[jwr404]

Summary

One heavy slice fits the factor-7 gate. For integers p ≥ 2, g ≥ 1, K ≥ 0 and C ≥ 0, whenever p ≥ max(64, 4K + 2C) and 7 g ceil(log2 p) ≤ p,

(p^{10})^{g-1} p^6 (1 + p^4) exp(K p - p^2) < 2^{-C p}.

Seven times the eighths is 56 (E + C p). The gate gives 560 g L = 80 · 7 g L ≤ 80 p, so that multiple is at most 80 p + 56 - 28 p^2. The polynomial 28 p^2 - 80 p - 112 equals 109456 at p = 64 and increases by 56 p - 52, so the exponent is at most -C p - 1.

At p = 98 and g = 2, 7 g L = 98 ≤ 98 while 8 g L = 112 > 98, and the exponent is -9463. The factor-8 line 80 g L ≤ 10 p fails there (1120 > 980). This 7 multiplies g L. The coefficient-9 monomials in #4495 still use 16 g L ≤ p, because 9 > 8. At p = 2 and g = 1 the factor-7 hypothesis fails, and e < 3 puts the slice above 1.

The g copies under this gate stay open. The trailing bound p^{10}, the section error, and the ceiling 9 p^2 stay hypotheses. This package does not prove ||H^{-1}||_2 ≤ p^{10}. No new κ. The selected record is 60403/20000000. This answers the open question of #4661. It branches from fetched main and does not replay that unmerged package.

Test plan

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

Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Upstream fetched at 2026-10-22T14:42:11Z was that commit. The pre-submit fetch, reread at 2026-10-22T14:43:15Z, was unchanged, and the pinned blobs match. κ remains 60403/20000000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.