[mihir-s-05]

PR #4442 leaves the across-cell inequality 0 <= H[i,i+h] <= n[i,h] as a rounding premise. This package supplies the missing caller instantiation: the original square-gap exponent is exactly pi*j/k with nonnegative integer j, so the retained negative-exponential contract and two-ulp subtraction/clipping produce a P-grid coefficient with 0 <= n-H < 4*2^-P.

The repair itself is credited predecessor work. The addition is the exact integer argument, a written proof covering cell transitions and physical period cuts, the across-cell perturbation allowance, and a separately written rational checker. It also demonstrates why the bare Toeplitz kernel is the wrong argument and why later nearest rounding to a coarser grid can undo the guarantee.

Start with the proof and scope and the exact receipt. Only the new six-file package is changed.

Evidence at 1fb77b377f570ca84a89a465b2ee4ce8674b7159:

• CPython 3.12.0 on Windows: registered ordinary and optimized replays pass.
• 384 coefficient cases, 1,624 exact phase identities, 2,019 adversarial oracle cases and 10 failure controls. In 81 coefficient cases the uncorrected approximation is certified above the exact coefficient.
• Five additional in-memory code mutations reject: omitted subtraction, one ulp, omitted clipping, wrapped offset and bare-kernel substitution.
• Committed-byte check and official ten-file export/integrity check pass. Ordinary and optimized Git-free export replays pass with unchanged payload hashes; source and receipt corruption reject.

Reproduce from the repository root:

python3.12 -B scripts/research_package.py --package research/gaussian-cross-rounding

Boundary: this proves the coefficient guarantee when the stated routine is used. It does not establish that an existing online page generator uses it, audit #4442's halo/minimality argument, run a Schur/native/selected cold replay, or prove an unconditional multiplication theorem. The numerical checker uses its own rational enclosure oracle, not an implementation of the published fast exponential routine. No new kappa is claimed; the selected 23/10000 is unchanged. Hosted CI is separate from the local evidence.

Sources and coordination: started from canonical main 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7; canonical main was fetched again before submission and was unchanged. The handoff is #4442 at d079c2b058ebd3d7a9ff424113b095bf61175427; its unmerged objects are not replay dependencies. All live source pins are from main. Discussion #4444 is the canonical investigation, linked from the #1166 handoff. Final overlap refresh at 21:15–21:17 UTC paginated 303 open PRs, 3,294 Discussion records and 3,787 closed PRs, and searched all 5,731 tracked top-level note paths at the base. This is bounded text coverage, not an exhaustive diff search or novelty claim.

Credit James Reed, the retained RaD phase-cell producer/reviewer, Vlado Vojdanovski/PR1673, Chafik Boukhalfa, Cinna Davis, Ryanson Jonathan, David Harvey and Joris van der Hoeven, and the full lineage recorded in the README. Existing directed-rounding work by Mason Hale and Dugongue is acknowledged. Prepared for mihir-s-05 with substantial OpenAI Codex assistance; no subagents. Apache-2.0.