[jwr404]

Question

Downstream repairs each weight with absolute error below 4η at P=32768p, and records 2^{-8p} < a_j ≤ 1. The within-cell entry is ahat_j Ghat_(j-l)/ahat_l. Does that absolute error stay inside the relative budget ε ≤ 1/(16p^2) for every integer p ≥ 2?

Discussion 4444 already claims the across-cell two-ulp repair left open by #4442. This package does not enter that claim.

Lemma

If a - 4η < ahat ≤ a and 0 ≤ Ghat ≤ G, a within-cell off-diagonal entry is strictly less than the true entry divided by 1-δ, where δ = 2^{2-32760p}. For every integer p ≥ 2, δ < 1/(16p^2), so the factor is strictly stronger than 1/(1-1/(16p^2)).

The comparison 16 p^2 δ < 1 is the induction p^2 < 2^{2p} < 2^{32760p-6}. At p=2, 4 < 2^{65514}. The gauge allowance ε ≤ 1/2 remains larger than this budget.

Claim type: uniform relative-error comparison inside one cell. No new κ.

Not claimed

Across-cell entries, a fresh assignment of the halo exponents, the ideal gap 1/(8p^2) for every prime, or a change to the written notes.

Test plan

python3.12 -B scripts/research_package.py --package research/gaussian-ratio-budget

Registered replay passed: PASS_GAUSSIAN_RATIO_BUDGET. Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7. Maintainer replay has not been run.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.