Exhibit 10.20

ACTUATE THERAPEUTICS, INC.

1751 River Run, Suite 400
Fort Worth, Texas 76107

October 2, 2026

Bios Partners, L.P,

1751 River Run, Suite 400

Fort Worth, TX 76107

Dear Dr. Fletcher:

We are writing with respect to the shares of common stock of Actuate Therapeutics, Inc., a Delaware corporation (the “Company”), beneficially owned by you and the various entities (“Bios Entities”) managed by Bios Partners, L.P. and that appear above your signature block.

As you are aware, Konik Capital Partners, LLC, a division of T.R. Winston (“Konik”), and Company propose to enter into an Underwriting Agreement (the “Underwriting Agreement”) providing for the public offering (the “Public Offering”) of shares of common stock, par value $0.000001 per share (“Common Stock”), of the Company. In connection with the Public Offering, you and the Bios Entities are being asked to lock-up all shares (“Shares”) of Common Stock of the Company held by either of you, both held as of the date of this letter agreement or hereafter acquired, for the period ending on the two-year anniversary of the close of the Public Offering, pursuant to the form of Lock-Up Agreement (“Lock-Up Agreement”) attached hereto as Exhibit A.

This will confirm the agreement of you and the Bios Entities to enter into the Lock-Up Agreements in consideration of the Company’s agreement to issue to those Bios Entities designated by you in writing, at and subject to the close of the Public Offering, warrants (“Bios Warrants”) to purchase, over a five-year term from the close of the Public Offering, an aggregate of 3,000,000 shares of Common Stock at an exercise price equal to 150% of the price offered to the public in the Public Offering and otherwise on the same terms as the warrant to be issued to Konik in the Public Offering.

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If this letter agreement accurately reflects our agreements and understandings, please acknowledge your agreements on behalf of you and the Bios Entities by executing this letter agreement in the locations provided below.

ACKNOWLEDGED AND AGREED:

Bios Actuate Co-Invest I, LP, Bios Actuate Co-Invest II, LP, Bios Actuate Co-Invest III, LP, Bios Fund I QP, LP, Bios Fund I, LP, Bios Fund II NT, LP, Bios Fund II QP, LP, Bios Fund II, LP, Bios Fund III NT, LP, Bios Fund III QP, LP, Bios Fund III, LP, BP Directors, LP, Bios Clinical Opportunity Fund, LP and Bios 2024 Co-Invest, LP

Exhibit A

Form of Lock-Up Agreement

October 2, 2026

Konik Capital Partners, LLC,

A Division of T.R. Winston and Company

7 World Trade Center, 46th Floor

New York, NY 10007

As Representative of the several Underwriters named on Schedule 1 to the Underwriting Agreement referenced below

and

Actuate Therapeutics, Inc.

1751 River Run, Suite 400
Fort Worth, Texas 76107

Ladies and Gentlemen:

The undersigned understands that Konik Capital Partners, LLC, a division of T.R. Winston and Company (the “Representative”), proposes to enter into an Underwriting Agreement (the “Underwriting Agreement”) with Actuate Therapeutics, Inc., a Delaware corporation (the “Company”), providing for the public offering (the “Public Offering”) of shares of common stock, par value $0.000001 per share, of the Company (the “Common Stock”).

To induce the Representative to continue its efforts in connection with the Public Offering, the undersigned hereby agrees that, without the prior written consent of the Representative, the undersigned will not, during the period commencing on the date hereof and ending on the two-year anniversary of the close of the Public Offering (the “Lock-Up Period”), (1) offer, pledge, sell, contract to sell, grant, lend, or otherwise transfer or dispose of, directly or indirectly, any shares of Common Stock or any securities convertible into or exercisable or exchangeable for Common Stock, whether now owned or hereafter acquired by the undersigned or with respect to which the undersigned has or hereafter acquires the power of disposition (collectively, the “Lock-Up Securities”); (2) enter into any swap or other arrangement that transfers to another, in whole or in part, any of the economic consequences of ownership of the Lock-Up Securities, whether any such transaction described in clause (1) or (2) above is to be settled by delivery of Lock-Up Securities, in cash or otherwise; (3) make any demand for or exercise any right with respect to the registration of any Lock-Up Securities; or (4) publicly disclose the intention to make any offer, sale, pledge or disposition, or to enter into any transaction, swap, hedge or other arrangement relating to any Lock-Up Securities. Notwithstanding the foregoing, and subject to the conditions below, the undersigned may transfer Lock-Up Securities without the prior written consent of the Representative in connection with (a) transfers of Lock-Up Securities as a bona fide gift, by will or intestacy or to a family member or trust for the benefit of the undersigned or a family member (for purposes of this lock-up agreement, “family member” means any relationship by blood, marriage or adoption, not more remote than first cousin); (b) transfers of Lock-Up Securities to a charity or educational institution; (c) if the undersigned is a corporation, partnership, limited liability company or other business entity, (i) any transfers of Lock-Up Securities to another corporation, partnership or other business entity that controls, is controlled by or is under common control with the undersigned or (ii) distributions of Lock-Up Securities to members, partners, stockholders, subsidiaries or affiliates (as defined in Rule 405 promulgated under the Securities Act of 1933, as amended) of the undersigned; provided that in the case of any transfer pursuant to the foregoing clauses (a), (b) or (c), (i) any such transfer shall not involve a disposition for value, (ii) each transferee shall sign and deliver to the Representative a lock-up agreement substantially in the form of this lock-up agreement and (iii) no filing under Section 13 or Section 16(a) of the Securities Exchange Act of 1934, as amended (the “Exchange Act”), or other public announcement shall be required or shall be voluntarily made; (d) the transfer of Lock-Up Securities that occurs by operation of law, such as pursuant to a qualified domestic order or in connection with a divorce settlement, provided that the transferee agrees to sign and deliver a lock-up agreement substantially in the form of this lock-up agreement for the balance of the Lock-Up Period, and provided further, that any filing under Section 13 or Section 16(a) of the Exchange Act that is required to be made during the Lock-Up Period as a result of such transfer shall include a statement that such transfer has occurred by operation of law; and (e) the transfer of Lock-Up Securities pursuant to a bona fide third party tender offer, merger, consolidation or other similar transaction made to all holders of the Common Stock involving a change of control (as defined below) of the Company after the closing of the Public Offering and approved by the Company’s board of directors; provided that in the event that the tender offer, merger, consolidation or other such transaction is not completed, the Lock-Up Securities owned by the undersigned shall remain subject to the restrictions contained in this lock-up agreement. For purposes of clause (e) above, “change of control” shall mean the consummation of any bona fide third party tender offer, merger, amalgamation, consolidation or other similar transaction the result of which is that any “person” (as defined in Section 13(d)(3) of the Exchange Act), or group of persons, becomes the beneficial owner (as defined in Rules 13d-3 and 13d-5 of the Exchange Act) of a majority of total voting power of the voting stock of the Company. The undersigned also agrees and consents to the entry of stop transfer instructions with the Company’s transfer agent and registrar against the transfer of the undersigned’s Lock-Up Securities except in compliance with this lock-up agreement.

If the undersigned is an officer or director of the Company, (i) the undersigned agrees that the foregoing restrictions shall be equally applicable to any issuer-directed or “friends and family” Securities that the undersigned may purchase in the Public Offering; (ii) at least three (3) business days before the effective date of any release or waiver by the Representative and the Company of the foregoing restrictions in connection with a transfer of Lock-Up Securities, the Company will announce the impending release or waiver by press release through a major news service at least two (2) business days before the effective date of the release or waiver. Any release or waiver granted by the Representative and the Company hereunder to any such officer or director shall only be effective two (2) business days after the publication date of such press release. The provisions of this paragraph will not apply if (a) the release or waiver is effected solely to permit a transfer of Lock-Up Securities not for consideration and (b) the transferee has agreed in writing to be bound by the same terms described in this lock-up agreement to the extent and for the duration that such terms remain in effect at the time of such transfer.

The undersigned understands that the Company and the Representative are relying upon this lock-up agreement in proceeding toward consummation of the Public Offering. The undersigned further understands that this lock-up agreement is irrevocable and shall be binding upon the undersigned’s heirs, legal representatives, successors and assigns.

The undersigned understands that, if the Underwriting Agreement is not executed by October 19, 2026, or if the Underwriting Agreement (other than the provisions thereof which survive termination) shall terminate or be terminated prior to payment for and delivery of the Common Stock to be sold thereunder, then this lock-up agreement shall be void and of no further force or effect.

This lock-up agreement may only be amended or waived by the mutual written consent of the Representative, the Company and the undersigned.

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Whether or not the Public Offering actually occurs depends on a number of factors, including market conditions. Any Public Offering will only be made pursuant to an Underwriting Agreement, the terms of which are subject to negotiation between the Company and the Representative.

AGREED AND ACCEPTED:

ACTUATE THERAPEUTICS, INC.