[jwr404]

Summary

The g copies of the heavy slice fit 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, g copies of

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

are strictly below 2^{-C p}. The gate forces g < 2^L, so the copies add one L. For p ≥ 64, 8 L ≤ p, and 32 L becomes 4 p. Four times the eighths is then at most 84 p + 32 - 16 p^2. The polynomial 16 p^2 - 84 p - 64 equals 60096 at p = 64 and increases by 32 p - 68, 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 copies exponent is -6838. 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 = 2 the hypothesis fails. Since e < 3, two copies are larger than 1.

A factor smaller than 4 stays 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 #4673. 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-copies
• Prints PASS_GAUSSIAN_SLICE_FOUR_COPIES and outcome=completed
• Maintainer replay was not run

Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Upstream fetched at 2026-10-22T15:27:47Z was that commit. The pre-submit fetches at 2026-10-22T15:29:17Z and 2026-10-22T15:29:33Z were the same commit, and the pinned blobs match. κ remains 60403/20000000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.