[petalas]

Summary

Scoped analytic proof: for every finite banded Gaussian chirp in the specified quadratic family, the unique real origin minimizing ||N(c)-I||_infinity is c*=(n-1)/2+beta/theta.

James Reed #4305: c=0, cited ten-vertex row-norm bound <97/100
New origin:      c*=-1435/2, proved row-norm bound <2^-38

The actual norm strictly decreases by uniqueness. This compares different members of the same diagonal-similarity family. It does not re-check #4305, change eigenvalues, improve kappa or admit a supplier. The new proof and verifier were produced by Nick Petalas. Credit James Reed / jwr404 #4304/#4305 for the origin question and fixed-origin sections, and Cinna Davis #4100 for the Gaussian chirp formula. Classical convexity, reflection and elementary series methods retain their credit.

Evidence

The written proof uses reflection and strict convexity. Exact finite corroboration passes 216 parameter cases, 4,140 exponent fixtures, 2,790 reflected-row equalities, 3,222 strict inequalities, all 34 prime-scale in-band exponent bounds and 15 controls. The prime-scale constant uses rational Machin and positive Taylor certificates.

From the repository root, with CPython 3.12.15 and no extra dependencies:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gaussian-origin-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/gaussian-origin-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gaussian-origin-optimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gaussian-origin-optimum --check-source-history

Normal and optimized direct checks each took about 0.2 seconds locally. Both registered commands passed. Committed bytes/source history, official export integrity and read-only Git-free empty-PATH normal/optimized replay passed; every exported file retained its bytes and mode. Expected markers are PASS_GAUSSIAN_ORIGIN_OPTIMUM, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES and PASS_RESEARCH_EXPORT. Another installed CPython 3.12 may replace the local interpreter path. Receipts are generated from the six package files, never handwritten.

Limitations

Only the origin varies; this is not optimization over arbitrary positive diagonal scalings. The example requires chirp diagonal range exp(81*pi). Conversion, magnitude, precision/tape and complete physical costs remain unpriced. A valid fixture also shows that optimality alone does not imply contraction for all parameters. No recursive Schur schedule, native supplier, compiler, stock, selected assumption or complete multiplication theorem is established. Lean, Docker, selected scientific replay, independent human review and formal proof-kernel review are unrun.

The original uniform-transfer-audit card is retained with an explicit changed scope. All predecessor pins are bibliographic provenance; no predecessor executable or receipt is consumed.

Open next questions

What origin is best under a specified logarithmic range budget for the chirp diagonal? First bound the permitted origins using an actual caller's scaling/precision allowance, then minimize the convex objective on that set. The missing physical allowance is the blocker.

Related open work

Refreshed after CI: inspected the titles and bodies of this account's ten other open PRs created since the advance claim at 2026-10-27T15:14:20Z. None studies this Gaussian origin-minimization problem. The merged #4304/#4305 fixed-origin results remain credited predecessors.

• #5952 studies spectral conditioning of signed rank transport in a finite-group register model. This is adjacent conditioning work with a different norm and object; it does not optimize Gaussian chirp origins, and no numerical comparison is made.

The earlier complete overlap snapshot covered 324 open PRs and 5,496 Discussion records. Sparse local notes and arbitrary code diffs remain incomplete coverage.

CI notes

Both hosted checks passed at dfc3b4196557eec8b7196ca881b8beb4e62cb8eb: plan and replay (research/gaussian-origin-optimum), in run 40530673686. The check listing was read once after completion. Hosted replay certifies the registered package checks, not the universal written theorem or physical supplier obligations. All local checks described above also passed.

Merge danger

Door: two-way. Blast radius: package. Six new isolated research files; no shared source, pinned predecessor, priority or selected-frontier changes.