Exhibit 23.1

CONSENT OF INDEPENDENT REGISTERED PUBLIC ACCOUNTING FIRM

We hereby consent to the incorporation by reference in this Registration Statement on Form S-1 pursuant to Rule 462(b) under the Securities Act of 1933, as amended, of our report dated August 11, 2025 (which includes an explanatory paragraph relating to Apex Treasury Corporation’s ability to continue as a going concern), relating to the financial statements of Apex Treasury Corporation as of June 30, 2025, and for the period from June 26, 2025 (inception) to June 30, 2025, appearing in the Registration Statement on Form S-1, File No. 333-289485.

/s/ WithumSmith+Brown, PC

New York, New York

October 27, 2025