[jwr404]

Summary

One heavy slice fits the factor-4 gate. For integers p ≥ 2, g ≥ 1, K ≥ 0 and C ≥ 0, whenever p ≥ max(64, 4K + 2C) and 4 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. Four times the eighths is 32 (E + C p). The gate gives 320 g L = 80 · 4 g L ≤ 80 p. The identity 32 (2 K + C) = 16 (4 K + 2 C) bounds the linear term by 16 p^2. The sum is at most -(16 p^2 - 80 p - 32). The polynomial 16 p^2 - 80 p - 64 equals 60352 at p = 64 and increases by 32 p - 64, so the exponent is at most -C p - 1.

At p = 84 and g = 3, 4 g L = 84 ≤ 84 while 5 g L = 105 > 84, and the exponent is -6845. The factor-5 comparison 400 g L ≤ 80 p fails there: 8400 > 6720. This 4 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 hypothesis fails. Since e < 3, the slice 1088 exp(-4) is larger than 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 #4670. 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-four-gate
• Prints PASS_GAUSSIAN_SLICE_FOUR_GATE and outcome=completed
• Maintainer replay was not run

Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Upstream fetched at 2026-10-22T15:14:16Z was that commit. The pre-submit fetches at 2026-10-22T15:16:03Z and 2026-10-22T15:16:22Z were the same commit, and the pinned blobs match. κ remains 60403/20000000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.