[jwr404]

Summary

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

g (9 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. The identity is

g (9 p^2)^{g-1} p^6 = g (9 p^2)^g (p^4 / 9).

The factor p^4 / 9 contributes 4 L - 3 in the exponent, because p^4 ≤ 2^{4 L} and 9 > 8. The folded exponent

2 g L + 4 g + 5 L - 3 + 2 K p - p^2

is at most -C p - 3. Eight times that shifted exponent is at most -4(p^2 - 2 p + 6), and the polynomial is at least 6.

The section ceiling 9 p^2 and the section error p^6 exp(-p^2) stay hypotheses. At g = p^2 the telescoping error is still greater than 1. No new κ. The selected record remains 23/10000.

This answers the open question of #4495. It does not replay #4495. The approximant base 9 p^2 + p^6 exp(-p^2) is left open.

Test plan

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

Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7. Upstream fetched at 2026-10-22T01:52:41Z was still that commit, and the pre-submit fetch was unchanged.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.