[jwr404]

Summary

A core row kept at distance ceil(p^{48}) inherits four facts and leaves one bound open:

1. Row identity, with no gauge. Place a core row past the agreement radius p^26 (no new κ) #4491 gives R < p^{26} < p^{48} = A. The section-polynomial row is the global polynomial row.
2. One section error, under the Lower the within-cell halo exponents to 11, 24, 26, 26 and 6 (no new κ) #4452 gauge. Place a core row past the agreement radius p^26 (no new κ) #4491 gives ||P - H_sec^{-1}||_2 < ε, with ε = p^6 exp(-p^2). That is ||B - P|| < ε for B = H_sec^{-1}.
3. Inverse ceiling, under the same gauge. Show the smallest Neumann coefficient is 9 (no new κ) #4493 proves U < 9 p^2 for every integer p ≥ 2. Under the gauge, the Neumann 1-norm and ∞-norm are strictly below U, so ||B|| ≤ 9 p^2.
4. Approximant norm, under the same gauge. Split one axis through the section polynomial at ceiling 9 (no new κ) #4509 turns that ceiling and that one error into ||P|| < 9 p^2 + ε. The approximant is the polynomial.
5. Two-inverse difference. ||B - C|| < 2 ε still needs ||C - P|| < ε. One core row does not supply it.

The integer shadow |8 - 5| = |8 - 6| + |6 - 5| separates the pieces. The ideal off-sums 1 - 1/(8 p^2) stay a hypothesis. This package does not re-prove the radius, the coefficient 9, or the triangle, and it does not claim a new κ. The selected record remains 23/10000.

This answers the open question of #4509. The next question is whether a second core row supplies the missing section error.

Test plan

• python3.12 -B scripts/research_package.py --package research/gaussian-axis-inheritance
• Prints PASS_GAUSSIAN_AXIS_INHERITANCE and outcome=completed
• Maintainer replay was not run

Base f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc, priority revision 7. Upstream fetched at 2026-10-22T02:57:30Z was that commit, and the pre-submit fetch was unchanged. κ remains 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.