[jwr404]

Question

#4462 transferred the exponents 11, 24, 26, 26 and 6 to the banded reference, with robust inverse bound 10 p^2. Its open question is whether those exponents survive a Schur complement. The named first step is to restate the elimination gap with 10 p^2 in place of 16 p^2 - 1.

Result

Suppose the diagonal is 1 and every row and column has absolute off-sum at most 1 - 1/(10 p^2). One elimination step preserves a margin of at least 1/(10 p^2) and does not increase an absolute row or column sum. After any principal elimination,

||S^{-1}||_1 ≤ 10 p^2,    ||S^{-1}||_∞ ≤ 10 p^2,    ||S||_2 < 2.

The #4452 polynomial then has error below p^6 exp(-p^2), bandwidth below p^{26}, and the same agreement distance. The five exponents apply to S. At p = 2, (10 p^2)^2 = 1600 < 2048, while (16 p^2 - 1)^2 = 3969 > 2048, so the older ceiling does not give spectral endpoint 11.

#4462's stronger margin (4p-1)/(32 p^3) is at least 1/(10 p^2), so those matrices meet the hypothesis. This package does not reprove that margin, the ideal gaps, or the core side 400.

Claim type: elimination lemma for one row-dominance margin. No new κ.

Test plan

python3.12 -B scripts/research_package.py --package research/gaussian-schur-gap

PASS_GAUSSIAN_SCHUR_GAP and outcome=completed on this machine. Maintainer replay has not been run.

Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7, selection refresh κ = 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.