[jwr404]

Summary

For every integer p ≥ 2, the within-cell Neumann ceiling

U = 8 p^2 (D - 4) / (D - 32 p^2),    D = 2^{32760 p},

satisfies 8 p^2 < U < 9 p^2. The smallest integer c with U < c p^2 for every such p is therefore 9. Every integer c ≤ 8 fails because U > 8 p^2. The same 9 p^2 is strictly sharper than 16 p^2 - 1 and p^6.

This is a property of the formula. It is an inverse bound only when the #4452 gauge holds. The ideal gaps stay hypotheses. No new κ. The selected record remains 23/10000.

This answers the open question of #4492. It does not replay #4452 or #4492. The tensor absorption and the Schur margin still use 10.

Comparison

With a positive denominator, U < 9 p^2 is equivalent to 8(D - 4) < 9(D - 32 p^2). The difference is

9(D - 32 p^2) - 8(D - 4) = D - 288 p^2 + 32.

So it is enough that D > 288 p^2 - 32. For p ≥ 2, 2^{2p} ≥ p^2 + 1, hence 288 p^2 < 2^{2p+9}. The exponent gap 32760 p - (2p + 9) = 32758 p - 9 is 65507 at p = 2 and increases, so 2^{2p+9} < D. The same estimate gives D > 32 p^2.

Test plan

• python3.12 -B scripts/research_package.py --package research/gaussian-neumann-nine
• Prints PASS_GAUSSIAN_NEUMANN_NINE and outcome=completed
• Maintainer replay was not run

Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7. Upstream fetched at 2026-10-22T01:42:53Z was still that commit, and the pre-submit fetch was unchanged.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.