[jwr404]

Summary

One heavy slice fits the factor-8 gate. For integers p ≥ 2, g ≥ 1, K ≥ 0 and C ≥ 0, whenever p ≥ max(64, 4K + 2C) and 8 g ceil(log2 p) ≤ p,

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

The exponent spends 10 g L, so the gate gives 80 g L ≤ 10 p. Eight times the exponent is then at most 10 p + 8 - 4 p^2. The polynomial 4 p^2 - 10 p - 16 equals 15728 at p = 64 and increases by 8 p - 6, so the exponent is at most -C p - 1.

At p = 112 and g = 2, 8 g L = 112 ≤ 112 while 9 g L = 126 > 112, and the exponent is -12403. This 8 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-8 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 on this base is 60403/20000000. This answers the open question of #4531. It branches from the fetched main b58ad9ef4 and does not replay that unmerged package. The predecessor was pinned to the older main f9f85c1e.

Test plan

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

Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Upstream fetched at 2026-10-22T14:29:21Z was that commit. The pre-submit fetch, reread at 2026-10-22T14:31:01Z, was unchanged, and the four pinned blobs match. κ remains 60403/20000000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.