[jwr404]

Question

#4439 left the exponents 12, 26, 28, 28 and 6 on the rounded block inside one cell. Do they survive on the banded reference, whose across-cell entries are the downward Gaussian entries rather than ahat_j Ghat_(j-k)/ahat_k?

Lemma

Assume the ideal row and column off-sums are at most 1-1/(8p^2), each within-cell off-diagonal entry is at most the matching entry of N divided by 1-ε with ε≤1/(16p^2), and each across-cell entry lies in [0, n]. For an integer bandwidth W≤p-1, both gaps of H are at least 1/(16p^2-1). Hence ||H||_2<2 and ||H^{-1}||_2≤16p^2-1.

The polynomial P=p^{-12} sum_{j<p^{26}} Q^j H^* then has error < p^6 exp(-p^2) and bandwidth < p^{28}.

Exponent	Verdict
12, 26, 28	sufficient, and still smallest
6, the normalizer	sufficient, and still smallest
5, 11, 25, 27	fail, at p=2 or for p≥3 at W=p-1

The across-cell bound is the reading of LOG-A’s “downward Gaussian entries”, downstream’s “downward dyadic approximation of the original entry”, and the review’s downward dyadic entries. FINE-PRIME says “rounded” in that place and does not contain the downward phrase. At p=2, an overshoot of one part in 64 already exceeds the ceiling, because 63·63=3969>3968=62·64. A step g=2 at θ=β=1/4 makes the bare kernel exp(-a ρ h^2) larger than n (square gaps 32 and 20 in sixteenths), so downward-of-Ghat is a different rule.

Claim type: conditional transfer for that banded reference. No new κ.

Not claimed

The four-ulp repair is stated for weights and truncated Toeplitz coefficients. This package does not prove a stored across-cell value lands in [0, n]. The ideal 1/(8p^2) gaps stay the eventual statements in GAUSSIAN and #4402. The Schur complement, the core exponent 400, and the written notes are untouched.

Test plan

python3.12 -B scripts/research_package.py --package research/gaussian-cross-exponents

Registered replay passed on this branch: PASS_GAUSSIAN_CROSS_EXPONENTS. Base 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7. Maintainer replay has not been run.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.