Exhibit 23.1

CONSENT OF INDEPENDENT REGISTERED PUBLIC ACCOUNTING FIRM

We hereby consent to the use in the Prospectus constituting a part of this Registration Statement on Form S-1 of our report dated June 18, 2026, relating to the financial statements of Bleichroeder Acquisition Corp. III as of April 10, 2026 and for the period from April 1, 2026 (inception) through April 10, 2026, which is contained in that Prospectus.

We also consent to the reference to us under the caption “Experts” in the Prospectus.

/s/ WithumSmith+Brown, PC

East Brunswick, New Jersey

June 18, 2026