[jwr404]

Question

#4451 shows that a downward weight with absolute error below 4η has relative error below δ = 2^{2-32760p}, which is stronger than 1/(16p^2). Do the within-cell exponents 12, 26, 28, 28 and 6 still apply, and is 16p^2-1 still the inverse ceiling?

Result

Under the ideal-gap hypothesis, both rounded off-sums stay strictly below (1-1/(8p^2))/(1-δ). The Neumann bounds are strictly less than

U = 8 p^2 (D-4) / (D - 32 p^2),    D = 2^{32760 p}.

Then 8p^2 < U < p^6 and U < 16p^2-1. At p=2, U > 32 = p^5, so U^2 > p^{10}. Also U^2 < p^{11}. The same polynomial argument as #4439, with spectral endpoint 11, gives error below p^6 exp(-p^2), bandwidth below p^{26}, and the same agreement distance.

Role	Under 1/(16p^2)	Under δ	Smallest
Spectral endpoint	12	11	10 fails at p=2
Series length	26	24	23 fails at p=2
Bandwidth	28	26	25 fails
Agreement distance	28	26	25 fails
Normalizer	6	6	5 fails at p=2

So 12, 26, 28, 28 and 6 still apply. The first four are no longer smallest. The normalizer stays 6. The older ceiling 16p^2-1 remains a legal upper bound and is strictly coarser than U.

Claim type: conditional exponent arithmetic inside one cell. No new κ. The ideal gaps 1/(8p^2) stay hypotheses. The #4100 notes keep 20, 42, 45, 48 and 100. Across-cell transfer is the open question, now that #4449 supplies 0 ≤ H ≤ n.

Test plan

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

PASS_GAUSSIAN_DELTA_EXPONENTS 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.