[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 U < 10 p^2. The coefficient 8 does not: U > 8 p^2 exactly when 32 p^2 > 4.

This is a property of the formula. It is an inverse bound only when the #4452 gauge holds. In that case the Neumann 1-norm and ∞-norm are already strictly below U, hence strictly below 10 p^2. The ideal gaps stay hypotheses. No new κ. The selected record remains 23/10000.

This answers the open question of #4491. It does not replay #4452. The smallest integer coefficient is left open.

Comparison

With a positive denominator, U < 10 p^2 is equivalent to 8(D - 4) < 10(D - 32 p^2). The difference is

10(D - 32 p^2) - 8(D - 4) = 2(D - 160 p^2 + 16).

So it is enough that D > 160 p^2 - 16. For p ≥ 2, 2^{2p} ≥ p^2 + 1, hence 160 p^2 < 2^{2p+8}. The exponent gap 32760 p - (2p + 8) = 32758 p - 8 is 65508 at p = 2 and increases, so 2^{2p+8} < D. The same estimate gives D > 32 p^2.

Test plan

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

Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.