[jwr404]

Summary

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

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

is strictly below 2^{-C p}. The exponent spends 10 g L. One times the eighths is 8 (E + C p). The gate gives 80 g L = 80 · g L ≤ 80 p. The identity 8 (2 K + C) = 4 (4 K + 2 C) bounds the linear term by 4 p^2. The sum is at most -(4 p^2 - 80 p - 8). The polynomial 4 p^2 - 80 p - 16 equals 11248 at p = 64 and increases by 8 p - 76, so the exponent is at most -C p - 1.

At p = 84 and g = 12, g L = 84 ≤ 84 while 2 g L = 168 > 84, and the exponent is -6215. The factor-2 comparison 160 g L ≤ 80 p fails there: 13440 > 6720. This 1 multiplies g L. It is not a one-bit majorant of the ceiling 9 and not a two-power majorant of C. The coefficient-9 monomials in #4495 still use 16 g L ≤ p, because 9 > 8. At p = 2 and g = 1 the factor holds, g L = 1 ≤ 2, but p < 64. Since e < 3, the slice 1088 exp(-4) is larger than 1.

The g copies under this gate stay open. The factor does not force g < 2^L when L = 1. 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 #4687. 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-one-gate
• Prints PASS_GAUSSIAN_SLICE_ONE_GATE and outcome=completed
• Maintainer replay was not run

Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Upstream fetched before the scaffold and again at 2026-10-22T16:08:45Z was that commit. The immediate pre-submit fetch at 2026-10-22T16:09:38Z was the same commit, and the pinned blobs match. κ remains 60403/20000000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.