[jwr404]

Summary

This adds research/gaussian-column-cut. It continues open question 1 of #4402 for the ideal narrow matrix only. #4402 bounded a column inside one constant-phase run and showed that the run identity fails across a cut. The same majorant holds for every column of the ideal entry law, including a step g = 2.

Claim type: a written lemma. No new κ. This does not bound the rounded reference H.

Lemma

With β ∈ [-1/2, 1/2) and every step in {1, 2},

||N-I||_1 <= exp(-2aθ) + exp(-2a(1-θ)) + 2 exp(-2a)/(1-exp(-4a)).

A g = 2 step makes the nearest contribution smaller than the both-steps-1 endpoint. Under GAUSSIAN.md's hypotheses 4 log(16p) <= a <= 5 log(16p) and θ ∈ [1/(4p^2), 1/p^2], the column gap is strictly above 1/(8p^2) for every integer p >= 2. Truncation does not increase the off-sum. If at most 2p-1 nonzeros each move by at most ε and 2p ε <= 1/(16p^2), the gap stays above 1/(16p^2).

The a and θ inequalities are hypotheses. The note's "eventually" is not re-proved, and this ε is not identified with the written rounding budget.

Not covered

• Column sums of diag(ahat) Ghat diag(ahat)^{-1}, and the downward entries used across a cell.
• ||N^{-1}||_1, HALO exponents 20, 42, 45, 48 and the normalizer 100 can fall under the ceiling 16 p^2 (no new κ) #4406's exponent replacements, and any new κ. The selected record remains 23/10000.

Evidence

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

Head: 7ca94e9762ee33edbfb2e527e7313bc579e463ee. Base: canonical main 2a13eb1827784206202db90e5dc8327cf8a250db, priority revision 7.

Test plan

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

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