The precise submission claim

The principal new deliverable requested by the plan is an explicit finite certificate for one frozen scalar optimization. That deliverable is Theorem thm:finite-certificate. Its proof is independent of the external multi-use entropy inequality.

The manuscript also supplies the elementary one-use results needed to identify the certified scalar maximum with $\chi ({\mathcal{A}}_{1/2,1/4})$. These are not presented as new discoveries: the one-use generalized amplitude-damping optimization has precedents in Hou–Fang and in the discussion of Khatri, Sharma, and Wilde HouFang,KhatriSharmaWilde. A local proof is included so that the finite certificate has no dependency on the unverified part of Family 276.

The identifier Oai276-A is used as requested. It does not assign authorship of this development to N. Cheung or assert its registration, endorsement, or acceptance in another series.