[jwr404]

Summary

Under the same gate p ≥ max(64, 4K + 2C) and 16 g ceil(log2 p) ≤ p,

g (9 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(K p - p^2) < 2^{-C p}

for integers p ≥ 2, g ≥ 1, K ≥ 0 and C ≥ 0. For every integer p ≥ 64, p^6 exp(-p^2) < 1/2, so

9 p^2 + p^6 exp(-p^2) < 9 p^2 + 1 < 16 p^2.

The comparison 9 p^2 + 1 < 16 p^2 is 1 < 7 p^2. The base stays strictly below 16 p^2, so the telescoping exponent is the same power of two already cleared for (16 p^2)^{g-1}, and it is at most -C p - 4.

The section ceiling 9 p^2 and the section error stay hypotheses. At g = p^2 the approximant monomial is still greater than 1. No new κ. The selected record remains 23/10000.

This answers the open question of #4496. It does not replay #4496. The face factor 2 is left open. #4486 is the same shape with coefficient 10.

Test plan

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

Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7. Upstream fetched at 2026-10-22T01:56:48Z was still that commit, and the pre-submit fetch was unchanged.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.