[jwr404]

Summary

Under HALO's written hypotheses, the global majorant is p^{10} exp(-p^2). It is not the within-cell bound p^6 exp(-p^2).

• The identity P - H^{-1} = -Q^J H^{-1} multiplies (1 - p^{-40})^J by the inverse factor p^{10}. For an integer p, J = p^{42}, so J p^{-40} = p^2. That power fills the exp(-p^2) slot and leaves the factor p^{10}.
• p^{10} = p^4 p^6 and p^4 ≥ 16 > 1, so the written majorant is strictly larger than p^6 exp(-p^2). The truncation exponent 2 is not the leftover exponent 4.
• Lower the within-cell halo exponents to 11, 24, 26, 26 and 6 (no new κ) #4452's p^6 bound is a different polynomial inside one cell. It does not move the global reference.
• HALO exponents 20, 42, 45, 48 and the normalizer 100 can fall under the ceiling 16 p^2 (no new κ) #4406's p^6 bound needs ||H^{-1}||_2 ≤ 16 p^2, which is not proved for the rounded reference. At p = 2, 16 p^2 = p^6.

This package does not put p^{10} into the #4520 gate. No new κ. The selected record remains 23/10000.

Test plan

• python3.12 -B scripts/research_package.py --package research/gaussian-global-tail
• Prints PASS_GAUSSIAN_GLOBAL_TAIL and outcome=completed
• Maintainer replay was not run

Base f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc, priority revision 7. Upstream fetched at 2026-10-22T03:23:31Z was that commit, and the pre-submit fetch at 2026-10-22T03:26:10Z was unchanged. κ remains 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.