[jwr404]

Summary

This adds research/gaussian-gauge-column. Inside one phase cell the rounded block is diag(ahat) Ghat diag(ahat)^{-1}. A relative weight error of 1/(16p^2) preserves a column gap above 1/(16p^2). The gauge audit's allowance ε ≤ 1/2 does not.

Claim type: a written transfer. No new κ. Across-cell entries are not bounded.

Lemma

With β_{i+h} = β_i + θ h, the ideal entry factors as (a_i/a_{i+h}) exp(-a ρ h^2). If Ghat is a downward rounding of that kernel and (1-ε) a ≤ ahat ≤ a, a within-cell column off-sum is at most the ideal off-sum divided by 1-ε.

When the ideal off-sum is at most 1 - 1/(8p^2), the budget ε ≤ 1/(16p^2) leaves gap at least 1/(16p^2 - 1). The smallest integer setup constant for which 2^{-C p} meets that budget at every integer p ≥ 2 is C = 3. The note's unnamed C is not claimed to be this large.

The downstream row bound does not substitute for this. Charging a band of 2p-1 entries with row mass 1/(16p) exceeds the ideal gap 1/(8p^2).

Not covered

• Across-cell entries, which use the downward original entry rather than this ratio.
• The exponents in HALO exponents 20, 42, 45, 48 and the normalizer 100 can fall under the ceiling 16 p^2 (no new κ) #4406, and any new κ. The selected record remains 23/10000.

Evidence

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

Head: 6157a47bfe169cd9c5f666f85ab8caa929e1174f. Base: canonical main 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7.

Test plan

python3.12 -B scripts/research_package.py --package research/gaussian-gauge-column --check-only
python3.12 -B scripts/research_package.py --package research/gaussian-gauge-column

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