[jwr404]

Summary

This adds research/gaussian-halo-exponents/. It answers the open question of #4403: if both 2-norms are at most 16 p^2, which of HALO's exponents 20, 42, 45 and 48, and GAUSSIAN's normalizer 100, can decrease?

Claim type: conditional exponent arithmetic. No new κ. The ceiling is not proved for the rounded reference H, and the #4100 notes are not edited.

All five can decrease. For every integer p >= 2 and every integer bandwidth W <= p - 1, the same locality argument with the stronger ceiling gives:

Written	Replacement
20	12
42	26
45	28
48	28
100	7

The error is then strictly below p^6 exp(-p^2), and a row at distance p^28 from a section edge matches the global polynomial row. Each replacement is the smallest uniform integer with that role. The written core exponent 400 remains sufficient and can fall to 86; this package does not retune it.

Test plan

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

Maintainer replay has not been run. Head: b1fa86ecc00307058cfd324f9d87042c5d951264. Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7, selected record unchanged. Rechecked 2026-10-21T16:51:10Z.

Not covered

• Phase cuts, truncation and rounding, so ||H^-1||_2 <= 16 p^2 is not claimed for the rounded matrix.
• An edit to HALO.md, GAUSSIAN.md or PROOF.md. The written exponents stay sufficient.

Claim: #1166 (reply in thread)

Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0. Maintainer edits are allowed; this PR does not merge anything or change the frontier.

Made with Cursor