Reconstructing the Bandwidth-One 68.185% Ceiling

Starting from a single Remark, this page goes directly into Zeta23/PairCeiling/ in Anthropic's official Lean companion repository and reads out a verifiable exact-rational constant: the simple-point fraction recorded in LawN256.lean is p0=0.681828687463832…, already very close to the 0.68185 used in the paper's Remark 1.1. The document reconstructs three layers of machinery — the stability identity that lifts discrete near-CUE rows to a continuous certificate value (Abel summation plus two integrations by parts), the near-CUE law that quantifies “near-CUE” into an error bound, and the signed ceiling inequality that combines the two into a directly citable form. The core insight is an information-indistinguishability obstruction: bandwidth-one pair observables being nearly identical does not mean the simple fraction approaches 1 — `LawN256.lean` is itself a deliberately constructed probability law whose average pair-correlation almost perfectly mimics CUE, yet whose average simple-point fraction is only about 68.18%. The document also honestly records the first gap: the external certificate cert_N256_blk_b128m.json specified by the official source (SHA-256 known) is listed as “available from the authors,” but the public repo does not currently include it, so the full chain — configuration generation → exact-rational LP solve → external JSON certificate — cannot yet be rerun end to end.