Exhibit 32.1
CERTIFICATION PURSUANT TO
18 U.S.C. SECTION 1350
AS ADOPTED PURSUANT TO
SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002
In connection with the Quarterly Report of HCM III Acquisition Corp. (the “Company”) on Form 10-Q for the quarterly period ended June 30, 2025, as filed with the Securities and Exchange Commission (the “Report”), I, Shawn Matthews , Chief Executive Officer and Director, certify, pursuant to 18 U.S.C. §1350, as adopted pursuant to §906 of the Sarbanes-Oxley Act of 2002, that, to the best of my knowledge:
| 1. | The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and |
| 2. | The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company. |
Dated: September 12, 2025
| /s/ Shawn Matthews | |
| Shawn Matthews | |
| Chief Executive Officer and Director | |
| (principal executive officer) |