[jwr404]

Summary

#4517 already answered the operator-norm question in #4515. Its open step is the scalar comparison. Replacing the base 9 p^2 + ε in #4506 by 9 p^2 gives

2 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, whenever p ≥ max(64, 4K + 2C) and 16 g ceil(log2 p) ≤ p.

The factor 2 is one bit added to the #4496 exponent. Eight times the new exponent is at most -16, so the exponent is at most -C p - 2. Four bits still majorize 9 because 9 < 16. Three bits still fail because 9 > 8. At g = p^2 the monomial is greater than 1.

This absorbs the scalar only. It does not prove ||C_i|| ≤ 9 p^2 or ||P_glob - C_i||_2 < p^6 exp(-p^2). Both ideal-gap gauges stay hypotheses. No new κ. The selected record remains 23/10000.

Test plan

• python3.12 -B scripts/research_package.py --package research/gaussian-restricted-gate
• Prints PASS_GAUSSIAN_RESTRICTED_GATE and outcome=completed
• Maintainer replay was not run

Base f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc, priority revision 7. Upstream fetched at 2026-10-22T03:18:46Z was that commit, and the pre-submit fetch at 2026-10-22T03:22:35Z was unchanged. κ remains 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.