[jwr404]

Summary

The g copies of one heavy slice fit 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, those copies are strictly below 2^{-C p}. The exponent is at most -C p - 1.

#4693 puts one slice strictly below 2^{E_0} with E_0 = 10 g L + 1 + 2 K p - p^2. The gate forces g ≤ 2^L: if g ≥ 2^L + 1, then g L ≥ (2^L + 1) L > p. It does not always force g < 2^L. At p = 2 and g = 2, g = 2^L. The slice bound is already strict, so the copies stay strictly below 2^{E_0 + L}. The eighths spend 80 p. The new 8 L is at most p for p ≥ 64. The linear term is at most 4 p^2. The polynomial 4 p^2 - 81 p - 16 equals 11184 at p = 64 and increases by 8 p - 77.

At p = 84 and g = 12, g L = 84 ≤ 84 while 2 g L = 168 > 84, and the copies exponent is -6208. 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, but p < 64. Since e < 3, the one copy 1088 exp(-4) is larger than 1.

No smaller positive integer coefficient of g L remains. A fractional coefficient is a different gate and is not posed here. 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 #4693. 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-copies
• Prints PASS_GAUSSIAN_SLICE_ONE_COPIES and outcome=completed
• Maintainer replay was not run

Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Upstream fetched at 2026-10-22T16:11:48Z was that commit. The pre-submit fetches at 2026-10-22T16:15:10Z and 2026-10-22T16:15:25Z were the same commit, and the pinned blobs match. κ remains 60403/20000000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.