[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: the sharp positive parameter range for the strict ordered-minor and Schur-fill sign criterion of the linearly truncated real Gaussian is 0<t<=1/2, uniformly over every finite size and bandwidth.

 G[i,j] = t^((i-j)^2) for |i-j|<=b, otherwise 0
-PR5207: sufficient range 0<t<=1/3
+new: exact uniform range 0<t<=1/2
+every t>1/2 has a negative width-one principal determinant

This settles Open next question 2 of PR5207. A finite alternating-tail estimate supplies negative-root factorization through the sharp endpoint, including the repeated root at b=1,t=1/2. A tridiagonal determinant recurrence excludes every larger t. The earlier path-factorization implication is cited and explained, with no predecessor census replay.

Only research/sharp-gaussian-schur-range/ is added. The contribution was produced by Nick Petalas. Credit Ryanson Jonathan / PR4407 for the actual-real-entry question, unpaidmoon / PR4373 for the separated carve, James Reed for the earlier Gaussian/bandwidth fixtures, Cinna Davis / PR4100 for the caller motivation, and the unchanged retained lineage. The alternating-sum, path and tridiagonal techniques are classical; no literature-priority claim is made for them.

Advance CLAIM.

Evidence

The written universal proof is in PROOF.md. The standalone exact verifier tests t=2/5,9/20,1/2, all in the newly covered range:

• 503 strict polynomial signs, one exceptional endpoint zero and 75 alternating tails.
• All 8,307 ordered minors of six-vertex kernels at b=1,2,3: 6,525 positive and 1,782 zero.
• All 2,223 nonempty-survivor Schur layouts through n=7: 25,335 entries checked by rational elimination and integer determinant ratios, with 16,503 positive, 4,632 negative and 4,200 zero.
• Six above-half counterexamples, including first negative principal order 157 at t=5001/10000; rational and denominator-cleared integer recurrences agree. The endpoint principal identity is checked through order 512.
• Eight invalid-input rejections, three altered-matrix rejections and two Schur support/parity fixtures.

On Darwin arm64, CPython 3.12.15, both exact registered commands pass locally in 0.270s and 0.271s. The full package replay passes in 0.781s. These are command durations, not performance-improvement claims. Committed bytes, modes and source ancestry pass at a294413. Official committed export and input-coverage checks pass; normal and optimized Git-free exported runs with empty PATH produce JSON identical to the working runs, including all six package hashes.

From the repository root:

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

Substitute another CPython 3.12 interpreter elsewhere. Expected statuses are PASS_SHARP_GAUSSIAN_SCHUR_RANGE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Default verification writes nothing; generated receipts bind all six package hashes. Source pins are historical provenance, with no imported predecessor executable or external scientific input.

Limitations and unrun obligations

This is a scoped proof, not supplier admission or a complete multiplication theorem. The sharpness is uniform over bandwidths; a fixed b>=2 is a different question. No concrete retained caller has been shown to satisfy the real linear-path/positive-gauge hypotheses. Periodic kernels, complex phases, conditioning, rounding, native realization, ownership, routing, stock, tape costs, compiler admission and exact recovery remain outside scope. The five selected framework assumptions remain. No new kappa or selected-frontier change.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor scientific replays and the selected full cold replay were not run. No local check was deferred to CI.

Open next questions

1. Bind one actual retained linear Gaussian caller to the sharp range, including its positive gauges, truncation, conditioning and paid physical contract. The missing caller and additional phases are blockers.
2. Determine the maximal range for a fixed bandwidth b>=2 using its finite Gaussian polynomial and an all-size argument. The width-one uniform obstruction does not settle that problem.

Related open work

• PR1754 proposes structured Gaussian boundary solves; this exact sign criterion neither prices nor audits that solver.

• PR1758 proposes shifted/gauged Gaussian interiors; the present positive-gauge corollary does not establish its concrete parameter or numerical caller obligations.

• PR3910 bounds physical Gaussian row norms and a local error transfer; this result concerns exact truncated-kernel minors and fill signs, with no norm comparison.

• PR371 accounts for final-grid Gaussian rounding; this exact algebraic result supplies no rounding or caller admission.

• PR6404 gives an exact cross-rim Schur determinant and full-rank obstruction on the same truncated kernel, including other fields and parameters. This contribution determines the sharp uniform real ordered-minor/sign range; it claims no new cross-rim determinant formula. No additional computation or numerical comparison of PR6404 was performed.

• PR6413 reduces periodic Gaussian expansion setup work while preserving interval outputs; this contribution concerns exact linearly truncated minors and makes no periodic setup or runtime claim. No common numerical output is compared.

No open PR on the exact uniform ordered-minor threshold was found in the live/cached scope. Searches cover public retained work, not unpublished work or arbitrary historical diffs.

CI notes

Hosted CI passed at exact head a2944133c5f43179df4033acc0dfd1cdc88af956 in run 41527666333. Both plan and replay (research/sharp-gaussian-schur-range) succeeded. The completed checks were read once; separate run metadata confirms the successful conclusion and matching head. Hosted replay covers the registered finite checks and package validation; the universal theorem rests on PROOF.md. All required local checks also pass. Related open work was refreshed against this account's open PRs created since the CLAIM, including PR6404 and PR6413. No extra computations checked other contributions, and no conflicting reported number was found.

Merge Danger

Door: two-way. The isolated package can be reverted.

Blast Radius: isolated. No shared implementation, predecessor evidence or selected-frontier file changes.