[jwr404]

Summary

A second output kept at distance ceil(p^{48}) inherits the same #4491 reading as the first. The distance comparison does not name a row, so both outputs satisfy R < p^{26} < A.

For each output:

• With no gauge, the section-polynomial row is the global polynomial row.
• Under that section's own Lower the within-cell halo exponents to 11, 24, 26, 26 and 6 (no new κ) #4452 gauge, ||P - H_sec^{-1}||_2 < ε with ε = p^6 exp(-p^2), and the core row is a row of that operator.

On the two matched rows the section polynomials equal the same global row, so

row(B) - row(C) = (row(B) - row(P)) + (row(P) - row(C)).

Two positive integers strictly below 4, namely 3 and 3, sum to 6, and 6 < 2 * 4. A polynomial gap of 2 spends that room: 3 + 2 + 3 = 8, which is not strictly below 8. One error alone is not the two-error sum.

This is the shared-row sum. It is not the operator norm ||B - C||_2 of the two section inverses, and it is not the global-inverse summand. Both ideal-gap gauges stay hypotheses. One section's gauge does not cover the other. No new κ. The selected record remains 23/10000.

This answers the open question of #4511 for the two row errors. Whether the sum lifts to the operator hypothesis in #4509 is left open.

Test plan

• python3.12 -B scripts/research_package.py --package research/gaussian-second-row
• Prints PASS_GAUSSIAN_SECOND_ROW and outcome=completed
• Maintainer replay was not run

Base f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc, priority revision 7. Upstream fetched at 2026-10-22T03:04:25Z was that commit, and the pre-submit fetch at 2026-10-22T03:06:24Z was unchanged. κ remains 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.