Exhibit 10.1

SPONSOR SUPPORT AGREEMENT

This SPONSOR SUPPORT AGREEMENT is made and entered into as of December 17, 2025 (this “Agreement”), by and between CGC III Sponsor LLC, a Cayman Islands limited liability company (“Sponsor”) and Factorial Inc., a Delaware corporation (the “Company”).

WHEREAS, Cartesian Growth Corporation III, an exempted company incorporated under the laws of the Cayman Islands (“CGC”), the Company and Fenway MS, Inc., a Delaware corporation (“Merger Sub”), propose to enter into, contemporaneously herewith, that certain Business Combination Agreement, dated as of the date hereof (the “BCA”; terms used but not defined in this Agreement shall have the meanings ascribed to them in the BCA); and

WHEREAS, as of the date hereof, Sponsor owns beneficially and of record 6,800,000 CGC Class B Shares (the “Sponsor Shares”).

WHEREAS, as of the date hereof, Sponsor owns beneficially and of record 4,400,000 CGC Warrants (the “CGC Private Warrants”).

NOW, THEREFORE, in consideration of the foregoing and of the mutual covenants and agreements contained herein, the receipt and sufficiency of which is hereby acknowledged, and intending to be legally bound, the parties hereto hereby agree as follows:

1.  Voting Obligations. Until the earlier of (a) the Closing or (b) termination of the BCA in accordance with its terms, Sponsor agrees that, at the CGC Shareholders Meeting and in connection with any written consent of the CGC Shareholders, Sponsor shall (A) appear at each such meeting or otherwise cause all of its Sponsor Shares to be counted as present thereat for purposes of calculating a quorum and (B) vote (or duly and promptly execute and deliver an action by written consent), or cause to be voted at such meeting (or cause such consent to be duly and promptly executed and delivered with respect to), all of the Sponsor Shares (i) in favor of the approval and adoption of the BCA, the transactions contemplated by the BCA, and any other proposal submitted by the CGC Board for approval by the CGC Shareholders in connection with the transactions contemplated by the BCA, including the Warrant Amendment, (ii) in favor of any other matter reasonably necessary to the consummation of the transactions contemplated by the BCA and considered and voted upon by the CGC Shareholders, including the Warrant Amendment, (iii) against any action, agreement or transaction or proposal that would (A) reasonably be expected to result in a breach of any covenant, representation or warranty or any other obligation or agreement of CGC under the BCA or Ancillary Documents, (B) reasonably be expected to result in the failure of the transactions contemplated by the BCA to be consummated, or (C) change the business, management, or CGC Board except as contemplated by the BCA and the Ancillary Documents, and (iv) against any merger agreement, merger, consolidation, combination, sale of substantial assets, reorganization, recapitalization, dissolution, liquidation or winding up of or by CGC (other than the BCA and the transactions contemplated by the BCA). Until the earlier of (a) the Closing or (b) termination of the BCA in accordance with its terms, Sponsor agrees that, at any meeting of the holders of CGC Warrants and in connection with any written consent of the holders of CGC Warrants, Sponsor shall (A) appear at each such meeting or otherwise cause all of its CGC Private Warrants to be counted as present thereat for purposes of calculating a quorum and (B) vote (or duly and promptly execute and deliver an action by written consent), or cause to be voted at such meeting (or cause such consent to be duly and promptly executed and delivered with respect to), all of the CGC Private Warrants in favor of the approval and adoption of the Warrant Amendment.

2.  CGC Private Warrant Exchange. At the Closing, Sponsor shall enter into the CGC Private Warrant Exchange Agreement pursuant to which it shall agree to exchange its CGC Private Warrants for shares of CGC Class A Common Stock at the same ratio as applied to the exchange of the CGC Public Warrants for shares of CGC Class A Common Stock pursuant to the Warrant Amendment.

3.  Waiver of Redemption Rights. Sponsor agrees not to (a) demand that CGC redeem the Sponsor Shares in connection with the transactions contemplated by the BCA or (b) otherwise participate in any such redemption by tendering or submitting any of the Sponsor Shares for redemption.

4.  Waiver of Anti-Dilution Rights. Sponsor hereby agrees to waive the provisions of Section 17.2 set forth in the Governing Document of CGC relating to the adjustment of the Initial Conversion Ratio (as defined in the Governing Document of CGC) in connection with the transactions contemplated by the BCA and agrees not to exercise, assert or perfect, any rights to adjustment or other anti-dilution protections with respect to the rate at which CGC Class B Shares held by such Sponsor convert into CGC Class A Shares, whether resulting from the transactions contemplated by the BCA or otherwise, so that each CGC Class B Share held by Sponsor issued and outstanding as of the Closing shall convert into one CGC Class A Share pursuant to, and in accordance with, the Governing Document of CGC.

5.  Transfer of Shares and CGC Private Warrants. Sponsor acknowledges and agrees that the Sponsor Shares and CGC Private Warrants are subject to the transfer restrictions under that certain Letter Agreement, dated May 1, 2025, between Sponsor and CGC (the “Letter Agreement”). Sponsor hereby agrees that the Company is an express third party beneficiary of the Letter Agreement with rights to enforce the provisions thereof against the Sponsor.

6.  Representations and Warranties. Sponsor represents and warrants to the Company as follows:

(a)            Organization. Sponsor is duly organized, validly existing and in good standing under the Laws of the jurisdiction in which it is incorporated, formed, organized or constituted, and the execution, delivery and performance of this Agreement and the consummation of the transactions contemplated hereby are within the Sponsor’s corporate, limited liability company or organizational powers and have been duly authorized by all necessary corporate, limited liability company or organizational actions on the part of the Sponsor.

(b)            No Conflicts. The execution, delivery and performance by Sponsor of this Agreement and the consummation by Sponsor of the transactions contemplated hereby do not and will not (i) conflict with or violate any United States or non-United States Law applicable to Sponsor, (ii) require any consent, approval or authorization of, declaration, filing or registration with, or notice to, any person or entity, (iii) result in the creation of any encumbrance on any Sponsor Shares or CGC Private Warrants (other than under this Agreement, the BCA and the agreements contemplated by the BCA, including the other Ancillary Documents), or (iv) conflict with or result in a breach of or constitute a default under any provision of Sponsor’s governing documents.

(c)            Ownership. As of the date of this Agreement, Sponsor owns exclusively and has good, valid and marketable title to the Sponsor Shares and the CGC Private Warrants free and clear of any Lien, proxy, option, right of first refusal, agreement, voting restriction, limitation on disposition, charge, adverse claim of ownership or use or other encumbrance of any kind, other than pursuant to (i) this Agreement, (ii) applicable securities Laws, and (iii) the Governing Document of CGC, and as of the date of this Agreement, Sponsor has the sole power (as currently in effect) to vote and right, power and authority to sell, transfer and deliver the Sponsor Shares and the CGC Private Warrants, as applicable, and Sponsor does not own, directly or indirectly, any other CGC Class B Shares.

(d)            Due Authorization. Sponsor has the power, authority and capacity to execute, deliver and perform this Agreement and this Agreement has been duly authorized, executed and delivered by Sponsor and, assuming due authorization, execution and delivery by the other parties to this Agreement, this Agreement constitutes a legally valid and binding obligation of the Sponsor, enforceable against the Sponsor in accordance with the terms hereof (except as enforceability may be limited by bankruptcy Laws, other similar Laws affecting creditors’ rights and general principles of equity affecting the availability of specific performance and other equitable remedies).

(e)            Litigation. There is no Proceeding pending against the Sponsor, or to the knowledge of the Sponsor threatened against Sponsor, before (or, in the case of threatened actions, that would be before) any arbitrator or any Governmental Entity, which in any manner challenges or seeks to prevent, enjoin or materially delay the performance by Sponsor of its obligations under this Agreement, the BCA or the transactions contemplated thereby.

(f)            Acknowledgment. Sponsor understands and acknowledges that the Company is entering into the BCA in reliance upon its execution and delivery of this Agreement.

7.  Termination. The obligations of the parties under this Agreement shall automatically terminate upon the earlier of (i) the Effective Time and (ii) the termination of the BCA in accordance with its terms, provided, that, if the Effective Time occurs, Section 5 shall terminate in accordance with the terms of the CGC Bylaws. Upon termination or expiration of this Agreement, no party shall have any further obligations or liabilities under this Agreement. Notwithstanding any termination of this Agreement, no such termination or expiration shall relieve any party hereto from liability for fraud or willful breach of this Agreement occurring prior to its termination.

8.  Miscellaneous.

(a)            All notices, requests, claims, demands and other communications hereunder shall be in writing and shall be given (and shall be deemed to have been duly given upon receipt) by delivery in person, by e-mail or by registered or certified mail (postage prepaid, return receipt requested) to the respective parties at the following addresses or e-mail addresses (or at such other address or e-mail address for a party as shall be specified in a notice given in accordance with this Section 8(a)):

If to CGC prior to or on the Closing Date, or to Sponsor, to:

505 Fifth Avenue, Suite 1500
New York, NY 10017

with a copy (which shall not constitute notice) to:

Greenberg Traurig, LLP
One Vanderbilt Avenue

New York, New York 10017

If to the Company:

Factorial Inc.
805 Middlesex Turnpike

Billerica, MA 01821

with a copy (which shall not constitute notice) to:

Goodwin Procter LLP
100 Northern Avenue
Boston, MA 02210

(b)            If any term or other provision of this Agreement is invalid, illegal or incapable of being enforced by any rule of law, or public policy, all other conditions and provisions of this Agreement shall nevertheless remain in full force and effect so long as the economic or legal substance of the transactions contemplated hereby is not affected in any manner materially adverse to any party hereto. Upon such determination that any term or other provision is invalid, illegal or incapable of being enforced, the parties hereto shall negotiate in good faith to modify this Agreement so as to effect the original intent of the parties as closely as possible in a mutually acceptable manner in order that the transactions contemplated hereby be consummated as originally contemplated to the fullest extent possible.

(c)            (i) The words “hereof”, “herein”, and “hereunder” and words of similar import, when used in this Agreement, shall refer to this Agreement as a whole and not to any particular provision of this Agreement; (ii) the words “date hereof,” when used in this Agreement, shall refer to the date set forth in the Preamble; (iii) the terms defined in the singular have a comparable meaning when used in the plural, and vice versa; (iv) the terms defined in the present tense have a comparable meaning when used in the past tense, and vice versa; (v) any references herein to a specific Section or Article shall refer, respectively, to Sections or Articles of this Agreement; (vi) references herein to any gender (including the neuter gender) includes each other gender; (vii) the word “or” shall not be exclusive; (viii) the headings herein are for convenience of reference only, do not constitute part of this Agreement and shall not be deemed to limit or otherwise affect any of the provisions hereof, and (ix) the parties hereto have participated jointly in the negotiation and drafting of this Agreement and, in the event that an ambiguity or question of intent or interpretation arises, this Agreement shall be construed as jointly drafted by the parties hereto and no presumption or burden of proof shall arise favoring or disfavoring any party by virtue of the authorship of any provision of this Agreement.

(d)            Sponsor agrees while this Agreement is in effect, not to take or agree or commit to take any action that would make any representation and warranty of Sponsor contained in this Agreement inaccurate or has the effect of preventing or disabling Sponsor from performing its obligations under this Agreement.

(e)            This Agreement is intended to create, and creates, a contractual relationship and is not intended to create, and does not create, any agency, partnership, joint venture or any like relationship between the parties hereto.

(f)            This Agreement, the BCA and Ancillary Documents constitute the entire agreement among the parties hereto with respect to the subject matter hereof and supersedes all prior agreements and undertakings, both written and oral, among the parties hereto, or any of them, with respect to the subject matter hereof. This Agreement shall not be assigned (whether pursuant to a merger, by operation of law or otherwise) by any party hereto without the prior express written consent of the other parties hereto.

(g)            This Agreement shall be binding upon and inure solely to the benefit of each party hereto, and nothing in this Agreement, express or implied, is intended to or shall confer upon any other person any right, benefit or remedy of any nature whatsoever under or by reason of this Agreement.

(h)            The parties hereto agree that irreparable damage would occur if any provision of this Agreement were not performed in accordance with the terms hereof, and, accordingly, that the parties hereto shall, to the fullest extent permitted by Law, be entitled to an injunction or injunctions to prevent breaches of this Agreement or to enforce specifically the performance of the terms and provisions hereof in any New York State court or Federal court of the United States of America sitting in New York City in the Borough of Manhattan without proof of actual damages or otherwise, in addition to any other remedy to which they are entitled at law or in equity. To the fullest extent permitted by applicable Law, each of the parties hereto hereby further waives (i) any defense in any Proceeding for specific performance that a remedy at law would be adequate and (ii) any requirement under any Law to post security or a bond as a prerequisite to obtaining equitable relief.

(i)            This Agreement shall be governed by, and construed in accordance with, the Laws of the State of New York applicable to contracts executed in and to be performed in that State. Any Proceeding arising out of or relating to this Agreement or the transactions contemplated hereby shall, to the fullest extent permitted by applicable Law, be heard and determined exclusively in any New York State court or Federal court of the United States of America sitting in New York City in the Borough of Manhattan. To the fullest extent permitted by applicable Law, the parties hereto hereby (i) irrevocably submit to the exclusive jurisdiction of the aforesaid courts for themselves and with respect to their respective properties for the purpose of any Proceeding arising out of or relating to this Agreement or the transactions contemplated hereby brought by any party, and (ii) agree not to commence any such Proceeding except in the courts described above in New York, other than any Proceeding in any court of competent jurisdiction to enforce any judgment, decree or award rendered by any such court in New York as described herein. To the fullest extent permitted by applicable Law, each of the parties hereto further agrees that notice as provided herein shall constitute sufficient service of process and the parties hereto further waive any argument that such service is insufficient. To the fullest extent permitted by applicable Law, each of the parties hereto hereby irrevocably and unconditionally waives, and agrees not to assert, by way of motion or as a defense, counterclaim or otherwise, in any Proceeding arising out of or relating to this Agreement or the transactions contemplated hereby, (A) any claim that it is not personally subject to the jurisdiction of the courts in New York as described herein for any reason, (B) that it or its property is exempt or immune from jurisdiction of any such court or from any legal process commenced in such courts (whether through service of notice, attachment prior to judgment, attachment in aid of execution of judgment, execution of judgment or otherwise), and (C) that (x) the Proceeding in any such court is brought in an inconvenient forum, (y) the venue of such Proceeding is improper, or (z) this Agreement or the transactions contemplated hereby, or the subject matter hereof, may not be enforced in or by such courts.

(j)            This Agreement may be executed and delivered (including by facsimile or portable document format (PDF) transmission) in one or more counterparts, and by the different parties hereto in separate counterparts, each of which when executed shall be deemed to be an original but all of which taken together shall constitute one and the same agreement.

(k)            Without further consideration, each party hereto shall execute and deliver or cause to be executed and delivered such additional documents and instruments and take all such further action as may be reasonably necessary or desirable to consummate the transactions contemplated by this Agreement.

(l)            This Agreement shall not be effective or binding upon any party hereto until after such time as the BCA is executed and delivered by CGC, the Company, and Merger Sub.

(m)            Each of the parties hereto hereby waives to the fullest extent permitted by applicable Law any right it may have to a trial by jury with respect to any Proceeding directly or indirectly arising out of or relating to this Agreement or the transactions contemplated hereby. Each of the parties hereto (i) certifies that no representative, agent or attorney of any other party has represented, expressly or otherwise, that such other party would not, in the event of any Proceeding, seek to enforce that foregoing waiver and (ii) acknowledges that it and the other parties hereto have been induced to enter into this Agreement and the transactions contemplated hereby, as applicable, by, among other things, the mutual waivers and certifications in this Section 8(m).

[Signature pages follow]

IN WITNESS WHEREOF, the parties have executed this Agreement as of the date first written above.