[jwr404]

Summary

This adds research/gaussian-cell-exponents. Inside one cell, once the relative weight error is at most 1/(16p^2) and the ideal off-sums are at most 1-1/(8p^2), both 2-norms of the rounded block meet the ceiling used by #4406. The exponents 12, 26, 28, 28 and 7 all apply. The normalizer 7 is not smallest: the inverse ceiling is 16p^2-1, so 6 already puts it strictly below one.

Claim type: conditional exponent arithmetic for that block. No new κ. Across-cell entries are not included.

Lemma

Both rounded gaps are at least 1/(16p^2-1). Hence ||A^{-1}||_2 ≤ 16p^2-1 and ||A||_2 < 2. The polynomial P = p^{-12} sum_{j<p^{26}} Q^j A^* then has error < p^6 exp(-p^2) and bandwidth < p^{28} when the integer bandwidth is at most p-1.

Exponent	Verdict
12, 26, 28	sufficient, and still smallest
7, the normalizer	sufficient, but 6 is smallest
5, as a normalizer or error multiplier	fails at p=2 (63>32)

Not covered

• The global reference. Across-cell entries are not the gauge ratio.
• Any new κ. The selected record remains 23/10000.

Evidence

python3.12 -B scripts/research_package.py --package research/gaussian-cell-exponents
  PASS_REGISTERED_RESEARCH_PACKAGE research/gaussian-cell-exponents outcome=completed

Head: 26af8afbadd7f8feb737d98b47ac9f4b45d35c7a. Base: canonical main 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7.

Test plan

python3.12 -B scripts/research_package.py --package research/gaussian-cell-exponents --check-only
python3.12 -B scripts/research_package.py --package research/gaussian-cell-exponents

Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0. This PR does not merge anything or change the frontier.