_DebtConversionChangeInControlPricePerUnitAsAPercentageOfLowestPriceOffered" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CommonClassAUnvestedIncentiveUnitsMember_872d41c2-00d7-49c4-81e4-fe080469ee84_terseLabel_en-US" xlink:label="lab_gemi_CommonClassAUnvestedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common Class A, Unvested Incentive Units</link:label>
    <link:label id="lab_gemi_CommonClassAUnvestedIncentiveUnitsMember_label_en-US" xlink:label="lab_gemi_CommonClassAUnvestedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Class A, Unvested Incentive Units [Member]</link:label>
    <link:label id="lab_gemi_CommonClassAUnvestedIncentiveUnitsMember_documentation_en-US" xlink:label="lab_gemi_CommonClassAUnvestedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Class A, Unvested Incentive Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CommonClassAUnvestedIncentiveUnitsMember" xlink:href="gemi-20250930.xsd#gemi_CommonClassAUnvestedIncentiveUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CommonClassAUnvestedIncentiveUnitsMember" xlink:to="lab_gemi_CommonClassAUnvestedIncentiveUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt_e2fc0152-4f98-425f-8f7c-f75f3a3ca8ed_terseLabel_en-US" xlink:label="lab_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit card receivables posted as collateral for funding debt</link:label>
    <link:label id="lab_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt_label_en-US" xlink:label="lab_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Card Receivables Posted As Collateral For Funding Debt</link:label>
    <link:label id="lab_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt_documentation_en-US" xlink:label="lab_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Credit Card Receivables Posted As Collateral For Funding Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt" xlink:href="gemi-20250930.xsd#gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt" xlink:to="lab_gemi_CreditCardReceivablesPostedAsCollateralForFundingDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongTermDebt_cbfc122d-576a-42c6-bf61-ff663266457a_terseLabel_en-US" xlink:label="lab_us-gaap_LongTermDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Principal outstanding</link:label>
    <link:label id="lab_us-gaap_LongTermDebt_label_en-US" xlink:label="lab_us-gaap_LongTermDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebt" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LongTermDebt" xlink:to="lab_us-gaap_LongTermDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_LegalEntityAxis_90e0eb05-514d-47c5-b178-b14c8256680e_terseLabel_en-US" xlink:label="lab_dei_LegalEntityAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Legal Entity [Axis]</link:label>
    <link:label id="lab_dei_LegalEntityAxis_label_en-US" xlink:label="lab_dei_LegalEntityAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Legal Entity [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_LegalEntityAxis" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_LegalEntityAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_LegalEntityAxis" xlink:to="lab_dei_LegalEntityAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RecognitionCriteriaMetMember_ff2d11c8-4e43-45e8-9ec1-d59087e47cbc_terseLabel_en-US" xlink:label="lab_gemi_RecognitionCriteriaMetMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Recognition Criteria, Met</link:label>
    <link:label id="lab_gemi_RecognitionCriteriaMetMember_label_en-US" xlink:label="lab_gemi_RecognitionCriteriaMetMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Recognition Criteria, Met [Member]</link:label>
    <link:label id="lab_gemi_RecognitionCriteriaMetMember_documentation_en-US" xlink:label="lab_gemi_RecognitionCriteriaMetMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Recognition Criteria, Met</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RecognitionCriteriaMetMember" xlink:href="gemi-20250930.xsd#gemi_RecognitionCriteriaMetMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RecognitionCriteriaMetMember" xlink:to="lab_gemi_RecognitionCriteriaMetMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_Assets_eb332af1-5ec6-41c2-a207-cf93970a18bd_totalLabel_en-US" xlink:label="lab_us-gaap_Assets" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total assets</link:label>
    <link:label id="lab_us-gaap_Assets_label_en-US" xlink:label="lab_us-gaap_Assets" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Assets</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Assets" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Assets"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_Assets" xlink:to="lab_us-gaap_Assets" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetAddition_b5f0df0c-9374-4add-b203-6f26b7ba79d5_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetAddition" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Addition of crypto assets</link:label>
    <link:label id="lab_us-gaap_CryptoAssetAddition_label_en-US" xlink:label="lab_us-gaap_CryptoAssetAddition" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Addition</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetAddition" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetAddition"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetAddition" xlink:to="lab_us-gaap_CryptoAssetAddition" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt_06974842-1787-4c2d-b7b6-2ce243041034_terseLabel_en-US" xlink:label="lab_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Conversion of related party loans into related party convertible notes</link:label>
    <link:label id="lab_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt_label_en-US" xlink:label="lab_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Conversion Of Related Party Debt To Convertible Debt</link:label>
    <link:label id="lab_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt_documentation_en-US" xlink:label="lab_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Conversion Of Related Party Debt To Convertible Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt" xlink:href="gemi-20250930.xsd#gemi_ConversionOfRelatedPartyDebtToConvertibleDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt" xlink:to="lab_gemi_ConversionOfRelatedPartyDebtToConvertibleDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset_eaa7ca62-354c-40cb-b90f-03c2d0bb183b_terseLabel_en-US" xlink:label="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party loans received in crypto assets</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset_7c4f8f69-af1c-4946-ae14-12ae89af1b5b_verboseLabel_en-US" xlink:label="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Amount Borrowed</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset_label_en-US" xlink:label="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Debt, Received In Crypto Asset</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset_documentation_en-US" xlink:label="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Debt, Received In Crypto Asset</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RelatedPartyDebtReceivedInCryptoAsset" xlink:href="gemi-20250930.xsd#gemi_RelatedPartyDebtReceivedInCryptoAsset"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RelatedPartyDebtReceivedInCryptoAsset" xlink:to="lab_gemi_RelatedPartyDebtReceivedInCryptoAsset" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths_4ac01e69-4b6b-4212-9dc3-4fe21d7c5671_terseLabel_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2026</link:label>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths_label_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Finite-Lived Intangible Asset, Expected Amortization, Year One</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths" xlink:to="lab_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember_d1de7cab-7074-4eb4-8076-456a2969cb3c_terseLabel_en-US" xlink:label="lab_us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Portion at Other than Fair Value Measurement</link:label>
    <link:label id="lab_us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember_label_en-US" xlink:label="lab_us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Portion at Other than Fair Value Measurement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember" xlink:to="lab_us-gaap_PortionAtOtherThanFairValueFairValueDisclosureMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue_dc5b2840-ea03-40d1-ae86-6821fb024321_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Vested (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Vested in Period, Weighted Average Grant Date Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsVestedInPeriodWeightedAverageGrantDateFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy_c1b67c01-9703-4b69-9131-63f08464455d_terseLabel_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Customer custodial funds and custodial funds due to customers</link:label>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy_label_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash and Cash Equivalents, Restricted Cash and Cash Equivalents, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy" xlink:to="lab_us-gaap_CashAndCashEquivalentsRestrictedCashAndCashEquivalentsPolicy" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OperatingLeaseMonthlyPayments_cd980f50-0fbe-47cf-9465-1c49a4adf178_terseLabel_en-US" xlink:label="lab_gemi_OperatingLeaseMonthlyPayments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Monthly base payments</link:label>
    <link:label id="lab_gemi_OperatingLeaseMonthlyPayments_label_en-US" xlink:label="lab_gemi_OperatingLeaseMonthlyPayments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Monthly Payments</link:label>
    <link:label id="lab_gemi_OperatingLeaseMonthlyPayments_documentation_en-US" xlink:label="lab_gemi_OperatingLeaseMonthlyPayments" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Operating Lease, Monthly Payments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OperatingLeaseMonthlyPayments" xlink:href="gemi-20250930.xsd#gemi_OperatingLeaseMonthlyPayments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OperatingLeaseMonthlyPayments" xlink:to="lab_gemi_OperatingLeaseMonthlyPayments" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_SegmentGeographicalDomain_b533b213-b91e-41b2-8346-38b778018262_terseLabel_en-US" xlink:label="lab_srt_SegmentGeographicalDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Geographical [Domain]</link:label>
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    <link:label id="lab_us-gaap_IncomeTaxPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_IncomeTaxPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income Tax, Policy [Policy Text Block]</link:label>
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    <link:label id="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_feda16f9-f5f3-47b8-8e35-4321c4d46ded_terseLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash flows from operating activities</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccountsNotesLoansAndFinancingReceivableByReceivableTypeAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccountsNotesLoansAndFinancingReceivableByReceivableTypeAxis"/>
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    <link:label id="lab_gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrency_2302ae15-2509-4303-97b8-e246ed30c863_terseLabel_en-US" xlink:label="lab_gemi_ProceedsFromIssuanceOfLongTermDebtCryptoCurrency" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets received in issuance of long-term debt (in units)</link:label>
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    <link:label id="lab_us-gaap_FairValueInputsLevel2Member_f1bcbbf1-9fd9-460d-9b93-a2750d919d5c_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueInputsLevel2Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Level 2</link:label>
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    <link:label id="lab_us-gaap_InvestmentIncomeInterest_7940eaa4-6d96-4fcf-9abd-2790c1eb538a_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentIncomeInterest" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest income</link:label>
    <link:label id="lab_us-gaap_InvestmentIncomeInterest_label_en-US" xlink:label="lab_us-gaap_InvestmentIncomeInterest" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Income, Interest</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InvestmentIncomeInterest" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentIncomeInterest"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InvestmentIncomeInterest" xlink:to="lab_us-gaap_InvestmentIncomeInterest" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NoncashInvestingAndFinancingItemsAbstract_14852c40-50f6-4254-bcb3-9b24c3da5e3a_terseLabel_en-US" xlink:label="lab_us-gaap_NoncashInvestingAndFinancingItemsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Supplemental schedule of non-cash investing and financing activities</link:label>
    <link:label id="lab_us-gaap_NoncashInvestingAndFinancingItemsAbstract_label_en-US" xlink:label="lab_us-gaap_NoncashInvestingAndFinancingItemsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Noncash Investing and Financing Items [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoncashInvestingAndFinancingItemsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoncashInvestingAndFinancingItemsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NoncashInvestingAndFinancingItemsAbstract" xlink:to="lab_us-gaap_NoncashInvestingAndFinancingItemsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_country_US_3a19ab56-7067-49d8-8949-cad836434c93_terseLabel_en-US" xlink:label="lab_country_US" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">U.S.</link:label>
    <link:label id="lab_country_US_label_en-US" xlink:label="lab_country_US" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">UNITED STATES</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_country_US" xlink:to="lab_country_US" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_7f7684f1-b8b0-4496-9984-033451ab19be_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Changes in operating assets and liabilities:</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOperatingCapitalAbstract_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Adjustment to Reconcile Net Income to Cash Provided by (Used in) Operating Activity, Increase (Decrease) in Operating Capital [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInOperatingCapitalAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:to="lab_us-gaap_IncreaseDecreaseInOperatingCapitalAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate_958acdf3-ba2a-4865-aa14-011b34fdd434_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Volatility</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Expected Volatility Rate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RetainedEarningsMember_cb0c4d98-09f7-459a-bf5a-76b131444451_terseLabel_en-US" xlink:label="lab_us-gaap_RetainedEarningsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accumulated Deficit</link:label>
    <link:label id="lab_us-gaap_RetainedEarningsMember_label_en-US" xlink:label="lab_us-gaap_RetainedEarningsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Retained Earnings [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RetainedEarningsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RetainedEarningsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RetainedEarningsMember" xlink:to="lab_us-gaap_RetainedEarningsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_StakingFeesReceivableMember_865a68b7-8c8e-4a48-af82-36633202fefb_terseLabel_en-US" xlink:label="lab_gemi_StakingFeesReceivableMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Staking fees receivable</link:label>
    <link:label id="lab_gemi_StakingFeesReceivableMember_label_en-US" xlink:label="lab_gemi_StakingFeesReceivableMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Staking Fees Receivable [Member]</link:label>
    <link:label id="lab_gemi_StakingFeesReceivableMember_documentation_en-US" xlink:label="lab_gemi_StakingFeesReceivableMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Staking Fees Receivable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_StakingFeesReceivableMember" xlink:href="gemi-20250930.xsd#gemi_StakingFeesReceivableMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_StakingFeesReceivableMember" xlink:to="lab_gemi_StakingFeesReceivableMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CommissionsPayableCurrent_6dbbc2db-0c6c-4a04-ba35-3fca3c0d354a_terseLabel_en-US" xlink:label="lab_gemi_CommissionsPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Commissions to NFT artists payable</link:label>
    <link:label id="lab_gemi_CommissionsPayableCurrent_label_en-US" xlink:label="lab_gemi_CommissionsPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Commissions Payable, Current</link:label>
    <link:label id="lab_gemi_CommissionsPayableCurrent_documentation_en-US" xlink:label="lab_gemi_CommissionsPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Commissions Payable, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CommissionsPayableCurrent" xlink:href="gemi-20250930.xsd#gemi_CommissionsPayableCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CommissionsPayableCurrent" xlink:to="lab_gemi_CommissionsPayableCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract_label_en-US" xlink:label="lab_us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract" xlink:to="lab_us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_6667f051-eee1-4987-a0b8-3c66cfc2f8ac_periodStartLabel_en-US" xlink:label="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Cash, cash equivalents, restricted cash and cash equivalents, beginning of period</link:label>
    <link:label id="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_3e51b541-8d81-4deb-a67f-dfa5ef810ae6_periodEndLabel_en-US" xlink:label="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Cash, cash equivalents, restricted cash and cash equivalents, end of period</link:label>
    <link:label id="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_01293ce5-d762-48ff-92cb-e9285c1c5ce7_totalLabel_en-US" xlink:label="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total cash, cash equivalents, restricted cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_label_en-US" xlink:label="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Continuing Operation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:to="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncomeStatementAbstract_label_en-US" xlink:label="lab_us-gaap_IncomeStatementAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income Statement [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeStatementAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeStatementAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncomeStatementAbstract" xlink:to="lab_us-gaap_IncomeStatementAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativeAssets_91b3eebb-fad9-4858-9e85-8d3fd5a38b86_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeAssets" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset options</link:label>
    <link:label id="lab_us-gaap_DerivativeAssets_6711463c-d241-4905-83e3-8d5c4fc89144_verboseLabel_en-US" xlink:label="lab_us-gaap_DerivativeAssets" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Assets</link:label>
    <link:label id="lab_us-gaap_DerivativeAssets_label_en-US" xlink:label="lab_us-gaap_DerivativeAssets" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Asset</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeAssets" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeAssets"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeAssets" xlink:to="lab_us-gaap_DerivativeAssets" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OfficeSpaceInChicagoMember_ff5eae31-dafe-4ff3-8c1c-cc1655b57d45_terseLabel_en-US" xlink:label="lab_gemi_OfficeSpaceInChicagoMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Office Space In Chicago</link:label>
    <link:label id="lab_gemi_OfficeSpaceInChicagoMember_label_en-US" xlink:label="lab_gemi_OfficeSpaceInChicagoMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Office Space In Chicago [Member]</link:label>
    <link:label id="lab_gemi_OfficeSpaceInChicagoMember_documentation_en-US" xlink:label="lab_gemi_OfficeSpaceInChicagoMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Office Space In Chicago</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OfficeSpaceInChicagoMember" xlink:href="gemi-20250930.xsd#gemi_OfficeSpaceInChicagoMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OfficeSpaceInChicagoMember" xlink:to="lab_gemi_OfficeSpaceInChicagoMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SaleOfStockNameOfTransactionDomain_525187ae-dd5e-4fba-9b88-4118ccda0a05_terseLabel_en-US" xlink:label="lab_us-gaap_SaleOfStockNameOfTransactionDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Sale of Stock [Domain]</link:label>
    <link:label id="lab_us-gaap_SaleOfStockNameOfTransactionDomain_label_en-US" xlink:label="lab_us-gaap_SaleOfStockNameOfTransactionDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Sale of Stock [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SaleOfStockNameOfTransactionDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SaleOfStockNameOfTransactionDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SaleOfStockNameOfTransactionDomain" xlink:to="lab_us-gaap_SaleOfStockNameOfTransactionDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_TemporaryEquityValueConversionOfConvertibleSecurities_be726e0f-7227-4cc8-ace4-577e58f55ad7_negatedTerseLabel_en-US" xlink:label="lab_gemi_TemporaryEquityValueConversionOfConvertibleSecurities" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Conversion of convertible preferred units to common stock in connection with IPO</link:label>
    <link:label id="lab_gemi_TemporaryEquityValueConversionOfConvertibleSecurities_label_en-US" xlink:label="lab_gemi_TemporaryEquityValueConversionOfConvertibleSecurities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Temporary Equity, Value, Conversion Of Convertible Securities</link:label>
    <link:label id="lab_gemi_TemporaryEquityValueConversionOfConvertibleSecurities_documentation_en-US" xlink:label="lab_gemi_TemporaryEquityValueConversionOfConvertibleSecurities" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Temporary Equity, Value, Conversion Of Convertible Securities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TemporaryEquityValueConversionOfConvertibleSecurities" xlink:href="gemi-20250930.xsd#gemi_TemporaryEquityValueConversionOfConvertibleSecurities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_TemporaryEquityValueConversionOfConvertibleSecurities" xlink:to="lab_gemi_TemporaryEquityValueConversionOfConvertibleSecurities" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_ProductsAndServicesDomain_cbc507be-9544-433e-bcc4-3101a45f9ff0_terseLabel_en-US" xlink:label="lab_srt_ProductsAndServicesDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Product and Service [Domain]</link:label>
    <link:label id="lab_srt_ProductsAndServicesDomain_label_en-US" xlink:label="lab_srt_ProductsAndServicesDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Product and Service [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_ProductsAndServicesDomain" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ProductsAndServicesDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_ProductsAndServicesDomain" xlink:to="lab_srt_ProductsAndServicesDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SecuredDebtRepurchaseAgreements_d37201a6-eada-48f6-ab0d-b769c7dfc366_verboseLabel_en-US" xlink:label="lab_us-gaap_SecuredDebtRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Repurchase agreement</link:label>
    <link:label id="lab_us-gaap_SecuredDebtRepurchaseAgreements_7581c916-e8f0-4931-b26f-ddc1cfeda923_terseLabel_en-US" xlink:label="lab_us-gaap_SecuredDebtRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Principal Outstanding</link:label>
    <link:label id="lab_us-gaap_SecuredDebtRepurchaseAgreements_label_en-US" xlink:label="lab_us-gaap_SecuredDebtRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Secured Debt, Repurchase Agreements</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SecuredDebtRepurchaseAgreements" xlink:to="lab_us-gaap_SecuredDebtRepurchaseAgreements" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember_8e9d2e56-0f60-4559-ae5e-0ab269be2591_terseLabel_en-US" xlink:label="lab_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Plan</link:label>
    <link:label id="lab_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember_label_en-US" xlink:label="lab_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Gemini Space Station, Inc. 2025 Omnibus Incentive Plan [Member]</link:label>
    <link:label id="lab_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember_documentation_en-US" xlink:label="lab_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Gemini Space Station, Inc. 2025 Omnibus Incentive Plan</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember" xlink:href="gemi-20250930.xsd#gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember" xlink:to="lab_gemi_GeminiSpaceStationInc.2025OmnibusIncentivePlanMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConversionOfStockAmountIssued1_9973c1e1-d027-485e-84a1-5a4fb05d2646_terseLabel_en-US" xlink:label="lab_us-gaap_ConversionOfStockAmountIssued1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Conversion of convertible preferred units to common stock in connection with initial public offering</link:label>
    <link:label id="lab_us-gaap_ConversionOfStockAmountIssued1_label_en-US" xlink:label="lab_us-gaap_ConversionOfStockAmountIssued1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Conversion of Stock, Amount Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConversionOfStockAmountIssued1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConversionOfStockAmountIssued1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConversionOfStockAmountIssued1" xlink:to="lab_us-gaap_ConversionOfStockAmountIssued1" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_RestatementDoesNotRequireRecoveryTextBlock_terseLabel_en-US" xlink:label="lab_ecd_RestatementDoesNotRequireRecoveryTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restatement does not require Recovery</link:label>
    <link:label id="lab_ecd_RestatementDoesNotRequireRecoveryTextBlock_label_en-US" xlink:label="lab_ecd_RestatementDoesNotRequireRecoveryTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restatement Does Not Require Recovery [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_RestatementDoesNotRequireRecoveryTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_RestatementDoesNotRequireRecoveryTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_RestatementDoesNotRequireRecoveryTextBlock" xlink:to="lab_ecd_RestatementDoesNotRequireRecoveryTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LiabilitiesFairValueDisclosure_18f54441-33f2-4940-8ac4-b235e4f52459_totalLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total liabilities</link:label>
    <link:label id="lab_us-gaap_LiabilitiesFairValueDisclosure_label_en-US" xlink:label="lab_us-gaap_LiabilitiesFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liabilities, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesFairValueDisclosure" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilitiesFairValueDisclosure" xlink:to="lab_us-gaap_LiabilitiesFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativeNotionalAmount_ffae381e-aef3-4f03-9479-5cb21ac66f1c_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeNotionalAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Gross Notional</link:label>
    <link:label id="lab_us-gaap_DerivativeNotionalAmount_label_en-US" xlink:label="lab_us-gaap_DerivativeNotionalAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative, Notional Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeNotionalAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeNotionalAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeNotionalAmount" xlink:to="lab_us-gaap_DerivativeNotionalAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract_label_en-US" xlink:label="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Organization, Consolidation and Presentation of Financial Statements [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:to="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementNumberOfPlans_ad1e11df-3ef2-4e97-a063-6d6d052bf0d7_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementNumberOfPlans" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of equity incentive plans</link:label>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementNumberOfPlans_label_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementNumberOfPlans" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Number Of Plans</link:label>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementNumberOfPlans_documentation_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementNumberOfPlans" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Number Of Plans</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedPaymentArrangementNumberOfPlans" xlink:href="gemi-20250930.xsd#gemi_ShareBasedPaymentArrangementNumberOfPlans"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedPaymentArrangementNumberOfPlans" xlink:to="lab_gemi_ShareBasedPaymentArrangementNumberOfPlans" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan10Member_41ac7576-54cf-486a-80ac-d4a8993c08ff_terseLabel_en-US" xlink:label="lab_gemi_Loan10Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 10</link:label>
    <link:label id="lab_gemi_Loan10Member_label_en-US" xlink:label="lab_gemi_Loan10Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 10 [Member]</link:label>
    <link:label id="lab_gemi_Loan10Member_documentation_en-US" xlink:label="lab_gemi_Loan10Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 10</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan10Member" xlink:href="gemi-20250930.xsd#gemi_Loan10Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan10Member" xlink:to="lab_gemi_Loan10Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RepurchaseAgreementsCollateralRate_b84f1677-4169-4978-9944-817df923da9f_verboseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsCollateralRate" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Collateral rate</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsCollateralRate_fee01c6f-ffc1-4c7a-b32c-b8d8191441b7_terseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsCollateralRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collateral Rate</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsCollateralRate_label_en-US" xlink:label="lab_gemi_RepurchaseAgreementsCollateralRate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Collateral Rate</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsCollateralRate_documentation_en-US" xlink:label="lab_gemi_RepurchaseAgreementsCollateralRate" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Collateral Rate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RepurchaseAgreementsCollateralRate" xlink:href="gemi-20250930.xsd#gemi_RepurchaseAgreementsCollateralRate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RepurchaseAgreementsCollateralRate" xlink:to="lab_gemi_RepurchaseAgreementsCollateralRate" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements_2e873c79-fcae-4daa-96e5-17a93be503a0_terseLabel_en-US" xlink:label="lab_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Exercisable increments, volume-weighted average price per share increasing increments per share (in dollars per share)</link:label>
    <link:label id="lab_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements_label_en-US" xlink:label="lab_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class Of Warrant Or Right, Exercise Increments, Volume-Weighted Average Price Per Share, Per Share Increasing Increments</link:label>
    <link:label id="lab_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements_documentation_en-US" xlink:label="lab_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Class Of Warrant Or Right, Exercise Increments, Volume-Weighted Average Price Per Share, Per Share Increasing Increments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements" xlink:href="gemi-20250930.xsd#gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements" xlink:to="lab_gemi_ClassOfWarrantOrRightExerciseIncrementsVolumeWeightedAveragePricePerSharePerShareIncreasingIncrements" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_CityAreaCode_2d701a08-c55a-4800-87f8-b20897bd817f_terseLabel_en-US" xlink:label="lab_dei_CityAreaCode" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">City Area Code</link:label>
    <link:label id="lab_dei_CityAreaCode_label_en-US" xlink:label="lab_dei_CityAreaCode" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">City Area Code</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_CityAreaCode" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_CityAreaCode"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_CityAreaCode" xlink:to="lab_dei_CityAreaCode" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardTmgHowMnpiCnsdrdTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AwardTmgHowMnpiCnsdrdTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Timing, How MNPI Considered</link:label>
    <link:label id="lab_ecd_AwardTmgHowMnpiCnsdrdTextBlock_label_en-US" xlink:label="lab_ecd_AwardTmgHowMnpiCnsdrdTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Timing, How MNPI Considered [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardTmgHowMnpiCnsdrdTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardTmgHowMnpiCnsdrdTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AwardTmgHowMnpiCnsdrdTextBlock" xlink:to="lab_ecd_AwardTmgHowMnpiCnsdrdTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DerivativeAssetMember_55db064b-3b6c-4f2b-b553-abf843554172_terseLabel_en-US" xlink:label="lab_gemi_DerivativeAssetMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative assets</link:label>
    <link:label id="lab_gemi_DerivativeAssetMember_label_en-US" xlink:label="lab_gemi_DerivativeAssetMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Asset [Member]</link:label>
    <link:label id="lab_gemi_DerivativeAssetMember_documentation_en-US" xlink:label="lab_gemi_DerivativeAssetMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Derivative Asset</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DerivativeAssetMember" xlink:href="gemi-20250930.xsd#gemi_DerivativeAssetMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DerivativeAssetMember" xlink:to="lab_gemi_DerivativeAssetMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CommonStockConvertibleConversionRatio_47e55826-0d01-4123-97c4-de855bca47c3_terseLabel_en-US" xlink:label="lab_gemi_CommonStockConvertibleConversionRatio" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common stock conversion ratio</link:label>
    <link:label id="lab_gemi_CommonStockConvertibleConversionRatio_label_en-US" xlink:label="lab_gemi_CommonStockConvertibleConversionRatio" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Convertible, Conversion Ratio</link:label>
    <link:label id="lab_gemi_CommonStockConvertibleConversionRatio_documentation_en-US" xlink:label="lab_gemi_CommonStockConvertibleConversionRatio" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Stock, Convertible, Conversion Ratio</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CommonStockConvertibleConversionRatio" xlink:href="gemi-20250930.xsd#gemi_CommonStockConvertibleConversionRatio"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CommonStockConvertibleConversionRatio" xlink:to="lab_gemi_CommonStockConvertibleConversionRatio" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ProceedsFromLongTermLinesOfCredit_268b04b5-d6fd-435d-801b-f0e88d025f45_terseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromLongTermLinesOfCredit" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from funding debt</link:label>
    <link:label id="lab_us-gaap_ProceedsFromLongTermLinesOfCredit_label_en-US" xlink:label="lab_us-gaap_ProceedsFromLongTermLinesOfCredit" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds from Long-Term Lines of Credit</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromLongTermLinesOfCredit" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromLongTermLinesOfCredit"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ProceedsFromLongTermLinesOfCredit" xlink:to="lab_us-gaap_ProceedsFromLongTermLinesOfCredit" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_26a29cd7-11bb-491a-af77-c20f1c71c50a_terseLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash flows from investing activities</link:label>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_label_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Investing Activity, Including Discontinued Operation [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:to="lab_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_GeneralAndAdministrativeExpense_18f5f5b3-7a73-40b3-8cf3-d6a9cc4ac408_verboseLabel_en-US" xlink:label="lab_us-gaap_GeneralAndAdministrativeExpense" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">General and administrative</link:label>
    <link:label id="lab_us-gaap_GeneralAndAdministrativeExpense_4f3bb022-d141-40eb-8141-958442127179_terseLabel_en-US" xlink:label="lab_us-gaap_GeneralAndAdministrativeExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">General and administrative expense</link:label>
    <link:label id="lab_us-gaap_GeneralAndAdministrativeExpense_label_en-US" xlink:label="lab_us-gaap_GeneralAndAdministrativeExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">General and Administrative Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_GeneralAndAdministrativeExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_GeneralAndAdministrativeExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_GeneralAndAdministrativeExpense" xlink:to="lab_us-gaap_GeneralAndAdministrativeExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PvpTable_terseLabel_en-US" xlink:label="lab_ecd_PvpTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pay vs Performance Disclosure</link:label>
    <link:label id="lab_ecd_PvpTable_label_en-US" xlink:label="lab_ecd_PvpTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pay vs Performance Disclosure [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PvpTable" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PvpTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PvpTable" xlink:to="lab_ecd_PvpTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InterestReceivable_15242e81-9670-40d4-8a89-95ae10cb77ca_terseLabel_en-US" xlink:label="lab_us-gaap_InterestReceivable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued interest receivable</link:label>
    <link:label id="lab_us-gaap_InterestReceivable_label_en-US" xlink:label="lab_us-gaap_InterestReceivable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Receivable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestReceivable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestReceivable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestReceivable" xlink:to="lab_us-gaap_InterestReceivable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RelatedPartyTransactionsAbstract_label_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transactions [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyTransactionsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyTransactionsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyTransactionsAbstract" xlink:to="lab_us-gaap_RelatedPartyTransactionsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LeaseCost_aae8fad2-bbf0-4974-8886-36c108247c4f_totalLabel_en-US" xlink:label="lab_us-gaap_LeaseCost" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total lease expense</link:label>
    <link:label id="lab_us-gaap_LeaseCost_label_en-US" xlink:label="lab_us-gaap_LeaseCost" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lease, Cost</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LeaseCost" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LeaseCost"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LeaseCost" xlink:to="lab_us-gaap_LeaseCost" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongTermDebtMeasurementInput_ab4c7400-43f4-4f94-b0b1-c673dab3fc4f_terseLabel_en-US" xlink:label="lab_us-gaap_LongTermDebtMeasurementInput" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Significant input utilized in analysis</link:label>
    <link:label id="lab_us-gaap_LongTermDebtMeasurementInput_label_en-US" xlink:label="lab_us-gaap_LongTermDebtMeasurementInput" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Measurement Input</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebtMeasurementInput" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebtMeasurementInput"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LongTermDebtMeasurementInput" xlink:to="lab_us-gaap_LongTermDebtMeasurementInput" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LeaseCostTableTextBlock_f8098f72-5e75-4d24-8682-e36cfef3aa3f_terseLabel_en-US" xlink:label="lab_us-gaap_LeaseCostTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Components of Lease Expense and Other Information</link:label>
    <link:label id="lab_us-gaap_LeaseCostTableTextBlock_label_en-US" xlink:label="lab_us-gaap_LeaseCostTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lease, Cost [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LeaseCostTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LeaseCostTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LeaseCostTableTextBlock" xlink:to="lab_us-gaap_LeaseCostTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OtherRevenueAndCorporateInterestMember_c4bfa7e7-ee4c-4e5a-ad5c-abacdb85abde_terseLabel_en-US" xlink:label="lab_gemi_OtherRevenueAndCorporateInterestMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other revenue</link:label>
    <link:label id="lab_gemi_OtherRevenueAndCorporateInterestMember_label_en-US" xlink:label="lab_gemi_OtherRevenueAndCorporateInterestMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Revenue And Corporate Interest [Member]</link:label>
    <link:label id="lab_gemi_OtherRevenueAndCorporateInterestMember_documentation_en-US" xlink:label="lab_gemi_OtherRevenueAndCorporateInterestMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Other Revenue And Interest Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OtherRevenueAndCorporateInterestMember" xlink:href="gemi-20250930.xsd#gemi_OtherRevenueAndCorporateInterestMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_OtherRevenueAndCorporateInterestMember" xlink:to="lab_gemi_OtherRevenueAndCorporateInterestMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_PreferredUnitsPricePerShare_171b0b8c-d905-4d1a-aa67-ce599e614826_terseLabel_en-US" xlink:label="lab_gemi_PreferredUnitsPricePerShare" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Original Price per Unit (in dollars per share)</link:label>
    <link:label id="lab_gemi_PreferredUnitsPricePerShare_label_en-US" xlink:label="lab_gemi_PreferredUnitsPricePerShare" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Units, Price Per Share</link:label>
    <link:label id="lab_gemi_PreferredUnitsPricePerShare_documentation_en-US" xlink:label="lab_gemi_PreferredUnitsPricePerShare" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Preferred Units, Price Per Share</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_PreferredUnitsPricePerShare" xlink:href="gemi-20250930.xsd#gemi_PreferredUnitsPricePerShare"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_PreferredUnitsPricePerShare" xlink:to="lab_gemi_PreferredUnitsPricePerShare" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LeaseContractualTermAxis_58c2c20d-8ebd-422f-9424-cbdac646a3f3_terseLabel_en-US" xlink:label="lab_us-gaap_LeaseContractualTermAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Lease Contractual Term [Axis]</link:label>
    <link:label id="lab_us-gaap_LeaseContractualTermAxis_label_en-US" xlink:label="lab_us-gaap_LeaseContractualTermAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lease Contractual Term [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LeaseContractualTermAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LeaseContractualTermAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LeaseContractualTermAxis" xlink:to="lab_us-gaap_LeaseContractualTermAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetTextBlock_ed47d790-b3eb-4c82-bb5d-0355b874b98b_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto Assets Held</link:label>
    <link:label id="lab_us-gaap_CryptoAssetTextBlock_label_en-US" xlink:label="lab_us-gaap_CryptoAssetTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetTextBlock" xlink:to="lab_us-gaap_CryptoAssetTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_TermLoan2024Member_81e5bf08-cc68-45fd-94ce-e27dd37a7a9f_terseLabel_en-US" xlink:label="lab_gemi_TermLoan2024Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Term Loan 2024</link:label>
    <link:label id="lab_gemi_TermLoan2024Member_label_en-US" xlink:label="lab_gemi_TermLoan2024Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Term Loan 2024 [Member]</link:label>
    <link:label id="lab_gemi_TermLoan2024Member_documentation_en-US" xlink:label="lab_gemi_TermLoan2024Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Term Loan 2024</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_TermLoan2024Member" xlink:href="gemi-20250930.xsd#gemi_TermLoan2024Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_TermLoan2024Member" xlink:to="lab_gemi_TermLoan2024Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_PerformanceBasedIncentiveUnitsMember_faa7123d-f621-4316-acb7-7bc5f851dd05_terseLabel_en-US" xlink:label="lab_gemi_PerformanceBasedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Performance Based Incentive Units</link:label>
    <link:label id="lab_gemi_PerformanceBasedIncentiveUnitsMember_label_en-US" xlink:label="lab_gemi_PerformanceBasedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Performance Based Incentive Units [Member]</link:label>
    <link:label id="lab_gemi_PerformanceBasedIncentiveUnitsMember_documentation_en-US" xlink:label="lab_gemi_PerformanceBasedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Performance Based Incentive Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_PerformanceBasedIncentiveUnitsMember" xlink:href="gemi-20250930.xsd#gemi_PerformanceBasedIncentiveUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_PerformanceBasedIncentiveUnitsMember" xlink:to="lab_gemi_PerformanceBasedIncentiveUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NonUsMember_19393f42-42cc-4a02-b310-a97c631e7c8f_terseLabel_en-US" xlink:label="lab_us-gaap_NonUsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">International</link:label>
    <link:label id="lab_us-gaap_NonUsMember_label_en-US" xlink:label="lab_us-gaap_NonUsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-US [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NonUsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NonUsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NonUsMember" xlink:to="lab_us-gaap_NonUsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NatureOfOperations_bdb41ad1-7234-4084-ad86-c73cead10d27_terseLabel_en-US" xlink:label="lab_us-gaap_NatureOfOperations" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Description of Business</link:label>
    <link:label id="lab_us-gaap_NatureOfOperations_label_en-US" xlink:label="lab_us-gaap_NatureOfOperations" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Nature of Operations [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NatureOfOperations" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NatureOfOperations"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NatureOfOperations" xlink:to="lab_us-gaap_NatureOfOperations" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue_36933fc8-8297-46e4-88c9-7a7ce6d676d0_terseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Nominal value</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue_label_en-US" xlink:label="lab_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Securities Transferred, Nominal Value</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue_documentation_en-US" xlink:label="lab_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Securities Transferred, Nominal Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue" xlink:href="gemi-20250930.xsd#gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue" xlink:to="lab_gemi_RepurchaseAgreementsSecuritiesTransferredNominalValue" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A340BTCOctober312023Member_bb39331d-fb59-4f57-aa68-1ff25e49b6ff_terseLabel_en-US" xlink:label="lab_gemi_A340BTCOctober312023Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">340 BTC October 31, 2023</link:label>
    <link:label id="lab_gemi_A340BTCOctober312023Member_label_en-US" xlink:label="lab_gemi_A340BTCOctober312023Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">340 BTC October 31, 2023 [Member]</link:label>
    <link:label id="lab_gemi_A340BTCOctober312023Member_documentation_en-US" xlink:label="lab_gemi_A340BTCOctober312023Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">340 BTC October 31, 2023</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A340BTCOctober312023Member" xlink:href="gemi-20250930.xsd#gemi_A340BTCOctober312023Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A340BTCOctober312023Member" xlink:to="lab_gemi_A340BTCOctober312023Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_LesseeOperatingSubleaseTermOfContract_0912ccf4-5860-4294-94fa-86a91cf995b5_terseLabel_en-US" xlink:label="lab_gemi_LesseeOperatingSubleaseTermOfContract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Sublease term</link:label>
    <link:label id="lab_gemi_LesseeOperatingSubleaseTermOfContract_label_en-US" xlink:label="lab_gemi_LesseeOperatingSubleaseTermOfContract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Sublease, Term Of Contract</link:label>
    <link:label id="lab_gemi_LesseeOperatingSubleaseTermOfContract_documentation_en-US" xlink:label="lab_gemi_LesseeOperatingSubleaseTermOfContract" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Lessee, Operating Sublease, Term Of Contract</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_LesseeOperatingSubleaseTermOfContract" xlink:href="gemi-20250930.xsd#gemi_LesseeOperatingSubleaseTermOfContract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_LesseeOperatingSubleaseTermOfContract" xlink:to="lab_gemi_LesseeOperatingSubleaseTermOfContract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SupplementalCashFlowInformationAbstract_ad95729d-c7e8-43c1-b35c-fbc76ea1338c_terseLabel_en-US" xlink:label="lab_us-gaap_SupplementalCashFlowInformationAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Supplemental disclosure of cash flow information</link:label>
    <link:label id="lab_us-gaap_SupplementalCashFlowInformationAbstract_label_en-US" xlink:label="lab_us-gaap_SupplementalCashFlowInformationAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Supplemental Cash Flow Information [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SupplementalCashFlowInformationAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SupplementalCashFlowInformationAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SupplementalCashFlowInformationAbstract" xlink:to="lab_us-gaap_SupplementalCashFlowInformationAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ServiceBasedCommonStockUnitsMember_008ce4de-a8ee-4ab8-baea-d9dc846837e4_terseLabel_en-US" xlink:label="lab_gemi_ServiceBasedCommonStockUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Service-Based Common Units</link:label>
    <link:label id="lab_gemi_ServiceBasedCommonStockUnitsMember_label_en-US" xlink:label="lab_gemi_ServiceBasedCommonStockUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Service-Based Common Stock Units [Member]</link:label>
    <link:label id="lab_gemi_ServiceBasedCommonStockUnitsMember_documentation_en-US" xlink:label="lab_gemi_ServiceBasedCommonStockUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Service-Based Common Stock Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ServiceBasedCommonStockUnitsMember" xlink:href="gemi-20250930.xsd#gemi_ServiceBasedCommonStockUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ServiceBasedCommonStockUnitsMember" xlink:to="lab_gemi_ServiceBasedCommonStockUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_OtherPerfMeasureAmt_terseLabel_en-US" xlink:label="lab_ecd_OtherPerfMeasureAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other Performance Measure, Amount</link:label>
    <link:label id="lab_ecd_OtherPerfMeasureAmt_label_en-US" xlink:label="lab_ecd_OtherPerfMeasureAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Performance Measure, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_OtherPerfMeasureAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_OtherPerfMeasureAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_OtherPerfMeasureAmt" xlink:to="lab_ecd_OtherPerfMeasureAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward_5f7a68b1-b449-4044-9588-f62b0557783a_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Weighted-Average Exercise Price</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Exercise Price [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsOutstandingWeightedAverageExercisePriceRollforward" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan15Member_52225172-aa77-42f0-acb9-b7f5bc3a4d8d_terseLabel_en-US" xlink:label="lab_gemi_Loan15Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 15</link:label>
    <link:label id="lab_gemi_Loan15Member_label_en-US" xlink:label="lab_gemi_Loan15Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 15 [Member]</link:label>
    <link:label id="lab_gemi_Loan15Member_documentation_en-US" xlink:label="lab_gemi_Loan15Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 15</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan15Member" xlink:href="gemi-20250930.xsd#gemi_Loan15Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan15Member" xlink:to="lab_gemi_Loan15Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RelatedPartyDomain_57f8f117-701f-41a2-b55d-a87cd3d9a522_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related and Nonrelated Parties [Domain]</link:label>
    <link:label id="lab_us-gaap_RelatedPartyDomain_label_en-US" xlink:label="lab_us-gaap_RelatedPartyDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related and Nonrelated Parties [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyDomain" xlink:to="lab_us-gaap_RelatedPartyDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureAbstract_label_en-US" xlink:label="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Deferred Costs, Capitalized, Prepaid, and Other Assets Disclosure [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureAbstract" xlink:to="lab_us-gaap_DeferredCostsCapitalizedPrepaidAndOtherAssetsDisclosureAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_GalaxyDigitalLLCMember_031cfff4-ecea-486d-a365-eaad1f7ca5e2_terseLabel_en-US" xlink:label="lab_gemi_GalaxyDigitalLLCMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Galaxy Digital LLC</link:label>
    <link:label id="lab_gemi_GalaxyDigitalLLCMember_label_en-US" xlink:label="lab_gemi_GalaxyDigitalLLCMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Galaxy Digital LLC [Member]</link:label>
    <link:label id="lab_gemi_GalaxyDigitalLLCMember_documentation_en-US" xlink:label="lab_gemi_GalaxyDigitalLLCMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Galaxy Digital LLC</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_GalaxyDigitalLLCMember" xlink:href="gemi-20250930.xsd#gemi_GalaxyDigitalLLCMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_GalaxyDigitalLLCMember" xlink:to="lab_gemi_GalaxyDigitalLLCMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativeInstrumentRiskAxis_4a906991-b7ab-4d42-9ea2-abe0d8a9f69b_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeInstrumentRiskAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative Instrument [Axis]</link:label>
    <link:label id="lab_us-gaap_DerivativeInstrumentRiskAxis_label_en-US" xlink:label="lab_us-gaap_DerivativeInstrumentRiskAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Instrument [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeInstrumentRiskAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeInstrumentRiskAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeInstrumentRiskAxis" xlink:to="lab_us-gaap_DerivativeInstrumentRiskAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan2Member_d7f4eb9b-b0f0-4401-acf9-bbe5d2cc507f_terseLabel_en-US" xlink:label="lab_gemi_Loan2Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 2</link:label>
    <link:label id="lab_gemi_Loan2Member_label_en-US" xlink:label="lab_gemi_Loan2Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 2 [Member]</link:label>
    <link:label id="lab_gemi_Loan2Member_documentation_en-US" xlink:label="lab_gemi_Loan2Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 2</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan2Member" xlink:href="gemi-20250930.xsd#gemi_Loan2Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan2Member" xlink:to="lab_gemi_Loan2Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CommonClassAVestedIncentiveUnitsMember_5ac55177-6ab6-4c92-8125-c15c718d6149_terseLabel_en-US" xlink:label="lab_gemi_CommonClassAVestedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common Class A, Vested Incentive Units</link:label>
    <link:label id="lab_gemi_CommonClassAVestedIncentiveUnitsMember_label_en-US" xlink:label="lab_gemi_CommonClassAVestedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Class A, Vested Incentive Units [Member]</link:label>
    <link:label id="lab_gemi_CommonClassAVestedIncentiveUnitsMember_documentation_en-US" xlink:label="lab_gemi_CommonClassAVestedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Class A, Vested Incentive Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CommonClassAVestedIncentiveUnitsMember" xlink:href="gemi-20250930.xsd#gemi_CommonClassAVestedIncentiveUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CommonClassAVestedIncentiveUnitsMember" xlink:to="lab_gemi_CommonClassAVestedIncentiveUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_VestingAxis_bb999294-1974-413d-b3d1-3e8c870252f5_terseLabel_en-US" xlink:label="lab_us-gaap_VestingAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Vesting [Axis]</link:label>
    <link:label id="lab_us-gaap_VestingAxis_label_en-US" xlink:label="lab_us-gaap_VestingAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Vesting [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_VestingAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_VestingAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_VestingAxis" xlink:to="lab_us-gaap_VestingAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherAssetsCurrent_04785ec5-ac6f-484d-a854-302d18fae729_terseLabel_en-US" xlink:label="lab_us-gaap_OtherAssetsCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other</link:label>
    <link:label id="lab_us-gaap_OtherAssetsCurrent_label_en-US" xlink:label="lab_us-gaap_OtherAssetsCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Assets, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherAssetsCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherAssetsCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherAssetsCurrent" xlink:to="lab_us-gaap_OtherAssetsCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityDomain_7fbc3ac6-1b10-4480-a2e5-8a30ec7e8c56_terseLabel_en-US" xlink:label="lab_dei_EntityDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity [Domain]</link:label>
    <link:label id="lab_dei_EntityDomain_label_en-US" xlink:label="lab_dei_EntityDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityDomain" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityDomain" xlink:to="lab_dei_EntityDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CustomerCustodialFundsFairValueDisclosure_87ffc0ad-c4b4-4d4b-9e0c-d342c6e593f3_terseLabel_en-US" xlink:label="lab_gemi_CustomerCustodialFundsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Customer custodial funds</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsFairValueDisclosure_label_en-US" xlink:label="lab_gemi_CustomerCustodialFundsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_CustomerCustodialFundsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CustomerCustodialFundsFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_CustomerCustodialFundsFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CustomerCustodialFundsFairValueDisclosure" xlink:to="lab_gemi_CustomerCustodialFundsFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInInvestingActivities_ed79192b-8e31-47f7-8f3d-526b11d1d8cc_totalLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net cash provided by investing activities</link:label>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInInvestingActivities_label_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Investing Activity, Including Discontinued Operation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:to="lab_us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember_terseLabel_en-US" xlink:label="lab_ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate Change in Present Value of Accumulated Benefit for All Pension Plans Reported in Summary Compensation Table</link:label>
    <link:label id="lab_ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember_label_en-US" xlink:label="lab_ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Aggregate Change in Present Value of Accumulated Benefit for All Pension Plans Reported in Summary Compensation Table [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember" xlink:to="lab_ecd_AggtChngPnsnValInSummryCompstnTblForAplblYrMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A3400ETHMarch102025Member_4bc6f32e-9219-4081-8883-18e0e631e222_terseLabel_en-US" xlink:label="lab_gemi_A3400ETHMarch102025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">3,400 ETH March 10, 2025</link:label>
    <link:label id="lab_gemi_A3400ETHMarch102025Member_label_en-US" xlink:label="lab_gemi_A3400ETHMarch102025Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">3,400 ETH March 10, 2025 [Member]</link:label>
    <link:label id="lab_gemi_A3400ETHMarch102025Member_documentation_en-US" xlink:label="lab_gemi_A3400ETHMarch102025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">3,400 ETH March 10, 2025</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A3400ETHMarch102025Member" xlink:href="gemi-20250930.xsd#gemi_A3400ETHMarch102025Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A3400ETHMarch102025Member" xlink:to="lab_gemi_A3400ETHMarch102025Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InterestPaidNet_de3b7a1b-9092-4d86-8840-187c7edbdda3_terseLabel_en-US" xlink:label="lab_us-gaap_InterestPaidNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash paid during the period for interest</link:label>
    <link:label id="lab_us-gaap_InterestPaidNet_label_en-US" xlink:label="lab_us-gaap_InterestPaidNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Paid, Excluding Capitalized Interest, Operating Activity</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestPaidNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestPaidNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestPaidNet" xlink:to="lab_us-gaap_InterestPaidNet" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_DocumentFiscalYearFocus_069a9037-4ab9-4efa-bfb8-2b650e81b700_terseLabel_en-US" xlink:label="lab_dei_DocumentFiscalYearFocus" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Document Fiscal Year Focus</link:label>
    <link:label id="lab_dei_DocumentFiscalYearFocus_label_en-US" xlink:label="lab_dei_DocumentFiscalYearFocus" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Document Fiscal Year Focus</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_DocumentFiscalYearFocus" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_DocumentFiscalYearFocus"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_DocumentFiscalYearFocus" xlink:to="lab_dei_DocumentFiscalYearFocus" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BasisOfAccountingPolicyPolicyTextBlock_bbd90282-8b07-4059-86fb-366142056779_terseLabel_en-US" xlink:label="lab_us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Basis of presentation</link:label>
    <link:label id="lab_us-gaap_BasisOfAccountingPolicyPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Basis of Accounting, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BasisOfAccountingPolicyPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xlink:to="lab_us-gaap_BasisOfAccountingPolicyPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseRightOfUseAsset_5ecb9099-a596-42e1-aaee-828913229006_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseRightOfUseAsset" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Lease right-of-use assets</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseRightOfUseAsset_b1ab8695-a22e-4fed-8afe-b7003c838f5f_verboseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseRightOfUseAsset" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Right-of-use asset</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseRightOfUseAsset_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseRightOfUseAsset" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Right-of-Use Asset</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseRightOfUseAsset" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseRightOfUseAsset"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseRightOfUseAsset" xlink:to="lab_us-gaap_OperatingLeaseRightOfUseAsset" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EffectiveIncomeTaxRateContinuingOperations_2c3166b8-5b47-49df-81ad-52fadfcc1731_terseLabel_en-US" xlink:label="lab_us-gaap_EffectiveIncomeTaxRateContinuingOperations" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Effective tax rate</link:label>
    <link:label id="lab_us-gaap_EffectiveIncomeTaxRateContinuingOperations_label_en-US" xlink:label="lab_us-gaap_EffectiveIncomeTaxRateContinuingOperations" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Effective Income Tax Rate Reconciliation, Percent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EffectiveIncomeTaxRateContinuingOperations" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EffectiveIncomeTaxRateContinuingOperations"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EffectiveIncomeTaxRateContinuingOperations" xlink:to="lab_us-gaap_EffectiveIncomeTaxRateContinuingOperations" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseCost_58b02c27-9d7e-462c-bf6e-fb18f036099e_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseCost" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating lease expense</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseCost_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseCost" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Cost</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseCost" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseCost"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseCost" xlink:to="lab_us-gaap_OperatingLeaseCost" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PaymentsForSoftware_c637f1da-2693-4b6c-bdbd-d4ed12a38afa_terseLabel_en-US" xlink:label="lab_us-gaap_PaymentsForSoftware" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Additions to software</link:label>
    <link:label id="lab_us-gaap_PaymentsForSoftware_label_en-US" xlink:label="lab_us-gaap_PaymentsForSoftware" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payments for Software</link:label>
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    <link:label id="lab_gemi_Loan1Member_label_en-US" xlink:label="lab_gemi_Loan1Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 1 [Member]</link:label>
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    <link:label id="lab_gemi_LesseeOperatingLeaseLiabilityToBePaidDueAfterYearFour_50283d5f-a4a6-4e7f-9835-c7827032ec9c_terseLabel_en-US" xlink:label="lab_gemi_LesseeOperatingLeaseLiabilityToBePaidDueAfterYearFour" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Thereafter</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_LesseeOperatingLeaseLiabilityToBePaidDueAfterYearFour" xlink:to="lab_gemi_LesseeOperatingLeaseLiabilityToBePaidDueAfterYearFour" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised_5191bf5b-d305-473f-934c-c96b3007dfbd_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Vested (in shares)</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockIssuedDuringPeriodSharesStockOptionsExercised"/>
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    <link:label id="lab_us-gaap_EquityAbstract_label_en-US" xlink:label="lab_us-gaap_EquityAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquityAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquityAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EquityAbstract" xlink:to="lab_us-gaap_EquityAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount_fa8e5ad6-eaad-434d-8a5a-10f8e765cf5e_terseLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cumulative upward adjustments for strategic investments</link:label>
    <link:label id="lab_us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount_label_en-US" xlink:label="lab_us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Securities without Readily Determinable Fair Value, Upward Price Adjustment, Cumulative Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount" xlink:to="lab_us-gaap_EquitySecuritiesWithoutReadilyDeterminableFairValueUpwardPriceAdjustmentCumulativeAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PurchaseObligation_c885eb89-c542-4bec-b744-f40385d0edd5_terseLabel_en-US" xlink:label="lab_us-gaap_PurchaseObligation" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Outstanding purchase commitments</link:label>
    <link:label id="lab_us-gaap_PurchaseObligation_label_en-US" xlink:label="lab_us-gaap_PurchaseObligation" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Purchase Obligation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PurchaseObligation" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PurchaseObligation"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PurchaseObligation" xlink:to="lab_us-gaap_PurchaseObligation" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativesPolicyTextBlock_30061ed8-638c-4bad-b565-3a1b08a36c57_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativesPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Warrants</link:label>
    <link:label id="lab_us-gaap_DerivativesPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_DerivativesPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivatives, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativesPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativesPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativesPolicyTextBlock" xlink:to="lab_us-gaap_DerivativesPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod_d29e83ae-eb09-4a09-891e-7bbffe5fd708_verboseLabel_en-US" xlink:label="lab_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Amount of crypto repaid (in units)</link:label>
    <link:label id="lab_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod_b1e1d9d1-9d51-4f39-a2c7-eca834a8f857_terseLabel_en-US" xlink:label="lab_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amount of crypto assets loaned, that were repaid during the period (in units)</link:label>
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    <link:label id="lab_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod_documentation_en-US" xlink:label="lab_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Instrument, Amount Of Crypto Currency Units Loaned, Repaid During The Period</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod" xlink:to="lab_gemi_DebtInstrumentAmountOfCryptoCurrencyUnitsLoanedRepaidDuringThePeriod" xlink:type="arc" order="1"/>
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    <link:label id="lab_gemi_A2000BTCMarch12023Member_71f3001b-6791-483f-b732-2f90af0f774a_terseLabel_en-US" xlink:label="lab_gemi_A2000BTCMarch12023Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2,000 BTC March 1, 2023</link:label>
    <link:label id="lab_gemi_A2000BTCMarch12023Member_label_en-US" xlink:label="lab_gemi_A2000BTCMarch12023Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">2,000 BTC March 1, 2023 [Member]</link:label>
    <link:label id="lab_gemi_A2000BTCMarch12023Member_documentation_en-US" xlink:label="lab_gemi_A2000BTCMarch12023Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">2,000 BTC March 1, 2023</link:label>
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    <link:label id="lab_us-gaap_IncreaseDecreaseInStockholdersEquityRollForward_5bd0ac23-78bd-4998-8876-ebb175ae6d94_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInStockholdersEquityRollForward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Stockholders' Equity [Roll Forward]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsLineItems" xlink:to="lab_us-gaap_FiniteLivedIntangibleAssetsLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure_cef903ac-c634-4e2a-82fe-75c71144c785_terseLabel_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Receivable, crypto assets pledged:</link:label>
    <link:label id="lab_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure_label_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Receivable, Crypto Assets Pledged, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Receivable, Crypto Assets Pledged, Fair Value Disclsoure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure" xlink:to="lab_gemi_ReceivableCryptoAssetsPledgedFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Total Shareholder Return</link:label>
    <link:label id="lab_ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock_label_en-US" xlink:label="lab_ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Total Shareholder Return [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock" xlink:to="lab_ecd_CompActuallyPaidVsTotalShareholderRtnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable_e4709b23-d589-4388-8c92-cc7c458a513e_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, Recurring and Nonrecurring [Table]</link:label>
    <link:label id="lab_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable_label_en-US" xlink:label="lab_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Recurring and Nonrecurring [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xlink:to="lab_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AmortizationOfIntangibleAssets_d65241a3-11fe-42c1-89a6-e9e80eec5f23_terseLabel_en-US" xlink:label="lab_us-gaap_AmortizationOfIntangibleAssets" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amortization of intangible assets</link:label>
    <link:label id="lab_us-gaap_AmortizationOfIntangibleAssets_label_en-US" xlink:label="lab_us-gaap_AmortizationOfIntangibleAssets" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Amortization of Intangible Assets</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AmortizationOfIntangibleAssets" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AmortizationOfIntangibleAssets"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AmortizationOfIntangibleAssets" xlink:to="lab_us-gaap_AmortizationOfIntangibleAssets" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue_68796cdd-42b3-440a-b9c8-74afebb322bd_terseLabel_en-US" xlink:label="lab_us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue_label_en-US" xlink:label="lab_us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Cash and Cash Equivalent, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue" xlink:to="lab_us-gaap_RestrictedCashAndCashEquivalentsAtCarryingValue" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost_529b8a5b-4d9c-4b8b-aceb-529a69f6d460_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cumulative compensation cost</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost_label_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Cumulative Compensation Cost</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost_documentation_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Cumulative Compensation Cost</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost" xlink:href="gemi-20250930.xsd#gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost" xlink:to="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardCumulativeCompensationCost" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DepreciationDepletionAndAmortization_ea1ff3f8-5eb5-49c6-bf63-1c2c305344f9_terseLabel_en-US" xlink:label="lab_us-gaap_DepreciationDepletionAndAmortization" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Depreciation and amortization</link:label>
    <link:label id="lab_us-gaap_DepreciationDepletionAndAmortization_label_en-US" xlink:label="lab_us-gaap_DepreciationDepletionAndAmortization" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Depreciation, Depletion and Amortization</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DepreciationDepletionAndAmortization" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DepreciationDepletionAndAmortization"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DepreciationDepletionAndAmortization" xlink:to="lab_us-gaap_DepreciationDepletionAndAmortization" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_NumberOfAssetAcquisitions_3ae363ba-625e-4833-b1d0-3c0b8778b60e_terseLabel_en-US" xlink:label="lab_gemi_NumberOfAssetAcquisitions" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of asset acquisitions</link:label>
    <link:label id="lab_gemi_NumberOfAssetAcquisitions_label_en-US" xlink:label="lab_gemi_NumberOfAssetAcquisitions" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Number Of Asset Acquisitions</link:label>
    <link:label id="lab_gemi_NumberOfAssetAcquisitions_documentation_en-US" xlink:label="lab_gemi_NumberOfAssetAcquisitions" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Number Of Asset Acquisitions</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_NumberOfAssetAcquisitions" xlink:href="gemi-20250930.xsd#gemi_NumberOfAssetAcquisitions"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_NumberOfAssetAcquisitions" xlink:to="lab_gemi_NumberOfAssetAcquisitions" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SaleOfStockConsiderationReceivedOnTransaction_9985d886-349e-4b58-911c-a76810ddfed0_terseLabel_en-US" xlink:label="lab_us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net proceeds from stock issued and sold</link:label>
    <link:label id="lab_us-gaap_SaleOfStockConsiderationReceivedOnTransaction_label_en-US" xlink:label="lab_us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Sale of Stock, Consideration Received on Transaction</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SaleOfStockConsiderationReceivedOnTransaction"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xlink:to="lab_us-gaap_SaleOfStockConsiderationReceivedOnTransaction" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CustomerCustodialFundsGeminiDollar_b593b93e-540d-41af-ad3c-6beb2bf063c2_verboseLabel_en-US" xlink:label="lab_gemi_CustomerCustodialFundsGeminiDollar" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Customer custodial funds related to GUSD</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsGeminiDollar_label_en-US" xlink:label="lab_gemi_CustomerCustodialFundsGeminiDollar" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds, Gemini Dollar</link:label>
    <link:label id="lab_gemi_CustomerCustodialFundsGeminiDollar_documentation_en-US" xlink:label="lab_gemi_CustomerCustodialFundsGeminiDollar" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Customer Custodial Funds, Gemini Dollar</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CustomerCustodialFundsGeminiDollar" xlink:href="gemi-20250930.xsd#gemi_CustomerCustodialFundsGeminiDollar"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CustomerCustodialFundsGeminiDollar" xlink:to="lab_gemi_CustomerCustodialFundsGeminiDollar" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet_3573f14a-8ead-4701-9c9b-83b07eb158b2_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets posted (returned) as collateral for third party loans, net</link:label>
    <link:label id="lab_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet_label_en-US" xlink:label="lab_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Assets Posted (Returned) As Collateral For Third Party Debt, Net</link:label>
    <link:label id="lab_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet_documentation_en-US" xlink:label="lab_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Assets Posted (Returned) As Collateral For Third Party Debt, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet" xlink:to="lab_gemi_CryptoAssetsPostedReturnedAsCollateralForThirdPartyDebtNet" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent_8f91c3c5-6968-4c88-820b-b7ae75e620b5_terseLabel_en-US" xlink:label="lab_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party loans</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent_label_en-US" xlink:label="lab_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Debt, Including Accrued Interest, Current</link:label>
    <link:label id="lab_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent_documentation_en-US" xlink:label="lab_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Debt, Including Accrued Interest, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent" xlink:href="gemi-20250930.xsd#gemi_RelatedPartyDebtIncludingAccruedInterestCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent" xlink:to="lab_gemi_RelatedPartyDebtIncludingAccruedInterestCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt_3826f110-f3ce-4f70-a91e-77f09b84622f_terseLabel_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value on related party convertible notes</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt_8116dd32-2827-4d67-9756-679fc9ce4a86_negatedTerseLabel_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value on related party convertible notes</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt_4dbe3df5-18f8-46a4-8f9c-004b2157a414_negatedLabel_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Unrealized Loss</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt_label_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Gain (Loss) On Change In Fair Value Of Related Party Convertible Debt</link:label>
    <link:label id="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt_documentation_en-US" xlink:label="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Gain (Loss) On Change In Fair Value Of Related Party Convertible Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:href="gemi-20250930.xsd#gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:to="lab_gemi_GainLossOnChangeInFairValueOfRelatedPartyConvertibleDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrDuration_terseLabel_en-US" xlink:label="lab_ecd_TrdArrDuration" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Arrangement Duration</link:label>
    <link:label id="lab_ecd_TrdArrDuration_label_en-US" xlink:label="lab_ecd_TrdArrDuration" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement Duration</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrDuration" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrDuration"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TrdArrDuration" xlink:to="lab_ecd_TrdArrDuration" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ServiceCustodialFeeRevenueMember_42ee111b-a698-4399-b4d4-01fb8499f0fe_terseLabel_en-US" xlink:label="lab_gemi_ServiceCustodialFeeRevenueMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Custodial fee revenue</link:label>
    <link:label id="lab_gemi_ServiceCustodialFeeRevenueMember_label_en-US" xlink:label="lab_gemi_ServiceCustodialFeeRevenueMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Service, Custodial Fee Revenue [Member]</link:label>
    <link:label id="lab_gemi_ServiceCustodialFeeRevenueMember_documentation_en-US" xlink:label="lab_gemi_ServiceCustodialFeeRevenueMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Service, Custodial Fee Revenue</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ServiceCustodialFeeRevenueMember" xlink:href="gemi-20250930.xsd#gemi_ServiceCustodialFeeRevenueMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ServiceCustodialFeeRevenueMember" xlink:to="lab_gemi_ServiceCustodialFeeRevenueMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain_c323fabe-3df4-4ce0-95ce-f24adf5c9d15_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value by Liability Class [Domain]</link:label>
    <link:label id="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain_label_en-US" xlink:label="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value by Liability Class [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain" xlink:to="lab_us-gaap_FairValueLiabilitiesMeasuredOnRecurringBasisUnobservableInputReconciliationByLiabilityClassDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RepaymentsOfRelatedPartyDebtAndRelatedPartyDebtDenominatedInCryptoAssets_545c7bf3-f808-4853-8966-9e5423e43cf8_verboseLabel_en-US" xlink:label="lab_gemi_RepaymentsOfRelatedPartyDebtAndRelatedPartyDebtDenominatedInCryptoAssets" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Amount Repaid</link:label>
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    <link:label id="lab_dei_LocalPhoneNumber_566db8f5-c477-49ac-b88e-9e7bb62d9557_terseLabel_en-US" xlink:label="lab_dei_LocalPhoneNumber" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Local Phone Number</link:label>
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    <link:label id="lab_ecd_Additional402vDisclosureTextBlock_terseLabel_en-US" xlink:label="lab_ecd_Additional402vDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Additional 402(v) Disclosure</link:label>
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    <link:label id="lab_us-gaap_ArrangementsAndNonarrangementTransactionsMember_a6ba8f56-55f3-4eed-abe9-bd3ccfebfaba_terseLabel_en-US" xlink:label="lab_us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement and Arrangement Other than Collaborative [Domain]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ArrangementsAndNonarrangementTransactionsMember"/>
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    <link:label id="lab_us-gaap_AccountsReceivableGrossCurrent_fceaa972-85f5-4547-a8b6-1070878460c5_terseLabel_en-US" xlink:label="lab_us-gaap_AccountsReceivableGrossCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounts receivable, before allowances</link:label>
    <link:label id="lab_us-gaap_AccountsReceivableGrossCurrent_label_en-US" xlink:label="lab_us-gaap_AccountsReceivableGrossCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts Receivable, before Allowance for Credit Loss, Current</link:label>
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    <link:label id="lab_us-gaap_ContractWithCustomerAssetNetNoncurrent_99c1fd2c-d9eb-4a73-b56e-10891839fb99_verboseLabel_en-US" xlink:label="lab_us-gaap_ContractWithCustomerAssetNetNoncurrent" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Warrant receivable</link:label>
    <link:label id="lab_us-gaap_ContractWithCustomerAssetNetNoncurrent_label_en-US" xlink:label="lab_us-gaap_ContractWithCustomerAssetNetNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Contract with Customer, Asset, after Allowance for Credit Loss, Noncurrent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ContractWithCustomerAssetNetNoncurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ContractWithCustomerAssetNetNoncurrent"/>
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    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPricePurchaseDate_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPricePurchaseDate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Discount from Market Price, Purchase Date</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPricePurchaseDate" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPricePurchaseDate"/>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock"/>
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    <link:label id="lab_us-gaap_AssetsAbstract_fdb28926-64cf-4fb5-8bd9-9baf686bc01e_terseLabel_en-US" xlink:label="lab_us-gaap_AssetsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Assets</link:label>
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    <link:label id="lab_us-gaap_CreditFacilityAxis_e8eee78c-fb17-45e6-8b5b-60e375f1800b_terseLabel_en-US" xlink:label="lab_us-gaap_CreditFacilityAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit Facility [Axis]</link:label>
    <link:label id="lab_us-gaap_CreditFacilityAxis_label_en-US" xlink:label="lab_us-gaap_CreditFacilityAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Facility [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CreditFacilityAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CreditFacilityAxis"/>
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    <link:label id="lab_us-gaap_CryptoAssetUnrealizedGainLossOperating_41f1ee34-bdc8-4246-ac42-d022a5ed0b51_verboseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetUnrealizedGainLossOperating" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Unrealized gain (loss) on crypto assets</link:label>
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    <link:label id="lab_us-gaap_DebtInstrumentAxis_c5f8b029-f35b-43df-8912-eb98cc267cf2_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Instrument [Axis]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentAxis"/>
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    <link:label id="lab_gemi_FiniteLivedIntangibleAssetExpectedAmortizationAfterYearFour_cade8efd-2d48-44a6-b249-69f3016af3a0_terseLabel_en-US" xlink:label="lab_gemi_FiniteLivedIntangibleAssetExpectedAmortizationAfterYearFour" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Thereafter</link:label>
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    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeituresWeightedAverageGrantDateFairValue_6d15b43b-5e7d-44e7-8b94-67fb040bdbe4_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsForfeituresWeightedAverageGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Forfeited (in dollars per share)</link:label>
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    <link:label id="lab_us-gaap_CreditFacilityDomain_label_en-US" xlink:label="lab_us-gaap_CreditFacilityDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Facility [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CreditFacilityDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CreditFacilityDomain"/>
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    <link:label id="lab_us-gaap_StockholdersEquityAbstract_8064e28d-ea21-420b-9b78-3db7b9703fee_terseLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquityAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stockholders' equity</link:label>
    <link:label id="lab_us-gaap_StockholdersEquityAbstract_label_en-US" xlink:label="lab_us-gaap_StockholdersEquityAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity, Attributable to Parent [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockholdersEquityAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockholdersEquityAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockholdersEquityAbstract" xlink:to="lab_us-gaap_StockholdersEquityAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativeLiabilitiesCurrent_2a4db887-db33-4a03-8c1a-e35fbae7f9a8_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative liabilities</link:label>
    <link:label id="lab_us-gaap_DerivativeLiabilitiesCurrent_label_en-US" xlink:label="lab_us-gaap_DerivativeLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Liability, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeLiabilitiesCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeLiabilitiesCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeLiabilitiesCurrent" xlink:to="lab_us-gaap_DerivativeLiabilitiesCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage_72dd6f4b-6540-406d-aeb4-3983a9db150f_terseLabel_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Vesting percentage</link:label>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage_label_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Award Vesting Rights, Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage" xlink:to="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardAwardVestingRightsPercentage" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CryptoAssetActivityTableTextBlock_d243eb81-2fe0-4fa2-80d2-dc26a211ca50_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Fair Value Roll-forward of Crypto Assets Held at Fair Value</link:label>
    <link:label id="lab_us-gaap_CryptoAssetActivityTableTextBlock_label_en-US" xlink:label="lab_us-gaap_CryptoAssetActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Activity [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetActivityTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetActivityTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetActivityTableTextBlock" xlink:to="lab_us-gaap_CryptoAssetActivityTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage_763124ee-d4db-4f2f-8e33-289ea87b6854_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stated interest rate</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage_0c629a64-ec0f-48c1-92ac-c35b357b476b_verboseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Interest Rate</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Interest Rate, Stated Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentInterestRateStatedPercentage"/>
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    <link:label id="lab_us-gaap_OperatingLeaseLiabilityNoncurrent_4f49bf58-6e7e-4f3b-b290-7d879e777b94_verboseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiabilityNoncurrent" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Lease liabilities</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseLiabilityNoncurrent_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiabilityNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Liability, Noncurrent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseLiabilityNoncurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseLiabilityNoncurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseLiabilityNoncurrent" xlink:to="lab_us-gaap_OperatingLeaseLiabilityNoncurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_WarrantFairValueDisclosure_25c7ff39-dad0-412e-b2c5-a25d1e3d57e8_terseLabel_en-US" xlink:label="lab_gemi_WarrantFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Warrants</link:label>
    <link:label id="lab_gemi_WarrantFairValueDisclosure_label_en-US" xlink:label="lab_gemi_WarrantFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Warrant, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_WarrantFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_WarrantFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Warrants, Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_WarrantFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_WarrantFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_WarrantFairValueDisclosure" xlink:to="lab_gemi_WarrantFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating_7ccc9530-fc87-4310-8394-6a5c30880073_terseLabel_en-US" xlink:label="lab_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Unrealized Gain (Loss)</link:label>
    <link:label id="lab_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating_label_en-US" xlink:label="lab_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Loans Payable, Unrealized Gain (Loss), Nonoperating</link:label>
    <link:label id="lab_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating_documentation_en-US" xlink:label="lab_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Loans Payable, Unrealized Gain (Loss), Nonoperating</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating" xlink:href="gemi-20250930.xsd#gemi_CryptoLoansPayableUnrealizedGainLossNonoperating"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating" xlink:to="lab_gemi_CryptoLoansPayableUnrealizedGainLossNonoperating" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod_6884b1e2-671f-43cb-aa9f-3620893f8b17_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accumulated payroll deduction offering period</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod_label_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Discount From Market Price, Offering Period</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod_documentation_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Discount From Market Price, Offering Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod" xlink:href="gemi-20250930.xsd#gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod" xlink:to="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardDiscountFromMarketPriceOfferingPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A4150ETHJuly52024Member_3c9d69e9-9a11-47ec-9673-d6606f5c1289_terseLabel_en-US" xlink:label="lab_gemi_A4150ETHJuly52024Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">4,150 ETH, July 5, 2024</link:label>
    <link:label id="lab_gemi_A4150ETHJuly52024Member_label_en-US" xlink:label="lab_gemi_A4150ETHJuly52024Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">4,150 ETH, July 5, 2024 [Member]</link:label>
    <link:label id="lab_gemi_A4150ETHJuly52024Member_documentation_en-US" xlink:label="lab_gemi_A4150ETHJuly52024Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">4,150 ETH, July 5, 2024</link:label>
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    <link:label id="lab_us-gaap_PropertyPlantAndEquipmentAbstract_label_en-US" xlink:label="lab_us-gaap_PropertyPlantAndEquipmentAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Property, Plant and Equipment [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PropertyPlantAndEquipmentAbstract" xlink:to="lab_us-gaap_PropertyPlantAndEquipmentAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityRegistrantName_d1862e4e-cd61-4e9d-be80-9d29fea7fd3b_terseLabel_en-US" xlink:label="lab_dei_EntityRegistrantName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Registrant Name</link:label>
    <link:label id="lab_dei_EntityRegistrantName_label_en-US" xlink:label="lab_dei_EntityRegistrantName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Registrant Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityRegistrantName" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityRegistrantName"/>
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    <link:label id="lab_us-gaap_LesseeOperatingLeaseDiscountRate_c2544b27-3f24-4321-9f7b-ec1a40065b87_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseDiscountRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Discount rate</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseDiscountRate_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseDiscountRate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Discount Rate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseDiscountRate" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseDiscountRate"/>
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    <link:label id="lab_us-gaap_TemporaryEquitySharesOutstanding_ca6bad13-cb6d-4f87-a6a6-f339740d7158_terseLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible preferred units, shares outstanding (in shares)</link:label>
    <link:label id="lab_us-gaap_TemporaryEquitySharesOutstanding_858b531a-98cb-44db-820c-357b8cff5995_periodStartLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Beginning balance (in shares)</link:label>
    <link:label id="lab_us-gaap_TemporaryEquitySharesOutstanding_ca783fc9-7b32-41c1-a12e-d4e63143dcbb_periodEndLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Ending balance (in shares)</link:label>
    <link:label id="lab_us-gaap_TemporaryEquitySharesOutstanding_label_en-US" xlink:label="lab_us-gaap_TemporaryEquitySharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Temporary Equity, Shares Outstanding</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TemporaryEquitySharesOutstanding" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TemporaryEquitySharesOutstanding"/>
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    <link:label id="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock_terseLabel_en-US" xlink:label="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock Price or TSR Estimation Method</link:label>
    <link:label id="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock_label_en-US" xlink:label="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Price or TSR Estimation Method [Text Block]</link:label>
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    <link:label id="lab_us-gaap_InterestExpenseOther_13055264-ee2e-4455-b814-0bd6601a9354_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_InterestExpenseOther" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Interest expense on funding debt</link:label>
    <link:label id="lab_us-gaap_InterestExpenseOther_label_en-US" xlink:label="lab_us-gaap_InterestExpenseOther" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Expense, Other</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestExpenseOther" xlink:to="lab_us-gaap_InterestExpenseOther" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan11Member_cc805cc3-e26a-4924-9b01-6be6ab89b8f0_terseLabel_en-US" xlink:label="lab_gemi_Loan11Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 11</link:label>
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    <link:label id="lab_gemi_Loan11Member_documentation_en-US" xlink:label="lab_gemi_Loan11Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 11</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan11Member" xlink:href="gemi-20250930.xsd#gemi_Loan11Member"/>
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    <link:label id="lab_us-gaap_CommonUnitIssuanceValue_2095875b-4cb0-4865-adb9-53aa2ca940b8_terseLabel_en-US" xlink:label="lab_us-gaap_CommonUnitIssuanceValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common Units: 54,201 Common Units authorized as of December&#160;31, 2024; 50,761 Common Units issued and outstanding as of December&#160;31, 2024. No shares authorized, issued, and outstanding as of September&#160;30, 2025.</link:label>
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    <link:label id="lab_gemi_ShareBasedPaymentArrangementOptionAndRestrictedStockUnitsRSUsMember_label_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementOptionAndRestrictedStockUnitsRSUsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Option And Restricted Stock Units (RSUs) [Member]</link:label>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementOptionAndRestrictedStockUnitsRSUsMember_documentation_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementOptionAndRestrictedStockUnitsRSUsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Option And Restricted Stock Units (RSUs)</link:label>
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    <link:label id="lab_us-gaap_CryptoAssetCost_0c3bfc8e-ccd9-4c85-b3a3-0ce8a35a74f0_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetCost" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cost Basis</link:label>
    <link:label id="lab_us-gaap_CryptoAssetCost_label_en-US" xlink:label="lab_us-gaap_CryptoAssetCost" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Cost</link:label>
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    <link:label id="lab_gemi_LitigationSettlementAggregateAmountReceived_5098a0ad-a948-413e-9396-2cd1dc16cccc_terseLabel_en-US" xlink:label="lab_gemi_LitigationSettlementAggregateAmountReceived" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Book value of aggregate amounts received</link:label>
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    <link:label id="lab_gemi_LitigationSettlementAggregateAmountReceived_documentation_en-US" xlink:label="lab_gemi_LitigationSettlementAggregateAmountReceived" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Litigation Settlement, Amount Received, Book Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_LitigationSettlementAggregateAmountReceived" xlink:href="gemi-20250930.xsd#gemi_LitigationSettlementAggregateAmountReceived"/>
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    <link:label id="lab_us-gaap_AccountsPayableAccruedLiabilitiesAndOtherLiabilitiesDisclosureCurrentTextBlock_4fd2b7a1-fe6a-4741-947f-7627ed32254f_terseLabel_en-US" xlink:label="lab_us-gaap_AccountsPayableAccruedLiabilitiesAndOtherLiabilitiesDisclosureCurrentTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued Expenses and Other Current Liabilities</link:label>
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    <link:label id="lab_us-gaap_StockIssuedDuringPeriodValueConversionOfConvertibleSecurities_208d40e1-e1e6-427e-a339-8776ce44961d_verboseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodValueConversionOfConvertibleSecurities" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Conversion in connection with IPO</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockIssuedDuringPeriodValueConversionOfConvertibleSecurities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockIssuedDuringPeriodValueConversionOfConvertibleSecurities"/>
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    <link:label id="lab_us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue_label_en-US" xlink:label="lab_us-gaap_AdjustmentsToAdditionalPaidInCapitalSharebasedCompensationRequisiteServicePeriodRecognitionValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">APIC, Share-Based Payment Arrangement, Increase for Cost Recognition</link:label>
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    <link:label id="lab_us-gaap_CryptoAssetPurchase_cfa0bcdc-d2c6-4648-81b4-063868584947_verboseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetPurchase" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets acquired</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetPurchase" xlink:to="lab_us-gaap_CryptoAssetPurchase" xlink:type="arc" order="1"/>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_LitigationCaseAxis" xlink:to="lab_srt_LitigationCaseAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueByLiabilityClassAxis_63b1e4fa-75b4-4a3a-8a10-76f4e4e28a06_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueByLiabilityClassAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Liability Class [Axis]</link:label>
    <link:label id="lab_us-gaap_FairValueByLiabilityClassAxis_label_en-US" xlink:label="lab_us-gaap_FairValueByLiabilityClassAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liability Class [Axis]</link:label>
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    <link:label id="lab_gemi_CryptoAssetCollateralPayableFairValueDisclosure_6a119ead-45a0-47c3-be20-40f428542cec_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetCollateralPayableFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset collateral payable</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetCollateralPayableFairValueDisclosure" xlink:to="lab_gemi_CryptoAssetCollateralPayableFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity_d1e35395-c952-43cf-a888-ccf0b25f336a_terseLabel_en-US" xlink:label="lab_us-gaap_LineOfCreditFacilityRemainingBorrowingCapacity" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Unused, available borrowing capacity</link:label>
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    <link:label id="lab_us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights_f5a5115f-b88f-4764-81a6-39e20bd8a39e_terseLabel_en-US" xlink:label="lab_us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of shares available to purchase from warrant rights (in shares)</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ClassOfWarrantOrRightNumberOfSecuritiesCalledByWarrantsOrRights"/>
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    <link:label id="lab_us-gaap_EarningsPerSharePolicyTextBlock_ad4649ca-edff-454f-9ab8-5aac4edfe478_terseLabel_en-US" xlink:label="lab_us-gaap_EarningsPerSharePolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net loss per share</link:label>
    <link:label id="lab_us-gaap_EarningsPerSharePolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_EarningsPerSharePolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Policy [Policy Text Block]</link:label>
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    <link:label id="lab_gemi_SeriesA1PreferredUnitsMember_0be2aa66-276c-45c3-a7a4-4b4ccf4ae286_terseLabel_en-US" xlink:label="lab_gemi_SeriesA1PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Series A-1</link:label>
    <link:label id="lab_gemi_SeriesA1PreferredUnitsMember_label_en-US" xlink:label="lab_gemi_SeriesA1PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Series A-1 Preferred Units [Member]</link:label>
    <link:label id="lab_gemi_SeriesA1PreferredUnitsMember_documentation_en-US" xlink:label="lab_gemi_SeriesA1PreferredUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Series A-1 Preferred Units</link:label>
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    <link:label id="lab_us-gaap_CryptoAssetNumberOfUnits_5f4bdf85-3dab-4a70-9e54-4863b39f0124_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Units (in unit)</link:label>
    <link:label id="lab_us-gaap_CryptoAssetNumberOfUnits_34efddc8-392f-4655-a384-16abe498dc02_verboseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Number of crypto assets held (in units)</link:label>
    <link:label id="lab_us-gaap_CryptoAssetNumberOfUnits_label_en-US" xlink:label="lab_us-gaap_CryptoAssetNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Number of Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CryptoAssetNumberOfUnits" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CryptoAssetNumberOfUnits"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CryptoAssetNumberOfUnits" xlink:to="lab_us-gaap_CryptoAssetNumberOfUnits" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PreferredStockValue_760ebae9-66d1-440f-88ee-70fcefe6a854_verboseLabel_en-US" xlink:label="lab_us-gaap_PreferredStockValue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Preferred stock, par value $0.001 per share; No shares authorized, issued and outstanding as of December&#160;31, 2024; 20,000 shares authorized, no shares issued and outstanding as of September&#160;30, 2025</link:label>
    <link:label id="lab_us-gaap_PreferredStockValue_label_en-US" xlink:label="lab_us-gaap_PreferredStockValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Stock, Value, Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredStockValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredStockValue"/>
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    <link:label id="lab_us-gaap_InterestPayableCurrentAndNoncurrent_66634c1c-9e32-4bbb-9f59-969cf2f350a7_terseLabel_en-US" xlink:label="lab_us-gaap_InterestPayableCurrentAndNoncurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest Payable</link:label>
    <link:label id="lab_us-gaap_InterestPayableCurrentAndNoncurrent_label_en-US" xlink:label="lab_us-gaap_InterestPayableCurrentAndNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Payable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestPayableCurrentAndNoncurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestPayableCurrentAndNoncurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestPayableCurrentAndNoncurrent" xlink:to="lab_us-gaap_InterestPayableCurrentAndNoncurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FinancingReceivablesPeriodPastDueAxis_557498e0-f92d-4607-ab20-e05c28ce1dba_terseLabel_en-US" xlink:label="lab_us-gaap_FinancingReceivablesPeriodPastDueAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Financial Asset, Aging [Axis]</link:label>
    <link:label id="lab_us-gaap_FinancingReceivablesPeriodPastDueAxis_label_en-US" xlink:label="lab_us-gaap_FinancingReceivablesPeriodPastDueAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Financial Asset, Aging [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivablesPeriodPastDueAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivablesPeriodPastDueAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FinancingReceivablesPeriodPastDueAxis" xlink:to="lab_us-gaap_FinancingReceivablesPeriodPastDueAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AllExecutiveCategoriesMember_terseLabel_en-US" xlink:label="lab_ecd_AllExecutiveCategoriesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Executive Categories</link:label>
    <link:label id="lab_ecd_AllExecutiveCategoriesMember_label_en-US" xlink:label="lab_ecd_AllExecutiveCategoriesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">All Executive Categories [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AllExecutiveCategoriesMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AllExecutiveCategoriesMember"/>
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    <link:label id="lab_us-gaap_CommonStockValue_878636b2-0052-40e5-bf19-14831344d30f_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common stock</link:label>
    <link:label id="lab_us-gaap_CommonStockValue_label_en-US" xlink:label="lab_us-gaap_CommonStockValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Value, Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockValue" xlink:to="lab_us-gaap_CommonStockValue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConvertibleDebtTableTextBlock_ac91c885-3061-4385-b559-ac312cbb0dc9_terseLabel_en-US" xlink:label="lab_us-gaap_ConvertibleDebtTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Convertible Notes</link:label>
    <link:label id="lab_us-gaap_ConvertibleDebtTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ConvertibleDebtTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Convertible Debt [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConvertibleDebtTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConvertibleDebtTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConvertibleDebtTableTextBlock" xlink:to="lab_us-gaap_ConvertibleDebtTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ProceedsFromRepurchaseAgreements_b6c6444d-b630-462c-bfe1-f8c301b3ba66_terseLabel_en-US" xlink:label="lab_gemi_ProceedsFromRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash proceeds</link:label>
    <link:label id="lab_gemi_ProceedsFromRepurchaseAgreements_label_en-US" xlink:label="lab_gemi_ProceedsFromRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds From Repurchase Agreements</link:label>
    <link:label id="lab_gemi_ProceedsFromRepurchaseAgreements_documentation_en-US" xlink:label="lab_gemi_ProceedsFromRepurchaseAgreements" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Proceeds From Repurchase Agreements</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ProceedsFromRepurchaseAgreements" xlink:href="gemi-20250930.xsd#gemi_ProceedsFromRepurchaseAgreements"/>
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    <link:label id="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTable_5c954a7d-9cbe-48ce-894b-5a8108d1faab_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, Assets Measured on Recurring Basis, Unobservable Input Reconciliation [Table]</link:label>
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    <link:label id="lab_srt_TitleOfIndividualAxis_ff0cd1a4-49b0-46d2-9955-1006ae68eb96_terseLabel_en-US" xlink:label="lab_srt_TitleOfIndividualAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Title and Position [Axis]</link:label>
    <link:label id="lab_srt_TitleOfIndividualAxis_label_en-US" xlink:label="lab_srt_TitleOfIndividualAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Title and Position [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_TitleOfIndividualAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_TitleOfIndividualAxis"/>
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    <link:label id="lab_us-gaap_CommitmentsAndContingencies_fa582a22-f5b0-4a28-8af8-51b4221b51b9_terseLabel_en-US" xlink:label="lab_us-gaap_CommitmentsAndContingencies" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Commitments and Contingencies</link:label>
    <link:label id="lab_us-gaap_CommitmentsAndContingencies_label_en-US" xlink:label="lab_us-gaap_CommitmentsAndContingencies" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Commitments and Contingencies</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommitmentsAndContingencies" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommitmentsAndContingencies"/>
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    <link:label id="lab_gemi_A240BTCMay12024Member_e7b52216-1c91-4a0e-9a07-70f49eb0764b_terseLabel_en-US" xlink:label="lab_gemi_A240BTCMay12024Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">240 BTC, May 1, 2024</link:label>
    <link:label id="lab_gemi_A240BTCMay12024Member_label_en-US" xlink:label="lab_gemi_A240BTCMay12024Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">240 BTC, May 1, 2024 [Member]</link:label>
    <link:label id="lab_gemi_A240BTCMay12024Member_documentation_en-US" xlink:label="lab_gemi_A240BTCMay12024Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">240 BTC, May 1, 2024</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A240BTCMay12024Member" xlink:href="gemi-20250930.xsd#gemi_A240BTCMay12024Member"/>
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    <link:label id="lab_gemi_A3000ETHFebruary72025Member_efd6d90b-dff2-4a1e-b7cb-38a7a3c0b160_terseLabel_en-US" xlink:label="lab_gemi_A3000ETHFebruary72025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">3,000 ETH, February 7, 2025</link:label>
    <link:label id="lab_gemi_A3000ETHFebruary72025Member_label_en-US" xlink:label="lab_gemi_A3000ETHFebruary72025Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">3,000 ETH, February 7, 2025 [Member]</link:label>
    <link:label id="lab_gemi_A3000ETHFebruary72025Member_documentation_en-US" xlink:label="lab_gemi_A3000ETHFebruary72025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">3,000 ETH, February 7, 2025</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A3000ETHFebruary72025Member" xlink:href="gemi-20250930.xsd#gemi_A3000ETHFebruary72025Member"/>
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    <link:loc xlink:type="locator" xlink:label="loc_gemi_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityConversions" xlink:href="gemi-20250930.xsd#gemi_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisLiabilityConversions"/>
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    <link:label id="lab_gemi_NonCashInstrumentSpecificCreditRiskGainLoss_02f07949-01fa-4e4a-98dc-00b2995c2038_terseLabel_en-US" xlink:label="lab_gemi_NonCashInstrumentSpecificCreditRiskGainLoss" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value attributable to instrument-specific credit risk</link:label>
    <link:label id="lab_gemi_NonCashInstrumentSpecificCreditRiskGainLoss_label_en-US" xlink:label="lab_gemi_NonCashInstrumentSpecificCreditRiskGainLoss" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-Cash Instrument-Specific Credit Risk, Gain (Loss)</link:label>
    <link:label id="lab_gemi_NonCashInstrumentSpecificCreditRiskGainLoss_documentation_en-US" xlink:label="lab_gemi_NonCashInstrumentSpecificCreditRiskGainLoss" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Non-Cash Instrument-Specific Credit Risk, Gain (Loss)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_NonCashInstrumentSpecificCreditRiskGainLoss" xlink:href="gemi-20250930.xsd#gemi_NonCashInstrumentSpecificCreditRiskGainLoss"/>
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    <link:label id="lab_us-gaap_LoansPayableMember_b9e43428-0de4-413b-aa1d-f211f74b8fbb_terseLabel_en-US" xlink:label="lab_us-gaap_LoansPayableMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Term Loans</link:label>
    <link:label id="lab_us-gaap_LoansPayableMember_label_en-US" xlink:label="lab_us-gaap_LoansPayableMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loans Payable [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LoansPayableMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LoansPayableMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LoansPayableMember" xlink:to="lab_us-gaap_LoansPayableMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NewAccountingPronouncementsOrChangeInAccountingPrincipleTable_2f5071de-31f8-4245-a670-f0ed160702a4_terseLabel_en-US" xlink:label="lab_us-gaap_NewAccountingPronouncementsOrChangeInAccountingPrincipleTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounting Standards Update and Change in Accounting Principle [Table]</link:label>
    <link:label id="lab_us-gaap_NewAccountingPronouncementsOrChangeInAccountingPrincipleTable_label_en-US" xlink:label="lab_us-gaap_NewAccountingPronouncementsOrChangeInAccountingPrincipleTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounting Standards Update and Change in Accounting Principle [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NewAccountingPronouncementsOrChangeInAccountingPrincipleTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NewAccountingPronouncementsOrChangeInAccountingPrincipleTable"/>
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    <link:label id="lab_us-gaap_RevenueFromExternalCustomersByGeographicAreasTableTextBlock_c974430a-f8ab-475a-ae4c-4207392ad421_terseLabel_en-US" xlink:label="lab_us-gaap_RevenueFromExternalCustomersByGeographicAreasTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Revenues Disaggregated by Geography</link:label>
    <link:label id="lab_us-gaap_RevenueFromExternalCustomersByGeographicAreasTableTextBlock_label_en-US" xlink:label="lab_us-gaap_RevenueFromExternalCustomersByGeographicAreasTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenue from External Customers by Geographic Areas [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RevenueFromExternalCustomersByGeographicAreasTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RevenueFromExternalCustomersByGeographicAreasTableTextBlock"/>
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    <link:label id="lab_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable_c3a64315-6b9b-494c-a067-0c50e99e5b94_terseLabel_en-US" xlink:label="lab_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued interest receivable included in write-offs</link:label>
    <link:label id="lab_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable_label_en-US" xlink:label="lab_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts Receivable, Allowance for Credit Loss, Writeoff, Interest Receivable</link:label>
    <link:label id="lab_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable_documentation_en-US" xlink:label="lab_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Accounts Receivable, Allowance for Credit Loss, Writeoff, Including Interest Receivable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable" xlink:href="gemi-20250930.xsd#gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable" xlink:to="lab_gemi_AccountsReceivableAllowanceForCreditLossWriteoffInterestReceivable" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetsReceivedAsCollateralForDerivatives_932c4c45-d745-4eb3-ba9b-33622dc37d1f_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetsReceivedAsCollateralForDerivatives" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets received as collateral for derivatives</link:label>
    <link:label id="lab_gemi_CryptoAssetsReceivedAsCollateralForDerivatives_label_en-US" xlink:label="lab_gemi_CryptoAssetsReceivedAsCollateralForDerivatives" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Assets Received As Collateral For Derivatives</link:label>
    <link:label id="lab_gemi_CryptoAssetsReceivedAsCollateralForDerivatives_documentation_en-US" xlink:label="lab_gemi_CryptoAssetsReceivedAsCollateralForDerivatives" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Assets Received As Collateral For Derivatives</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetsReceivedAsCollateralForDerivatives" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetsReceivedAsCollateralForDerivatives"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetsReceivedAsCollateralForDerivatives" xlink:to="lab_gemi_CryptoAssetsReceivedAsCollateralForDerivatives" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_OriginalPhantomUnitsMember_cf28b8b1-cfd6-4303-ae83-0cd9f1e2dc5d_terseLabel_en-US" xlink:label="lab_gemi_OriginalPhantomUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Original Phantom Units</link:label>
    <link:label id="lab_gemi_OriginalPhantomUnitsMember_label_en-US" xlink:label="lab_gemi_OriginalPhantomUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Original Phantom Units [Member]</link:label>
    <link:label id="lab_gemi_OriginalPhantomUnitsMember_documentation_en-US" xlink:label="lab_gemi_OriginalPhantomUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Original Phantom Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_OriginalPhantomUnitsMember" xlink:href="gemi-20250930.xsd#gemi_OriginalPhantomUnitsMember"/>
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    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract_094420e3-1fab-4ea5-945f-895e137992c4_terseLabel_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Numerator:</link:label>
    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract_label_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Net Income (Loss) Available to Common Stockholders, Basic [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:to="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShortTermDebtLineItems_93823b94-42ce-48a8-a968-6b1bde9748c5_terseLabel_en-US" xlink:label="lab_us-gaap_ShortTermDebtLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Short-Term Debt [Line Items]</link:label>
    <link:label id="lab_us-gaap_ShortTermDebtLineItems_label_en-US" xlink:label="lab_us-gaap_ShortTermDebtLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Short-Term Debt [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShortTermDebtLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShortTermDebtLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShortTermDebtLineItems" xlink:to="lab_us-gaap_ShortTermDebtLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches_1794bd58-8e6e-4f7a-8c77-e2514a8ad6eb_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of vesting tranches</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches_label_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Number Of Vesting Tranches</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches_documentation_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Number Of Vesting Tranches</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches" xlink:to="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfVestingTranches" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ThirdPartyDebtFairValueDisclosure_9e5abda5-7911-4a50-a95b-12d82f08e24e_terseLabel_en-US" xlink:label="lab_gemi_ThirdPartyDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Third party loans</link:label>
    <link:label id="lab_gemi_ThirdPartyDebtFairValueDisclosure_label_en-US" xlink:label="lab_gemi_ThirdPartyDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Third Party Debt, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_ThirdPartyDebtFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_ThirdPartyDebtFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Third Party Debt, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ThirdPartyDebtFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_ThirdPartyDebtFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ThirdPartyDebtFairValueDisclosure" xlink:to="lab_gemi_ThirdPartyDebtFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan7Member_52fec713-7349-43f6-a078-1c740e26f316_terseLabel_en-US" xlink:label="lab_gemi_Loan7Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 7</link:label>
    <link:label id="lab_gemi_Loan7Member_label_en-US" xlink:label="lab_gemi_Loan7Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 7 [Member]</link:label>
    <link:label id="lab_gemi_Loan7Member_documentation_en-US" xlink:label="lab_gemi_Loan7Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 7</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan7Member" xlink:href="gemi-20250930.xsd#gemi_Loan7Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan7Member" xlink:to="lab_gemi_Loan7Member" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PeoActuallyPaidCompAmt_terseLabel_en-US" xlink:label="lab_ecd_PeoActuallyPaidCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">PEO Actually Paid Compensation Amount</link:label>
    <link:label id="lab_ecd_PeoActuallyPaidCompAmt_label_en-US" xlink:label="lab_ecd_PeoActuallyPaidCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">PEO Actually Paid Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeoActuallyPaidCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeoActuallyPaidCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PeoActuallyPaidCompAmt" xlink:to="lab_ecd_PeoActuallyPaidCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetCollateralHeldFairValueDisclosure_03ac3ba8-aa5e-4107-8b71-3fd07599ad66_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetCollateralHeldFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset collateral held</link:label>
    <link:label id="lab_gemi_CryptoAssetCollateralHeldFairValueDisclosure_label_en-US" xlink:label="lab_gemi_CryptoAssetCollateralHeldFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Collateral Held, Fair Value Disclosure</link:label>
    <link:label id="lab_gemi_CryptoAssetCollateralHeldFairValueDisclosure_documentation_en-US" xlink:label="lab_gemi_CryptoAssetCollateralHeldFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Collateral Held, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetCollateralHeldFairValueDisclosure" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetCollateralHeldFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetCollateralHeldFairValueDisclosure" xlink:to="lab_gemi_CryptoAssetCollateralHeldFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DerivativeInstrumentRecognitionCriteriaAxis_7e430f65-3d98-4054-8db4-df88af285af8_terseLabel_en-US" xlink:label="lab_gemi_DerivativeInstrumentRecognitionCriteriaAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative Instrument, Recognition Criteria [Axis]</link:label>
    <link:label id="lab_gemi_DerivativeInstrumentRecognitionCriteriaAxis_label_en-US" xlink:label="lab_gemi_DerivativeInstrumentRecognitionCriteriaAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Instrument, Recognition Criteria [Axis]</link:label>
    <link:label id="lab_gemi_DerivativeInstrumentRecognitionCriteriaAxis_documentation_en-US" xlink:label="lab_gemi_DerivativeInstrumentRecognitionCriteriaAxis" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Derivative Instrument, Recognition Criteria</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DerivativeInstrumentRecognitionCriteriaAxis" xlink:href="gemi-20250930.xsd#gemi_DerivativeInstrumentRecognitionCriteriaAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DerivativeInstrumentRecognitionCriteriaAxis" xlink:to="lab_gemi_DerivativeInstrumentRecognitionCriteriaAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits_941e218f-ec7c-4723-8700-b73336b907cc_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto assets used to repay debt (in units)</link:label>
    <link:label id="lab_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits_label_en-US" xlink:label="lab_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Disposition For Debt Repayments, Number Of Units</link:label>
    <link:label id="lab_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits_documentation_en-US" xlink:label="lab_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset, Pledged As Collateral</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits" xlink:href="gemi-20250930.xsd#gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits" xlink:to="lab_gemi_CryptoAssetDispositionForDebtRepaymentsNumberOfUnits" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A5200ETHJanuary272025Member_d8f8a945-9d91-4172-800f-71bb4e23d824_terseLabel_en-US" xlink:label="lab_gemi_A5200ETHJanuary272025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">5,200 ETH January 27, 2025</link:label>
    <link:label id="lab_gemi_A5200ETHJanuary272025Member_label_en-US" xlink:label="lab_gemi_A5200ETHJanuary272025Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">5,200 ETH January 27, 2025 [Member]</link:label>
    <link:label id="lab_gemi_A5200ETHJanuary272025Member_documentation_en-US" xlink:label="lab_gemi_A5200ETHJanuary272025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">5,200 ETH January 27, 2025</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A5200ETHJanuary272025Member" xlink:href="gemi-20250930.xsd#gemi_A5200ETHJanuary272025Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A5200ETHJanuary272025Member" xlink:to="lab_gemi_A5200ETHJanuary272025Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RevenueFromContractWithCustomerAbstract_label_en-US" xlink:label="lab_us-gaap_RevenueFromContractWithCustomerAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenue from Contract with Customer [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RevenueFromContractWithCustomerAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RevenueFromContractWithCustomerAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RevenueFromContractWithCustomerAbstract" xlink:to="lab_us-gaap_RevenueFromContractWithCustomerAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_IncreaseDecreaseInDueToThirdParties_cd60aef9-822e-4cc3-b458-e5facfeab00a_terseLabel_en-US" xlink:label="lab_gemi_IncreaseDecreaseInDueToThirdParties" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Payables due to third parties</link:label>
    <link:label id="lab_gemi_IncreaseDecreaseInDueToThirdParties_label_en-US" xlink:label="lab_gemi_IncreaseDecreaseInDueToThirdParties" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) In Due To Third Parties</link:label>
    <link:label id="lab_gemi_IncreaseDecreaseInDueToThirdParties_documentation_en-US" xlink:label="lab_gemi_IncreaseDecreaseInDueToThirdParties" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Increase (Decrease) In Due To Third Parties</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_IncreaseDecreaseInDueToThirdParties" xlink:href="gemi-20250930.xsd#gemi_IncreaseDecreaseInDueToThirdParties"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_IncreaseDecreaseInDueToThirdParties" xlink:to="lab_gemi_IncreaseDecreaseInDueToThirdParties" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A235BTCOctober312024Member_f251da4c-721a-40e6-a251-8003676daf05_terseLabel_en-US" xlink:label="lab_gemi_A235BTCOctober312024Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">235 BTC, October 31, 2024</link:label>
    <link:label id="lab_gemi_A235BTCOctober312024Member_label_en-US" xlink:label="lab_gemi_A235BTCOctober312024Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">235 BTC, October 31, 2024 [Member]</link:label>
    <link:label id="lab_gemi_A235BTCOctober312024Member_documentation_en-US" xlink:label="lab_gemi_A235BTCOctober312024Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">235 BTC, October 31, 2024</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A235BTCOctober312024Member" xlink:href="gemi-20250930.xsd#gemi_A235BTCOctober312024Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A235BTCOctober312024Member" xlink:to="lab_gemi_A235BTCOctober312024Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CapitalCommonAUnitsMember_49d46785-ae9e-4b88-9dfe-29f2f106b63f_terseLabel_en-US" xlink:label="lab_gemi_CapitalCommonAUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Capital Common A Units</link:label>
    <link:label id="lab_gemi_CapitalCommonAUnitsMember_label_en-US" xlink:label="lab_gemi_CapitalCommonAUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Capital Common A Units [Member]</link:label>
    <link:label id="lab_gemi_CapitalCommonAUnitsMember_documentation_en-US" xlink:label="lab_gemi_CapitalCommonAUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Capital Common A Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CapitalCommonAUnitsMember" xlink:href="gemi-20250930.xsd#gemi_CapitalCommonAUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CapitalCommonAUnitsMember" xlink:to="lab_gemi_CapitalCommonAUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent_f2b2a58c-86bd-4943-9789-145469286299_totalLabel_en-US" xlink:label="lab_us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total accrued expenses and other current liabilities</link:label>
    <link:label id="lab_us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent_label_en-US" xlink:label="lab_us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts Payable and Other Accrued Liabilities, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent" xlink:to="lab_us-gaap_AccountsPayableAndOtherAccruedLiabilitiesCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted_71a88c79-b93c-49f2-b4c3-65477f581503_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Units converted (in shares)</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted_label_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Equity Instruments Other Than Options, Converted</link:label>
    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted_documentation_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement By Share-Based Payment Award, Equity Instruments Other Than Options, Converted</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted" xlink:href="gemi-20250930.xsd#gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted" xlink:to="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsConverted" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits_047f3328-0c1f-4028-b31d-52ebf0951e5f_terseLabel_en-US" xlink:label="lab_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pledged crypto subject to a margin call (in units)</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits_label_en-US" xlink:label="lab_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Crypto Asset, Pledged And Subject To Margin Call, Number Of Units</link:label>
    <link:label id="lab_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits_documentation_en-US" xlink:label="lab_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Repurchase Agreements, Crypto Asset, Pledged And Subject To Margin Call, Number Of Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits" xlink:href="gemi-20250930.xsd#gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits" xlink:to="lab_gemi_RepurchaseAgreementsCryptoAssetPledgedAndSubjectToMarginCallNumberOfUnits" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A5280ETHFebruary282025Member_55a08c4b-b6fe-4836-b3b4-a7971ed9927f_terseLabel_en-US" xlink:label="lab_gemi_A5280ETHFebruary282025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">5,280 ETH, February 28, 2025</link:label>
    <link:label id="lab_gemi_A5280ETHFebruary282025Member_label_en-US" xlink:label="lab_gemi_A5280ETHFebruary282025Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">5,280 ETH, February 28, 2025 [Member]</link:label>
    <link:label id="lab_gemi_A5280ETHFebruary282025Member_documentation_en-US" xlink:label="lab_gemi_A5280ETHFebruary282025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">5,280 ETH, February 28, 2025</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A5280ETHFebruary282025Member" xlink:href="gemi-20250930.xsd#gemi_A5280ETHFebruary282025Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A5280ETHFebruary282025Member" xlink:to="lab_gemi_A5280ETHFebruary282025Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DecemberNoteUSDMember_cf472bd4-5e96-4de8-9b5f-85e65c77217d_terseLabel_en-US" xlink:label="lab_gemi_DecemberNoteUSDMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">December Note (USD)</link:label>
    <link:label id="lab_gemi_DecemberNoteUSDMember_label_en-US" xlink:label="lab_gemi_DecemberNoteUSDMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">December Note (USD) [Member]</link:label>
    <link:label id="lab_gemi_DecemberNoteUSDMember_documentation_en-US" xlink:label="lab_gemi_DecemberNoteUSDMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">December Note (USD)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DecemberNoteUSDMember" xlink:href="gemi-20250930.xsd#gemi_DecemberNoteUSDMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DecemberNoteUSDMember" xlink:to="lab_gemi_DecemberNoteUSDMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DerivativeContractTypeDomain_d9c96789-7a04-49cd-b780-3049e971875b_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeContractTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Derivative Contract [Domain]</link:label>
    <link:label id="lab_us-gaap_DerivativeContractTypeDomain_label_en-US" xlink:label="lab_us-gaap_DerivativeContractTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Contract [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DerivativeContractTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DerivativeContractTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DerivativeContractTypeDomain" xlink:to="lab_us-gaap_DerivativeContractTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1_24b5052b-a83b-4cee-a398-014f642aece2_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Performance period</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Award Requisite Service Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardRequisiteServicePeriod1" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AllIndividualsMember_terseLabel_en-US" xlink:label="lab_ecd_AllIndividualsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Individuals</link:label>
    <link:label id="lab_ecd_AllIndividualsMember_label_en-US" xlink:label="lab_ecd_AllIndividualsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">All Individuals [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AllIndividualsMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AllIndividualsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AllIndividualsMember" xlink:to="lab_ecd_AllIndividualsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongtermDebtTypeDomain_1135582c-5588-4934-afeb-cbdc23983aac_terseLabel_en-US" xlink:label="lab_us-gaap_LongtermDebtTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Type [Domain]</link:label>
    <link:label id="lab_us-gaap_LongtermDebtTypeDomain_label_en-US" xlink:label="lab_us-gaap_LongtermDebtTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongtermDebtTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongtermDebtTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LongtermDebtTypeDomain" xlink:to="lab_us-gaap_LongtermDebtTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueDisclosuresAbstract_label_en-US" xlink:label="lab_us-gaap_FairValueDisclosuresAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Disclosures [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueDisclosuresAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueDisclosuresAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueDisclosuresAbstract" xlink:to="lab_us-gaap_FairValueDisclosuresAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_1542d660-6efe-4a03-a9ca-c45c8a87ef58_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Purchase of software, property and equipment</link:label>
    <link:label id="lab_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_label_en-US" xlink:label="lab_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payments to Acquire Property, Plant, and Equipment</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsToAcquirePropertyPlantAndEquipment"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xlink:to="lab_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ForgoneRecoveryIndName_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Name</link:label>
    <link:label id="lab_ecd_ForgoneRecoveryIndName_label_en-US" xlink:label="lab_ecd_ForgoneRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Forgone Recovery, Individual Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ForgoneRecoveryIndName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ForgoneRecoveryIndName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ForgoneRecoveryIndName" xlink:to="lab_ecd_ForgoneRecoveryIndName" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod_fbcdbd22-6804-4be1-b1c1-a967e531ec3b_terseLabel_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Contractual period</link:label>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod_label_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Expiration Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xlink:to="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardExpirationPeriod" xlink:type="arc" order="1"/>
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    <link:label id="lab_gemi_TransactionAndServiceRevenueAndInterestIncomeMember_58ff0480-174d-4a25-b146-a9d08279adf8_terseLabel_en-US" xlink:label="lab_gemi_TransactionAndServiceRevenueAndInterestIncomeMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net revenue</link:label>
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    <link:label id="lab_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember_terseLabel_en-US" xlink:label="lab_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Year-over-Year Change in Fair Value of Equity Awards Granted in Prior Years That are Outstanding and Unvested</link:label>
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    <link:label id="lab_gemi_OverAllotmentOptionCompanyMember_19e3aa2b-02c8-458e-9055-8981708e0128_terseLabel_en-US" xlink:label="lab_gemi_OverAllotmentOptionCompanyMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Over-Allotment Option, Company</link:label>
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    <link:label id="lab_us-gaap_DebtWeightedAverageInterestRate_d2a88c3e-cda2-4882-b1d9-ed31ab4ab140_terseLabel_en-US" xlink:label="lab_us-gaap_DebtWeightedAverageInterestRate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Blended interest rate</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtWeightedAverageInterestRate" xlink:to="lab_us-gaap_DebtWeightedAverageInterestRate" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract_42d31933-a97f-4c04-83dc-308d42d1a50e_terseLabel_en-US" xlink:label="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash, cash equivalents, restricted cash and cash equivalents consisted of the following:</link:label>
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    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardPurchasePriceOfCommonStockPercent_9ea0e4ab-b7ab-4f1c-a9b5-3151d471e4a9_terseLabel_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardPurchasePriceOfCommonStockPercent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Purchase price of common stock percentage</link:label>
    <link:label id="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardPurchasePriceOfCommonStockPercent_label_en-US" xlink:label="lab_us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardPurchasePriceOfCommonStockPercent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Purchase Price of Common Stock, Percent</link:label>
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    <link:label id="lab_us-gaap_PrivatePlacementMember_2dd7611e-6306-48b7-bf5e-d7e92947916e_terseLabel_en-US" xlink:label="lab_us-gaap_PrivatePlacementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Private Placement</link:label>
    <link:label id="lab_us-gaap_PrivatePlacementMember_label_en-US" xlink:label="lab_us-gaap_PrivatePlacementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Private Placement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PrivatePlacementMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PrivatePlacementMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PrivatePlacementMember" xlink:to="lab_us-gaap_PrivatePlacementMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CryptoAssetFuturesAssetMember_9b0e8d64-0477-4a9f-95f8-c6ffc995a1f3_terseLabel_en-US" xlink:label="lab_gemi_CryptoAssetFuturesAssetMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Crypto asset futures</link:label>
    <link:label id="lab_gemi_CryptoAssetFuturesAssetMember_label_en-US" xlink:label="lab_gemi_CryptoAssetFuturesAssetMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset Futures, Asset [Member]</link:label>
    <link:label id="lab_gemi_CryptoAssetFuturesAssetMember_documentation_en-US" xlink:label="lab_gemi_CryptoAssetFuturesAssetMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Crypto Asset Futures, Asset</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CryptoAssetFuturesAssetMember" xlink:to="lab_gemi_CryptoAssetFuturesAssetMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfDerivativeInstrumentsTextBlock_7df1fe8c-33a9-40b0-8b7a-a32be19c2c38_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfDerivativeInstrumentsTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Breakdown of Crypto Asset Collateral Associated with Derivative Positions</link:label>
    <link:label id="lab_us-gaap_ScheduleOfDerivativeInstrumentsTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfDerivativeInstrumentsTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Derivative Instruments [Table Text Block]</link:label>
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    <link:label id="lab_us-gaap_AccruedLiabilitiesCurrentAbstract_599e25c5-20b1-4c45-9742-3840536ed562_terseLabel_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesCurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued expenses:</link:label>
    <link:label id="lab_us-gaap_AccruedLiabilitiesCurrentAbstract_label_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesCurrentAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accrued Liabilities, Current [Abstract]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccruedLiabilitiesCurrentAbstract" xlink:to="lab_us-gaap_AccruedLiabilitiesCurrentAbstract" xlink:type="arc" order="1"/>
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    <link:label id="lab_ecd_ErrCompAnalysisTextBlock_label_en-US" xlink:label="lab_ecd_ErrCompAnalysisTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Erroneous Compensation Analysis [Text Block]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ErrCompAnalysisTextBlock" xlink:to="lab_ecd_ErrCompAnalysisTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTable_1119db96-8f6b-41ea-a826-5bb6efc41077_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Intangible Asset, Finite-Lived [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Intangible Asset, Finite-Lived [Table]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTable" xlink:to="lab_us-gaap_ScheduleOfFiniteLivedIntangibleAssetsTable" xlink:type="arc" order="1"/>
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    <link:label id="lab_us-gaap_LesseeLeaseDescriptionLineItems_07877a40-14fd-402c-a2f0-721e38236e6e_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeLeaseDescriptionLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating Leased Assets [Line Items]</link:label>
    <link:label id="lab_us-gaap_LesseeLeaseDescriptionLineItems_label_en-US" xlink:label="lab_us-gaap_LesseeLeaseDescriptionLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Lease, Description [Line Items]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeLeaseDescriptionLineItems" xlink:to="lab_us-gaap_LesseeLeaseDescriptionLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtConversionConvertedInstrumentSharesIssued1_965cf6a3-6fa9-4f47-b0f0-b5375ef1063f_terseLabel_en-US" xlink:label="lab_us-gaap_DebtConversionConvertedInstrumentSharesIssued1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of shares resulting from conversion of debt (in shares)</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_Rule10b51ArrTrmntdFlag" xlink:to="lab_ecd_Rule10b51ArrTrmntdFlag" xlink:type="arc" order="1"/>
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    <link:label id="lab_gemi_A10000ETHApril72025Member_documentation_en-US" xlink:label="lab_gemi_A10000ETHApril72025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">10,000 ETH, April 7, 2025</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A10000ETHApril72025Member" xlink:href="gemi-20250930.xsd#gemi_A10000ETHApril72025Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A10000ETHApril72025Member" xlink:to="lab_gemi_A10000ETHApril72025Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CreditCardReceivablesMember_636847c2-d412-46e3-97c4-ff6a2ad04230_terseLabel_en-US" xlink:label="lab_us-gaap_CreditCardReceivablesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit card receivables</link:label>
    <link:label id="lab_us-gaap_CreditCardReceivablesMember_label_en-US" xlink:label="lab_us-gaap_CreditCardReceivablesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Card Receivable [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CreditCardReceivablesMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CreditCardReceivablesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CreditCardReceivablesMember" xlink:to="lab_us-gaap_CreditCardReceivablesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A35000ETHMarch12023Member_00b32fcb-f1fe-400d-828a-3a3d28e565c8_terseLabel_en-US" xlink:label="lab_gemi_A35000ETHMarch12023Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">35,000 ETH March 1, 2023</link:label>
    <link:label id="lab_gemi_A35000ETHMarch12023Member_label_en-US" xlink:label="lab_gemi_A35000ETHMarch12023Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">35,000 ETH March 1, 2023 [Member]</link:label>
    <link:label id="lab_gemi_A35000ETHMarch12023Member_documentation_en-US" xlink:label="lab_gemi_A35000ETHMarch12023Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">35,000 ETH March 1, 2023</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A35000ETHMarch12023Member" xlink:href="gemi-20250930.xsd#gemi_A35000ETHMarch12023Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A35000ETHMarch12023Member" xlink:to="lab_gemi_A35000ETHMarch12023Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan5Member_749ddc02-c25c-4909-b10d-d5ee5cdce890_terseLabel_en-US" xlink:label="lab_gemi_Loan5Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 5</link:label>
    <link:label id="lab_gemi_Loan5Member_label_en-US" xlink:label="lab_gemi_Loan5Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 5 [Member]</link:label>
    <link:label id="lab_gemi_Loan5Member_documentation_en-US" xlink:label="lab_gemi_Loan5Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 5</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan5Member" xlink:href="gemi-20250930.xsd#gemi_Loan5Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan5Member" xlink:to="lab_gemi_Loan5Member" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ErrCompRecoveryTable_terseLabel_en-US" xlink:label="lab_ecd_ErrCompRecoveryTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Erroneously Awarded Compensation Recovery</link:label>
    <link:label id="lab_ecd_ErrCompRecoveryTable_label_en-US" xlink:label="lab_ecd_ErrCompRecoveryTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Erroneously Awarded Compensation Recovery [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ErrCompRecoveryTable" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ErrCompRecoveryTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ErrCompRecoveryTable" xlink:to="lab_ecd_ErrCompRecoveryTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InterestIncomeOperating_94176c76-70e2-4426-976d-00bb0b78552d_terseLabel_en-US" xlink:label="lab_us-gaap_InterestIncomeOperating" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Corporate interest</link:label>
    <link:label id="lab_us-gaap_InterestIncomeOperating_label_en-US" xlink:label="lab_us-gaap_InterestIncomeOperating" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Income, Operating</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeOperating" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeOperating"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestIncomeOperating" xlink:to="lab_us-gaap_InterestIncomeOperating" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_Security12bTitle_6ed62785-39f6-4a13-9563-523611b25475_terseLabel_en-US" xlink:label="lab_dei_Security12bTitle" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Title of 12(b) Security</link:label>
    <link:label id="lab_dei_Security12bTitle_label_en-US" xlink:label="lab_dei_Security12bTitle" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Title of 12(b) Security</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_Security12bTitle" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_Security12bTitle"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_Security12bTitle" xlink:to="lab_dei_Security12bTitle" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A250BTCJuly52024Member_6ab5a083-be17-43a7-8404-57ea012fdeae_terseLabel_en-US" xlink:label="lab_gemi_A250BTCJuly52024Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">250 BTC, July 5, 2024</link:label>
    <link:label id="lab_gemi_A250BTCJuly52024Member_label_en-US" xlink:label="lab_gemi_A250BTCJuly52024Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">250 BTC, July 5, 2024 [Member]</link:label>
    <link:label id="lab_gemi_A250BTCJuly52024Member_documentation_en-US" xlink:label="lab_gemi_A250BTCJuly52024Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">250 BTC, July 5, 2024</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_A250BTCJuly52024Member" xlink:href="gemi-20250930.xsd#gemi_A250BTCJuly52024Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_A250BTCJuly52024Member" xlink:to="lab_gemi_A250BTCJuly52024Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeLeaseDescriptionTable_4cf1590c-22ee-4360-adca-671b443da4ee_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeLeaseDescriptionTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Operating Leased Assets [Table]</link:label>
    <link:label id="lab_us-gaap_LesseeLeaseDescriptionTable_label_en-US" xlink:label="lab_us-gaap_LesseeLeaseDescriptionTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Lease, Description [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeLeaseDescriptionTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeLeaseDescriptionTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeLeaseDescriptionTable" xlink:to="lab_us-gaap_LesseeLeaseDescriptionTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TemporaryEquityLiquidationPreference_9293ea90-0da9-418c-8f55-846bb94a99f2_terseLabel_en-US" xlink:label="lab_us-gaap_TemporaryEquityLiquidationPreference" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible preferred units, liquidation preference</link:label>
    <link:label id="lab_us-gaap_TemporaryEquityLiquidationPreference_label_en-US" xlink:label="lab_us-gaap_TemporaryEquityLiquidationPreference" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Temporary Equity, Liquidation Preference</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TemporaryEquityLiquidationPreference" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TemporaryEquityLiquidationPreference"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TemporaryEquityLiquidationPreference" xlink:to="lab_us-gaap_TemporaryEquityLiquidationPreference" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ServiceOtherMember_97841d84-1847-4841-8545-32593a50bcc1_terseLabel_en-US" xlink:label="lab_us-gaap_ServiceOtherMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other services revenue</link:label>
    <link:label id="lab_us-gaap_ServiceOtherMember_label_en-US" xlink:label="lab_us-gaap_ServiceOtherMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Service, Other [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ServiceOtherMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ServiceOtherMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ServiceOtherMember" xlink:to="lab_us-gaap_ServiceOtherMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EarningsPerShareBasicLineItems_44267140-e519-40c5-8e67-96039249bda4_terseLabel_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasicLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Basic, by Common Class, Including Two Class Method [Line Items]</link:label>
    <link:label id="lab_us-gaap_EarningsPerShareBasicLineItems_label_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasicLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Basic, by Common Class, Including Two Class Method [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EarningsPerShareBasicLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EarningsPerShareBasicLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EarningsPerShareBasicLineItems" xlink:to="lab_us-gaap_EarningsPerShareBasicLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CreditCardReceivableInTransitMember_401cc996-1ae7-4e38-9c32-a3e45e0f3192_terseLabel_en-US" xlink:label="lab_gemi_CreditCardReceivableInTransitMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">In-transit receivables</link:label>
    <link:label id="lab_gemi_CreditCardReceivableInTransitMember_label_en-US" xlink:label="lab_gemi_CreditCardReceivableInTransitMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Card Receivable, In-Transit [Member]</link:label>
    <link:label id="lab_gemi_CreditCardReceivableInTransitMember_documentation_en-US" xlink:label="lab_gemi_CreditCardReceivableInTransitMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Credit Card Receivable, In-Transit</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CreditCardReceivableInTransitMember" xlink:href="gemi-20250930.xsd#gemi_CreditCardReceivableInTransitMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CreditCardReceivableInTransitMember" xlink:to="lab_gemi_CreditCardReceivableInTransitMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EarningsPerShareAbstract_fd42188a-94ea-4d7f-a8fd-00f546e42f5e_terseLabel_en-US" xlink:label="lab_us-gaap_EarningsPerShareAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net loss per share attributable to common stockholders</link:label>
    <link:label id="lab_us-gaap_EarningsPerShareAbstract_label_en-US" xlink:label="lab_us-gaap_EarningsPerShareAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EarningsPerShareAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EarningsPerShareAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EarningsPerShareAbstract" xlink:to="lab_us-gaap_EarningsPerShareAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConversionOfStockSharesConverted1_cf99c329-d0e8-4f19-b831-fecec08d1e2e_terseLabel_en-US" xlink:label="lab_us-gaap_ConversionOfStockSharesConverted1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of outstanding units converted (in shares)</link:label>
    <link:label id="lab_us-gaap_ConversionOfStockSharesConverted1_label_en-US" xlink:label="lab_us-gaap_ConversionOfStockSharesConverted1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Conversion of Stock, Shares Converted</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConversionOfStockSharesConverted1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConversionOfStockSharesConverted1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConversionOfStockSharesConverted1" xlink:to="lab_us-gaap_ConversionOfStockSharesConverted1" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NonoperatingIncomeExpense_7ec34fa1-d3b6-4345-ae65-efd46aefd4c4_totalLabel_en-US" xlink:label="lab_us-gaap_NonoperatingIncomeExpense" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total other income (expense), net</link:label>
    <link:label id="lab_us-gaap_NonoperatingIncomeExpense_label_en-US" xlink:label="lab_us-gaap_NonoperatingIncomeExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Nonoperating Income (Expense)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NonoperatingIncomeExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NonoperatingIncomeExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NonoperatingIncomeExpense" xlink:to="lab_us-gaap_NonoperatingIncomeExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid_cd4fea35-2c32-4e4e-a4a5-349af7b27504_terseLabel_en-US" xlink:label="lab_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-qualified financing event, conversion price as a percentage of lowest price paid per unit by the purchasers of the convertible preferred units</link:label>
    <link:label id="lab_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid_label_en-US" xlink:label="lab_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Conversion, Non-Qualifying Financing Event, Option To Convert Notes, Price Per Unit As A Percentage Of Lowest Price Per Unit Paid</link:label>
    <link:label id="lab_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid_documentation_en-US" xlink:label="lab_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Conversion, Non-Qualifying Financing Event, Option To Convert Notes, Price Per Unit As A Percentage Of Lowest Price Per Unit Paid</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid" xlink:href="gemi-20250930.xsd#gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid" xlink:to="lab_gemi_DebtConversionNonQualifyingFinancingEventOptionToConvertNotesPricePerUnitAsAPercentageOfLowestPricePerUnitPaid" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock_563ec921-4fa4-4b1f-8f2f-61f9d7b813da_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Potentially Dilutive Units Not Included in Calculation of Diluted Shares Outstanding</link:label>
    <link:label id="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Antidilutive Securities Excluded from Computation of Earnings Per Share [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock" xlink:to="lab_us-gaap_ScheduleOfAntidilutiveSecuritiesExcludedFromComputationOfEarningsPerShareTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShortTermDebtTypeAxis_0017216b-9c7f-4774-bf73-c85314ed4a8b_terseLabel_en-US" xlink:label="lab_us-gaap_ShortTermDebtTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Short-Term Debt, Type [Axis]</link:label>
    <link:label id="lab_us-gaap_ShortTermDebtTypeAxis_label_en-US" xlink:label="lab_us-gaap_ShortTermDebtTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Short-Term Debt, Type [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShortTermDebtTypeAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShortTermDebtTypeAxis"/>
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    <link:loc xlink:type="locator" xlink:label="loc_gemi_AccountsReceivableAfterAllowanceForCreditLossCurrentMember" xlink:href="gemi-20250930.xsd#gemi_AccountsReceivableAfterAllowanceForCreditLossCurrentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_AccountsReceivableAfterAllowanceForCreditLossCurrentMember" xlink:to="lab_gemi_AccountsReceivableAfterAllowanceForCreditLossCurrentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward_31d5358c-ef51-43c0-9c34-82fe888e133d_terseLabel_en-US" xlink:label="lab_us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounts Receivable, Allowance for Credit Loss [Roll Forward]</link:label>
    <link:label id="lab_us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward_label_en-US" xlink:label="lab_us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accounts Receivable, Allowance for Credit Loss [Roll Forward]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward" xlink:to="lab_us-gaap_AllowanceForDoubtfulAccountsReceivableRollforward" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1_0bede81d-d8b8-4987-8117-c2e210a490d6_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Vesting period</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Award Vesting Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RevenueNotFromContractWithCustomer_6271643d-4dde-4b0f-9633-bd6764d7b7c2_totalLabel_en-US" xlink:label="lab_us-gaap_RevenueNotFromContractWithCustomer" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total revenue not from contracts with customers</link:label>
    <link:label id="lab_us-gaap_RevenueNotFromContractWithCustomer_9e665d97-784c-446e-ab43-5fa11f209464_verboseLabel_en-US" xlink:label="lab_us-gaap_RevenueNotFromContractWithCustomer" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Total revenue not from contracts with customers</link:label>
    <link:label id="lab_us-gaap_RevenueNotFromContractWithCustomer_label_en-US" xlink:label="lab_us-gaap_RevenueNotFromContractWithCustomer" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenue Not from Contract with Customer</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RevenueNotFromContractWithCustomer" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RevenueNotFromContractWithCustomer"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RevenueNotFromContractWithCustomer" xlink:to="lab_us-gaap_RevenueNotFromContractWithCustomer" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PreferredStockSharesAuthorized_38c57977-c323-4e22-9d36-a9d49814a050_terseLabel_en-US" xlink:label="lab_us-gaap_PreferredStockSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Preferred stock, shares authorized (in shares)</link:label>
    <link:label id="lab_us-gaap_PreferredStockSharesAuthorized_label_en-US" xlink:label="lab_us-gaap_PreferredStockSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Stock, Shares Authorized</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredStockSharesAuthorized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredStockSharesAuthorized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PreferredStockSharesAuthorized" xlink:to="lab_us-gaap_PreferredStockSharesAuthorized" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrAdoptionDate_terseLabel_en-US" xlink:label="lab_ecd_TrdArrAdoptionDate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adoption Date</link:label>
    <link:label id="lab_ecd_TrdArrAdoptionDate_label_en-US" xlink:label="lab_ecd_TrdArrAdoptionDate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement Adoption Date</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrAdoptionDate" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrAdoptionDate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TrdArrAdoptionDate" xlink:to="lab_ecd_TrdArrAdoptionDate" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PriorPeriodReclassificationAdjustmentDescription_3dfaa550-1485-42b8-bd21-5929ee0e3ddc_terseLabel_en-US" xlink:label="lab_us-gaap_PriorPeriodReclassificationAdjustmentDescription" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reclassification</link:label>
    <link:label id="lab_us-gaap_PriorPeriodReclassificationAdjustmentDescription_label_en-US" xlink:label="lab_us-gaap_PriorPeriodReclassificationAdjustmentDescription" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reclassification, Comparability Adjustment [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PriorPeriodReclassificationAdjustmentDescription" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PriorPeriodReclassificationAdjustmentDescription"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PriorPeriodReclassificationAdjustmentDescription" xlink:to="lab_us-gaap_PriorPeriodReclassificationAdjustmentDescription" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonUnitIssued_3c57c931-df8a-41c8-8060-c08afe956b79_terseLabel_en-US" xlink:label="lab_us-gaap_CommonUnitIssued" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common units, issued (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonUnitIssued_label_en-US" xlink:label="lab_us-gaap_CommonUnitIssued" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Unit, Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonUnitIssued" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonUnitIssued"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonUnitIssued" xlink:to="lab_us-gaap_CommonUnitIssued" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ConvertibleDebtAndLoansPayableMember_acf45573-e59f-4d5d-a3f2-6c041b3237a8_terseLabel_en-US" xlink:label="lab_gemi_ConvertibleDebtAndLoansPayableMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Convertible Debt And Loans Payable</link:label>
    <link:label id="lab_gemi_ConvertibleDebtAndLoansPayableMember_label_en-US" xlink:label="lab_gemi_ConvertibleDebtAndLoansPayableMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Convertible Debt And Loans Payable [Member]</link:label>
    <link:label id="lab_gemi_ConvertibleDebtAndLoansPayableMember_documentation_en-US" xlink:label="lab_gemi_ConvertibleDebtAndLoansPayableMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Convertible Debt And Loans Payable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ConvertibleDebtAndLoansPayableMember" xlink:href="gemi-20250930.xsd#gemi_ConvertibleDebtAndLoansPayableMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ConvertibleDebtAndLoansPayableMember" xlink:to="lab_gemi_ConvertibleDebtAndLoansPayableMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseTermOfContract_db3bbd48-991b-4276-9b06-fc8aed3e5c4e_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseTermOfContract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Lease term</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseTermOfContract_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseTermOfContract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Term of Contract</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseTermOfContract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseTermOfContract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseTermOfContract" xlink:to="lab_us-gaap_LesseeOperatingLeaseTermOfContract" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan9Member_d17f6251-5c1b-4d77-b1b4-240ae4fbbdfa_terseLabel_en-US" xlink:label="lab_gemi_Loan9Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 9</link:label>
    <link:label id="lab_gemi_Loan9Member_label_en-US" xlink:label="lab_gemi_Loan9Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 9 [Member]</link:label>
    <link:label id="lab_gemi_Loan9Member_documentation_en-US" xlink:label="lab_gemi_Loan9Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 9</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan9Member" xlink:href="gemi-20250930.xsd#gemi_Loan9Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan9Member" xlink:to="lab_gemi_Loan9Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CustomerRelationshipsMember_c4466144-4d33-4cf6-83f7-62d8bf342b6d_terseLabel_en-US" xlink:label="lab_us-gaap_CustomerRelationshipsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Customer relationships</link:label>
    <link:label id="lab_us-gaap_CustomerRelationshipsMember_label_en-US" xlink:label="lab_us-gaap_CustomerRelationshipsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Customer Relationships [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CustomerRelationshipsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CustomerRelationshipsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CustomerRelationshipsMember" xlink:to="lab_us-gaap_CustomerRelationshipsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingIncomeLoss_8e208ae0-4734-490e-8be9-e69d6d7db6f2_totalLabel_en-US" xlink:label="lab_us-gaap_OperatingIncomeLoss" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Operating loss</link:label>
    <link:label id="lab_us-gaap_OperatingIncomeLoss_label_en-US" xlink:label="lab_us-gaap_OperatingIncomeLoss" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Income (Loss)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingIncomeLoss" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingIncomeLoss"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingIncomeLoss" xlink:to="lab_us-gaap_OperatingIncomeLoss" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LineOfCreditMember_bb5c9218-03da-444e-aa33-44dd39968235_terseLabel_en-US" xlink:label="lab_us-gaap_LineOfCreditMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Line of Credit</link:label>
    <link:label id="lab_us-gaap_LineOfCreditMember_label_en-US" xlink:label="lab_us-gaap_LineOfCreditMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Line of Credit [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LineOfCreditMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LineOfCreditMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LineOfCreditMember" xlink:to="lab_us-gaap_LineOfCreditMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RetainedEarningsAccumulatedDeficit_802b2dd1-2d3d-46d8-8768-29c800722748_terseLabel_en-US" xlink:label="lab_us-gaap_RetainedEarningsAccumulatedDeficit" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accumulated deficit</link:label>
    <link:label id="lab_us-gaap_RetainedEarningsAccumulatedDeficit_label_en-US" xlink:label="lab_us-gaap_RetainedEarningsAccumulatedDeficit" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Retained Earnings (Accumulated Deficit)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RetainedEarningsAccumulatedDeficit" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RetainedEarningsAccumulatedDeficit"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RetainedEarningsAccumulatedDeficit" xlink:to="lab_us-gaap_RetainedEarningsAccumulatedDeficit" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet_5814bd55-1edc-4fca-ac0f-3e6a28d04069_terseLabel_en-US" xlink:label="lab_us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Increase (decrease) in fair value of embedded derivatives</link:label>
    <link:label id="lab_us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet_label_en-US" xlink:label="lab_us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Embedded Derivative, Gain (Loss) on Embedded Derivative, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet" xlink:to="lab_us-gaap_EmbeddedDerivativeGainLossOnEmbeddedDerivativeNet" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CustodialFundsDueToCustomersCurrent_1210f58e-9acf-49f7-bd2d-1e869d1fb5e5_terseLabel_en-US" xlink:label="lab_gemi_CustodialFundsDueToCustomersCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Custodial funds due to customers</link:label>
    <link:label id="lab_gemi_CustodialFundsDueToCustomersCurrent_label_en-US" xlink:label="lab_gemi_CustodialFundsDueToCustomersCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Custodial Funds Due To Customers, Current</link:label>
    <link:label id="lab_gemi_CustodialFundsDueToCustomersCurrent_documentation_en-US" xlink:label="lab_gemi_CustodialFundsDueToCustomersCurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Custodial Funds Due To Customers, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CustodialFundsDueToCustomersCurrent" xlink:href="gemi-20250930.xsd#gemi_CustodialFundsDueToCustomersCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CustodialFundsDueToCustomersCurrent" xlink:to="lab_gemi_CustodialFundsDueToCustomersCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember_terseLabel_en-US" xlink:label="lab_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Year-end Fair Value of Equity Awards Granted in Covered Year that are Outstanding and Unvested</link:label>
    <link:label id="lab_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember_label_en-US" xlink:label="lab_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Year-end Fair Value of Equity Awards Granted in Covered Year that are Outstanding and Unvested [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember" xlink:to="lab_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_InterestPaidAmountPreviouslyAccrued_4667e3f6-c53a-4b68-a091-cf7032b377bf_terseLabel_en-US" xlink:label="lab_gemi_InterestPaidAmountPreviouslyAccrued" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Prior year interest accrued that was paid during the year</link:label>
    <link:label id="lab_gemi_InterestPaidAmountPreviouslyAccrued_label_en-US" xlink:label="lab_gemi_InterestPaidAmountPreviouslyAccrued" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Paid, Amount Previously Accrued</link:label>
    <link:label id="lab_gemi_InterestPaidAmountPreviouslyAccrued_documentation_en-US" xlink:label="lab_gemi_InterestPaidAmountPreviouslyAccrued" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Interest Paid, Amount Previously Accrued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_InterestPaidAmountPreviouslyAccrued" xlink:href="gemi-20250930.xsd#gemi_InterestPaidAmountPreviouslyAccrued"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_InterestPaidAmountPreviouslyAccrued" xlink:to="lab_gemi_InterestPaidAmountPreviouslyAccrued" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock_7685a978-0ec5-4f19-be30-96676c2b84de_terseLabel_en-US" xlink:label="lab_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit card receivables pledged</link:label>
    <link:label id="lab_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock_label_en-US" xlink:label="lab_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Card Receivables Pledged, Policy [Policy Text Block]</link:label>
    <link:label id="lab_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock_documentation_en-US" xlink:label="lab_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Credit Card Receivables Pledged, Policy</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock" xlink:href="gemi-20250930.xsd#gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock" xlink:to="lab_gemi_CreditCardReceivablesPledgedPolicyPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementClassOfStockAxis_1f08010a-45a7-4d62-92bb-573407b9448e_terseLabel_en-US" xlink:label="lab_us-gaap_StatementClassOfStockAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Class of Stock [Axis]</link:label>
    <link:label id="lab_us-gaap_StatementClassOfStockAxis_label_en-US" xlink:label="lab_us-gaap_StatementClassOfStockAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Class of Stock [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementClassOfStockAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementClassOfStockAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementClassOfStockAxis" xlink:to="lab_us-gaap_StatementClassOfStockAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock_ae560e37-eeac-4045-80d6-1e5fdd74c1d0_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Phantom Units Activity</link:label>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock_label_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Phantom Units, Outstanding Activity [Table Text Block]</link:label>
    <link:label id="lab_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock_documentation_en-US" xlink:label="lab_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Phantom Units, Outstanding Activity</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock" xlink:href="gemi-20250930.xsd#gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock" xlink:to="lab_gemi_ShareBasedPaymentArrangementPhantomUnitsOutstandingActivityTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SubsequentEventTypeDomain_038724a8-aa96-46d7-99ff-a0eb03a57d6e_terseLabel_en-US" xlink:label="lab_us-gaap_SubsequentEventTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Subsequent Event Type [Domain]</link:label>
    <link:label id="lab_us-gaap_SubsequentEventTypeDomain_label_en-US" xlink:label="lab_us-gaap_SubsequentEventTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Subsequent Event Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubsequentEventTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubsequentEventTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SubsequentEventTypeDomain" xlink:to="lab_us-gaap_SubsequentEventTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInDueToRelatedParties_b9c1c84d-3b15-4cf5-8f61-1cf85f16f6ae_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInDueToRelatedParties" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Payables due to related parties</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInDueToRelatedParties_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInDueToRelatedParties" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Due to Related Parties</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInDueToRelatedParties" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInDueToRelatedParties"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInDueToRelatedParties" xlink:to="lab_us-gaap_IncreaseDecreaseInDueToRelatedParties" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember_terseLabel_en-US" xlink:label="lab_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate Grant Date Fair Value of Equity Award Amounts Reported in Summary Compensation Table</link:label>
    <link:label id="lab_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember_label_en-US" xlink:label="lab_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Aggregate Grant Date Fair Value of Equity Award Amounts Reported in Summary Compensation Table [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember" xlink:to="lab_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PayablesAndAccrualsAbstract_label_en-US" xlink:label="lab_us-gaap_PayablesAndAccrualsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payables and Accruals [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PayablesAndAccrualsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PayablesAndAccrualsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PayablesAndAccrualsAbstract" xlink:to="lab_us-gaap_PayablesAndAccrualsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherNonoperatingIncomeExpense_8396adf2-cbb0-4f3d-9896-cbfddeaabc82_terseLabel_en-US" xlink:label="lab_us-gaap_OtherNonoperatingIncomeExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other income (expense), net</link:label>
    <link:label id="lab_us-gaap_OtherNonoperatingIncomeExpense_label_en-US" xlink:label="lab_us-gaap_OtherNonoperatingIncomeExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Nonoperating Income (Expense)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherNonoperatingIncomeExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherNonoperatingIncomeExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherNonoperatingIncomeExpense" xlink:to="lab_us-gaap_OtherNonoperatingIncomeExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementTable_de4661a1-81ad-464b-9438-2d5643c2b2f8_terseLabel_en-US" xlink:label="lab_us-gaap_StatementTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement [Table]</link:label>
    <link:label id="lab_us-gaap_StatementTable_label_en-US" xlink:label="lab_us-gaap_StatementTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementTable" xlink:to="lab_us-gaap_StatementTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_3ecd7166-758f-45f1-9e11-fb7380f7d83e_terseLabel_en-US" xlink:label="lab_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustments to reconcile net loss to net cash used in operating activities</link:label>
    <link:label id="lab_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_label_en-US" xlink:label="lab_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Adjustment to Reconcile Net Income to Cash Provided by (Used in) Operating Activity [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:to="lab_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_58629676-4412-48cd-a278-f66a465db8b8_terseLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash flows from financing activities</link:label>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_label_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Financing Activity, Including Discontinued Operation [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:to="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_Loan8Member_1ac24d2a-82bc-4267-83e3-fdc3c9bf926a_terseLabel_en-US" xlink:label="lab_gemi_Loan8Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loan 8</link:label>
    <link:label id="lab_gemi_Loan8Member_label_en-US" xlink:label="lab_gemi_Loan8Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loan 8 [Member]</link:label>
    <link:label id="lab_gemi_Loan8Member_documentation_en-US" xlink:label="lab_gemi_Loan8Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loan 8</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_Loan8Member" xlink:href="gemi-20250930.xsd#gemi_Loan8Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_Loan8Member" xlink:to="lab_gemi_Loan8Member" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock_240b0024-a078-4db3-844a-81780ccee7c2_terseLabel_en-US" xlink:label="lab_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Prepaid Expenses and Other Assets</link:label>
    <link:label id="lab_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock_label_en-US" xlink:label="lab_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Prepaid Expense And Other Assets Disclosure [Text Block]</link:label>
    <link:label id="lab_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock_documentation_en-US" xlink:label="lab_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Prepaid Expense And Other Assets Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock" xlink:href="gemi-20250930.xsd#gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock" xlink:to="lab_gemi_PrepaidExpenseAndOtherAssetsDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ServiceAdvisoryFeeMember_d281fd78-4e23-46d9-bed4-849a5bf4f61d_terseLabel_en-US" xlink:label="lab_gemi_ServiceAdvisoryFeeMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Advisory fee revenue</link:label>
    <link:label id="lab_gemi_ServiceAdvisoryFeeMember_label_en-US" xlink:label="lab_gemi_ServiceAdvisoryFeeMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Service, Advisory Fee [Member]</link:label>
    <link:label id="lab_gemi_ServiceAdvisoryFeeMember_documentation_en-US" xlink:label="lab_gemi_ServiceAdvisoryFeeMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Service, Advisory Fee</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ServiceAdvisoryFeeMember" xlink:href="gemi-20250930.xsd#gemi_ServiceAdvisoryFeeMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ServiceAdvisoryFeeMember" xlink:to="lab_gemi_ServiceAdvisoryFeeMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AllAdjToCompMember_terseLabel_en-US" xlink:label="lab_ecd_AllAdjToCompMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Adjustments to Compensation</link:label>
    <link:label id="lab_ecd_AllAdjToCompMember_label_en-US" xlink:label="lab_ecd_AllAdjToCompMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">All Adjustments to Compensation [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AllAdjToCompMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AllAdjToCompMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AllAdjToCompMember" xlink:to="lab_ecd_AllAdjToCompMember" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_AmendmentFlag_0c8c5630-702a-4309-adb9-fad2801e956a_terseLabel_en-US" xlink:label="lab_dei_AmendmentFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amendment Flag</link:label>
    <link:label id="lab_dei_AmendmentFlag_label_en-US" xlink:label="lab_dei_AmendmentFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Amendment Flag</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_AmendmentFlag" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_AmendmentFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_AmendmentFlag" xlink:to="lab_dei_AmendmentFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SaleOfStockPricePerShare_9011a7bb-5b90-48f5-b658-2d1e46fd1e03_verboseLabel_en-US" xlink:label="lab_us-gaap_SaleOfStockPricePerShare" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Public offering price of shares issued and sold (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_SaleOfStockPricePerShare_a8a1a54d-3624-4909-891e-042b92bd8b04_terseLabel_en-US" xlink:label="lab_us-gaap_SaleOfStockPricePerShare" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Share price per share (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_SaleOfStockPricePerShare_label_en-US" xlink:label="lab_us-gaap_SaleOfStockPricePerShare" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Sale of Stock, Price Per Share</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SaleOfStockPricePerShare" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SaleOfStockPricePerShare"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SaleOfStockPricePerShare" xlink:to="lab_us-gaap_SaleOfStockPricePerShare" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent_913c86f4-919d-43e9-8dbc-6279718444fc_terseLabel_en-US" xlink:label="lab_us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value attributable to instrument-specific credit risk</link:label>
    <link:label id="lab_us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent_label_en-US" xlink:label="lab_us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">OCI, Market Risk Benefit, Instrument-Specific Credit Risk, Gain (Loss), after Adjustments and Tax, Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent" xlink:to="lab_us-gaap_OciMarketRiskBenefitInstrumentSpecificCreditRiskGainLossAfterAdjustmentsAndTaxParent" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_PaymentsToAcquireCreditCardReceivable_aa7c1e97-b7ac-407a-b36b-0d1e0bd59bd9_negatedTerseLabel_en-US" xlink:label="lab_gemi_PaymentsToAcquireCreditCardReceivable" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Purchases of credit card receivables</link:label>
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    <link:label id="lab_us-gaap_TreasuryStockCommonValue_5b9942bb-92ea-49ac-8b06-695be43d9eab_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonValue" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Treasury stock at cost; 0 and 23 shares as of December&#160;31, 2024 and September&#160;30, 2025, respectively</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonValue_label_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Treasury Stock, Common, Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TreasuryStockCommonValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TreasuryStockCommonValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TreasuryStockCommonValue" xlink:to="lab_us-gaap_TreasuryStockCommonValue" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonUnitAuthorized_edeb55fd-2599-430f-a3f5-f31419236b0d_terseLabel_en-US" xlink:label="lab_us-gaap_CommonUnitAuthorized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common units, authorized (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonUnitAuthorized_label_en-US" xlink:label="lab_us-gaap_CommonUnitAuthorized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Unit, Authorized</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonUnitAuthorized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonUnitAuthorized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonUnitAuthorized" xlink:to="lab_us-gaap_CommonUnitAuthorized" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrTerminationDate_terseLabel_en-US" xlink:label="lab_ecd_TrdArrTerminationDate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Termination Date</link:label>
    <link:label id="lab_ecd_TrdArrTerminationDate_label_en-US" xlink:label="lab_ecd_TrdArrTerminationDate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement Termination Date</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrTerminationDate" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrTerminationDate"/>
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    <link:label id="lab_us-gaap_PreferredUnitsOutstanding_4b5fea04-8d49-4f46-a59a-4ee0ac8aaeab_terseLabel_en-US" xlink:label="lab_us-gaap_PreferredUnitsOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Preferred units, outstanding (in shares)</link:label>
    <link:label id="lab_us-gaap_PreferredUnitsOutstanding_28c09838-0aff-433e-85bb-ae11dcca07aa_verboseLabel_en-US" xlink:label="lab_us-gaap_PreferredUnitsOutstanding" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Units Outstanding (in shares)</link:label>
    <link:label id="lab_us-gaap_PreferredUnitsOutstanding_label_en-US" xlink:label="lab_us-gaap_PreferredUnitsOutstanding" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Units, Outstanding</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredUnitsOutstanding" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredUnitsOutstanding"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PreferredUnitsOutstanding" xlink:to="lab_us-gaap_PreferredUnitsOutstanding" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag_terseLabel_en-US" xlink:label="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures Adopted</link:label>
    <link:label id="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag_label_en-US" xlink:label="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures Adopted [Flag]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_InsiderTrdPoliciesProcAdoptedFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:to="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_A2500ETHMarch282025Member_b7f53f33-4166-4580-968c-b53e9c7b1419_terseLabel_en-US" xlink:label="lab_gemi_A2500ETHMarch282025Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2,500 ETH, March 28, 2025</link:label>
    <link:label id="lab_gemi_A2500ETHMarch282025Member_label_en-US" xlink:label="lab_gemi_A2500ETHMarch282025Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">2,500 ETH, March 28, 2025 [Member]</link:label>
    <link:label id="lab_gemi_A2500ETHMarch282025Member_documentation_en-US" xlink:label="lab_gemi_A2500ETHMarch282025Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">2,500 ETH, March 28, 2025</link:label>
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    <link:label id="lab_us-gaap_ReceivableTypeDomain_8f488d1d-e328-4aed-a2eb-8b56afbd1463_terseLabel_en-US" xlink:label="lab_us-gaap_ReceivableTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Receivable [Domain]</link:label>
    <link:label id="lab_us-gaap_ReceivableTypeDomain_label_en-US" xlink:label="lab_us-gaap_ReceivableTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Receivable [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReceivableTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReceivableTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReceivableTypeDomain" xlink:to="lab_us-gaap_ReceivableTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent_346d8a30-d110-4c2c-89ec-90e45becdb2e_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Weighted-average discount rate</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Weighted Average Discount Rate, Percent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:to="lab_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock_616807df-3a8c-4c05-b895-be54ab11e8a3_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Stock Option Activity</link:label>
    <link:label id="lab_us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Option, Activity [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock" xlink:to="lab_us-gaap_ScheduleOfShareBasedCompensationStockOptionsActivityTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ServiceBasedIncentiveUnitsMember_14f4bb6e-30a3-4699-bbe5-31de6dd63c72_terseLabel_en-US" xlink:label="lab_gemi_ServiceBasedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Service-Based Incentive Units</link:label>
    <link:label id="lab_gemi_ServiceBasedIncentiveUnitsMember_label_en-US" xlink:label="lab_gemi_ServiceBasedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Service Based Incentive Units [Member]</link:label>
    <link:label id="lab_gemi_ServiceBasedIncentiveUnitsMember_documentation_en-US" xlink:label="lab_gemi_ServiceBasedIncentiveUnitsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Service Based Incentive Units</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ServiceBasedIncentiveUnitsMember" xlink:href="gemi-20250930.xsd#gemi_ServiceBasedIncentiveUnitsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ServiceBasedIncentiveUnitsMember" xlink:to="lab_gemi_ServiceBasedIncentiveUnitsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember_7bcbcf25-b1c0-41ec-b8e8-ef7828137856_terseLabel_en-US" xlink:label="lab_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Executive Officers, Other Employees And Board Members</link:label>
    <link:label id="lab_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember_label_en-US" xlink:label="lab_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Executive Officers, Other Employees And Board Members [Member]</link:label>
    <link:label id="lab_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember_documentation_en-US" xlink:label="lab_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Executive Officers, Other Employees And Board Members</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember" xlink:href="gemi-20250930.xsd#gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember" xlink:to="lab_gemi_ExecutiveOfficersOtherEmployeesAndBoardMembersMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_c454db39-46a8-4ec2-bb45-336f993838a9_terseLabel_en-US" xlink:label="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Weighted average shares outstanding - basic (in shares)</link:label>
    <link:label id="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_d4a649e6-eb76-4770-be7e-cf8574cdf63d_verboseLabel_en-US" xlink:label="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Weighted average common stock outstanding, basic (in shares)</link:label>
    <link:label id="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_label_en-US" xlink:label="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Weighted Average Number of Shares Outstanding, Basic</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_WeightedAverageNumberOfSharesOutstandingBasic"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:to="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock_43de864a-f60b-470f-88ae-c77a0fbf4b43_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Basic and Diluted Net Loss per Share</link:label>
    <link:label id="lab_us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Earnings Per Share, Basic and Diluted [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:to="lab_us-gaap_ScheduleOfEarningsPerShareBasicAndDilutedTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber_aa16c732-f38d-447e-a035-eeb3827dff4e_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of stock options exercisable (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Number</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsExercisableNumber" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SubsequentEventsAbstract_label_en-US" xlink:label="lab_us-gaap_SubsequentEventsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Subsequent Events [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubsequentEventsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubsequentEventsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SubsequentEventsAbstract" xlink:to="lab_us-gaap_SubsequentEventsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtDisclosureAbstract_label_en-US" xlink:label="lab_us-gaap_DebtDisclosureAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Disclosure [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtDisclosureAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtDisclosureAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtDisclosureAbstract" xlink:to="lab_us-gaap_DebtDisclosureAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NonoperatingIncomeExpenseAbstract_dca295a1-3f7e-4945-a0c3-4120aea108bd_terseLabel_en-US" xlink:label="lab_us-gaap_NonoperatingIncomeExpenseAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other income (expense):</link:label>
    <link:label id="lab_us-gaap_NonoperatingIncomeExpenseAbstract_label_en-US" xlink:label="lab_us-gaap_NonoperatingIncomeExpenseAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Nonoperating Income (Expense) [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NonoperatingIncomeExpenseAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NonoperatingIncomeExpenseAbstract"/>
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    <link:label id="lab_us-gaap_CryptoAssetSale_0bb57237-848c-4e78-9adf-f093a1488438_terseLabel_en-US" xlink:label="lab_us-gaap_CryptoAssetSale" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Disposition</link:label>
    <link:label id="lab_us-gaap_CryptoAssetSale_label_en-US" xlink:label="lab_us-gaap_CryptoAssetSale" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Crypto Asset, Sale</link:label>
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    <link:label id="lab_gemi_DeferredOfferingCostsIncurredButNotYetPaid_documentation_en-US" xlink:label="lab_gemi_DeferredOfferingCostsIncurredButNotYetPaid" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Deferred Offering Costs, Incurred But Not Yet Paid</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_gemi_DeferredOfferingCostsIncurredButNotYetPaid" xlink:to="lab_gemi_DeferredOfferingCostsIncurredButNotYetPaid" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_Liabilities_7250e71a-4558-42e0-ba09-b399a9da17ab_totalLabel_en-US" xlink:label="lab_us-gaap_Liabilities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total liabilities</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Liabilities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Liabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_Liabilities" xlink:to="lab_us-gaap_Liabilities" xlink:type="arc" order="1"/>
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    <link:label id="lab_us-gaap_AntidilutiveSecuritiesNameDomain_label_en-US" xlink:label="lab_us-gaap_AntidilutiveSecuritiesNameDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Antidilutive Securities, Name [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AntidilutiveSecuritiesNameDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AntidilutiveSecuritiesNameDomain"/>
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    <link:label id="lab_gemi_WarrantNoncurrent_documentation_en-US" xlink:label="lab_gemi_WarrantNoncurrent" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Warrant Derivative, Noncurrent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_gemi_WarrantNoncurrent" xlink:href="gemi-20250930.xsd#gemi_WarrantNoncurrent"/>
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    <link:label id="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversion_e1e7ca91-354c-41cb-a21b-0643c9bd9cf6_terseLabel_en-US" xlink:label="lab_gemi_ShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsInitialPublicOfferingConversion" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Conversion upon IPO (in shares)</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_gemi_IncreaseDecreaseInDueToFundingDebt" xlink:href="gemi-20250930.xsd#gemi_IncreaseDecreaseInDueToFundingDebt"/>
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    <link:label id="lab_gemi_NonCashLeaseExpense_6dab66bd-9e9e-4f7a-b432-e7befb8e3ce0_terseLabel_en-US" xlink:label="lab_gemi_NonCashLeaseExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-cash lease expense</link:label>
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    <link:label id="lab_gemi_NonCashLeaseExpense_documentation_en-US" xlink:label="lab_gemi_NonCashLeaseExpense" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Non-Cash Lease Expense</link:label>
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    <link:label id="lab_us-gaap_StockIssuedDuringPeriodValueNewIssues_df90c4bf-a7e5-4463-a9d6-41f4f7497295_terseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodValueNewIssues" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Issuance of common stock</link:label>
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    <link:label id="lab_us-gaap_ScheduleOfShareBasedCompensationRestrictedStockUnitsAwardActivityTableTextBlock_9be84f5c-e710-4b7d-b281-b974e50cc437_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfShareBasedCompensationRestrictedStockUnitsAwardActivityTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Restricted Stock Unit Activity</link:label>
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    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtSecurities_471e9425-102c-4550-a622-e29b26417074_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtSecurities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Corporate bonds</link:label>
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    <link:label id="lab_gemi_ConvertibleDebtReceivedInCryptoAssets_ead24996-3708-4161-a66f-3e6d2c810870_terseLabel_en-US" xlink:label="lab_gemi_ConvertibleDebtReceivedInCryptoAssets" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party convertible notes received in crypto assets</link:label>
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