[petalas]

Summary

Scoped proof with exact finite corroboration. For the ideal nearest-grid cyclic Gaussian matrix with ratio 1+1/s, even s, and every bandwidth 2 <= w < s/2, the continuous concentration threshold for strict row dominance is

$$A_{s,w}=\frac{\log s-\log\log s+\log 2}{2}+o(1),$$

uniformly in the bandwidth. The proof gives a matching sufficient condition and a literal three-entry half-tie obstruction. No new κ is claimed.

- Retained row majorant: exp(-2aθ) + 5 exp(-2a)
+ New majorant: exp(-2aθ) + 2 exp(-2a) + 2 exp(-6a)/(1-exp(-6a))
+ Matching threshold, including the log-log term and additive log(2)/2

The new farther-tail estimate and matching obstruction are the contribution. The nearest-neighbor estimate is cited from Vlado Vojdanovski's retained PR1673 phase-cell audit, with the RaD/hipotures lineage; the matrix convention comes from Jose Manuel's PR3851. Existing sources and credit are preserved. This contribution was produced by Nick Petalas.

Only six new files under research/gaussian-concentration-threshold/ are added. Exact base and source pins are in SOURCES.json; the universal argument is in PROOF.md.

Evidence

• Before: the retained coefficient-five estimate does not locate the sharp additive transition constant.
• After: the proof matches two leading tail contributions with three literal entries. At s=2^20, exact certificates give failure at a=8 log(2) and dominance at a=9 log(2). At s=2^19, a=8 log(2), omitting one leading tail contribution falsely admits the matrix.
• The verifier enumerates 522 geometry equalities, 336 complete band rows with 4,368 retained off-diagonal entries, and all 231 declared dyadic concentration/period pairs: 136 certified dominant, 95 certified nondominant, none unresolved. Four invalid arithmetic domains are rejected.
• All finite enclosures use integer arithmetic on a 192-bit dyadic grid. No floating point, transcendental oracle or predecessor verifier is used.
• CPython 3.12.15 registered normal/optimized checks pass. Timed invocations took 0.072388s and 0.072748s locally, each below the 60s timeout. The direct isolated warnings-as-errors/development-mode run passes.
• Metadata, committed bytes, source ancestry and executable modes pass. Official export integrity and declared-input coverage pass for eight package/input files. Normal and optimized Git-free replay pass with empty PATH and read-only files. Source corruption is rejected; all nine exported files, including the generated export receipt, retain their hashes.
• The selected binding-only orientation passes after materializing its sparse inputs. No selected scientific replay is claimed.

Exact reproduction from the repository root:

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

Use another CPython 3.12 executable on other systems. The verifier prints PASS_GAUSSIAN_CONCENTRATION_THRESHOLD and a generated hash-bound report. The finite checks corroborate the written theorem; they do not establish its asymptotic quantifiers.

Limitations

Concentration varies outside the unchanged production guard. This is a theorem about ideal row dominance, not a reduced-concentration physical caller or a new inverse implementation. Numerical coefficient errors, infinite periodization comparison, LU growth, preparation/tape costs, tensor assembly and the complete recursive invoice remain separate obligations. All five selected framework assumptions remain. No full selected cold replay, proof kernel or independent human review ran. The proof does not determine the sign at the exact critical asymptotic offset; integer u retains bounded quantization uncertainty.

Open next questions

• Price one complete physical parameter tuple at reduced concentration, including its coefficient-error and inverse allowances. The explicit majorant supplies the first test; physical geometry and the complete invoice are blockers.
• Determine the first vanishing correction to the continuous threshold. Start with the same half-tie entries and a sharper uniform remainder; an exact maximizing-row characterization remains open.

Related open work

• Prove production Gaussian inverse bound 2 + 2p/u² (no new κ) #6126 studies the same nearest-grid Gaussian kernel and strengthens the rounded inverse bound under unchanged production guards. This result instead varies concentration and locates a matching dominance/failure threshold. It computes no competing inverse value, so no numerical agreement or disagreement is asserted.
• Prove production Gaussian positivity and retained matrix perturbation #6099 proves positivity and a retained-band perturbation estimate for the physical Gaussian producer. It is context, not a theorem dependency or replay target here.
• Audit uniform directed bounds for actual encoded Gaussian coefficients #6085 supplies the directed coefficient-enclosure theorem used by Prove production Gaussian positivity and retained matrix perturbation #6099. This ideal-kernel threshold makes no claim about those rounded coefficients.
• Add physical Gaussian row norm transfer audit #3910 bounds the normalized Gaussian expansion row mass. Its expansion norm concerns a different map from the original-coordinate dominance threshold here.
• Account for final-grid rounding in the Gaussian block enclosure #371 accounts for final-grid Gaussian-block rounding. This proof uses ideal coefficients and supplies no replacement rounding budget.

CI notes

Uncertified by CI. For contribution commit 537caa285a813a5024ec4d95aaa2710eed76a841, run 40693507009 failed in plan at the onboarding helper tests. The package replay job was skipped and did not run in CI. Local replay success is not hosted certification.

The exact cause is two existing navigation mismatches on canonical main 721c03822c421b7f7fcbb2598993b947c4d88419, whose PR6030 admission commit was pushed with [skip ci]:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md. The file instead bolds 0.003125804 = 781451/250000000 together, so the required literal is absent.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md. That section still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

The failed logs were inspected. PR6120 run 40691776027 shows the same two failures. This contribution changes neither file and does not cause those failures. It needs a maintainer re-run after main's documentation is fixed; no hosted scientific check is claimed.

Merge Danger

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

Blast Radius: research. No shared implementation, frozen source, or selected record is changed.