[petalas]

Summary

Scoped written proof with exact finite corroboration. For every normalized constant-shift Gaussian section with N>=2, gamma>0 and |s|<=1/2, improve the finite-cap inverse spectral-norm bound of Jose Manuel / PR4193:

- N / P_(N-1)^3
+ sqrt(D_N(q^4)) / P_(N-1)^3
  D_N(x) = N(N+1)/2 + ((N-1)x-Nx^2+x^(N+1))/(1-x)^2
  q = exp(-gamma), P_(N-1) = product_(r=1)^(N-1)(1-q^(2r))

The new bound is strictly smaller at every N>=2. For N=2,q=1/2 it is 112/27 instead of 128/27, exactly one eighth smaller. For fixed q the ratio of the upper bounds tends to 1/sqrt(2). No sharp actual inverse-norm asymptotic is claimed.

The new step retains the complete inverse-entry decay in the Frobenius sum. PR4193's entry theorem is a cited premise, not an audit deliverable. Only six files in research/gaussian-finite-cap-frobenius/ are added. No new kappa or selected-frontier change.

This contribution was produced by Nick Petalas. Credit Jose Manuel / jmbarrancoml PR4193 for the entry theorem, finite-cap framework and explicit open question; Vlado Vojdanovski / PR1673 for the retained q-LDU identities; Jeremy Kothe and PR1758 contributors for the shifted-kernel lineage; and Cinna Davis for the later caller. Frobenius domination, convexity and finite geometric sums are classical. Predecessor packages and credit remain unchanged.

Advance CLAIM. Base canonical main 7c49fe3, refreshed unchanged before submission, priority revision 8, selected PR6030. The original card is explicitly changed to this analytic side result; no selected assumption is discharged.

Evidence

Before, the cited entry estimate was bounded by N/P_(N-1)^3. After, the written proof supplies the smaller closed-form bound uniformly through both half-shift endpoints. Universal quantifiers come from PROOF.md and the cited entry premise.

Local CPython 3.12.15, standard library only:

• 896 exact geometric cases, including 889 strict improvements at nontrivial sizes.
• 180 literal rational inverses, 2,376 entries and 180 shift-envelope cases. Gauss-Jordan inversion uses no predecessor executable; both inverse products are checked before testing the full squared Frobenius norm.
• Fourteen invalid inputs reject and nine enumerated semantic control groups pass. A disposable optimized mutation omitting the distance contribution rejects.
• Warning-strict normal and optimized complete commands pass with identical generated receipts. Individually timed local runs take 0.359s and 0.243s, under their 120-second limits. These are check durations, not performance-improvement claims.
• Registered replay, exact committed bytes, modes and source ancestry pass at 587475d.
• Official committed export and input-coverage checks pass. Read-only Git-free normal/optimized replay with empty PATH preserves all seven exported files, including the manifest, and produces receipts identical to the committed checkout.
• Source compilation, function-size inspection and whitespace checks pass. Maximum function length is 33 lines.

Reproduce from the repository root:

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

Another CPython 3.12 path may be substituted. Expected markers are PASS_GAUSSIAN_FINITE_CAP_FROBENIUS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Direct verification writes no files and emits a generated receipt bound to all six package hashes.

Limitations and unrun obligations

This is an improved analytic upper bound under PR4193's entry theorem, not an optimal norm, physical inverse algorithm, native supplier or complete multiplication theorem. The asymptotic ratio keeps q fixed. Actual diagonal gauges, coefficient preparation, rounded-prefix error, movement, restoration, stock, tape/compiler costs and complete caller integration remain separate.

Every registered command ran locally. No predecessor scientific suite, selected cold replay, native suite, Lean or Docker build was run. Independent human and formal proof-kernel review are unclaimed. All five selected framework assumptions remain.

Open next questions

• Find a matching finite-size lower bound or the sharp fixed-q half-shift inverse-norm asymptotic, starting with a trial vector for the positive checkerboard inverse and retaining the boundary rows.
• Bind a named rounded constant-shift caller to the new allowance, retaining its exact perturbation inequality, diagonal gauges and complete physical costs.

The first verified improvement is shipped without pursuing these follow-ons.

Related open work

• Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1758 supplies the shifted Gaussian/gauge-free interior lineage. This adds a smaller finite-section inverse bound, without admitting its complete construction or comparing kappa values.

• Conditional κ = 0.001678344171736538: structured (displacement-rank) boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1754 studies structured Gaussian boundary solves. This does not change their displacement representation, solve program or complete cost.

• Add physical Gaussian row norm transfer audit #3910 proves a forward Gaussian row-mass and domain-transfer bound. This concerns a constant-shift inverse spectral norm, so no common numerical norm is computed.

• Prove an exact Gaussian boundary determinant and rank obstruction (no κ) #6404 proves an exact cross-rim determinant after eliminating a truncated Gaussian interior. That boundary determinant is a different quantity from this untruncated input inverse bound.

• Prove the sharp t=1/2 boundary for Gaussian Schur fill (no κ) #6405 proves a sharp positivity range for truncated Gaussian Schur fill. This retains the untruncated constant-shift inverse-entry contract and supplies no Schur-fill claim.

• Bound separated circular Schur bandwidth; exact 3w-1 family (no κ) #6420 bounds circular Schur bandwidth after multiple deletions. This supplies no layout, deletion or circular-bandwidth result.

• Prove sharp critical-tail parity-cut bound N (no κ) #6470 proves an infinity-norm envelope for critical-tail triangular Toeplitz compiler cuts. This bounds the spectral norm of a completed constant-shift Gaussian inverse; the symbol class and norm differ, and no common numerical norm is computed.

The overlap helper reports 406 open PRs and 6,358 Discussion records with complete remote pagination. Local sparse-note coverage is incomplete. Fresh open PRs, claims, main README formula/object searches and merged-PR searches were checked before committing; no same-bound result was identified in that coverage. Unpublished work and arbitrary old diffs remain outside it.

The completion refresh inspected all fifteen account open PRs created since the advance CLAIM, including this PR, and added #6470 above. The other new submissions study separate quantities. No competing value for this same Gaussian bound was identified in those stated results; no extra computations checked other contributions.

CI notes

Both hosted checks passed at exact contribution head 587475d in Registered research packages run 41550349850: plan and registered package replay. The completed checks were read once; run metadata confirms completed status, success and the matching head. The replay executes both registered warning-strict normal and optimized commands. CI certifies execution of those bounded checks; the universal result rests on the written proof and the credited entry theorem. All registered commands also passed locally. Independent human and formal proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated research package. Six added files; shared code, frozen evidence and selected records are unchanged.