[jwr404]

Summary

The shared-row sum does not lift to ||B - C||_2 < 2 ε.

• Split one axis through the section polynomial (no new κ) #4490 requires both inverses to lie within ε of one polynomial in the norm of the whole object. Split one axis through the section polynomial at ceiling 9 (no new κ) #4509 repeats that sentence at ceiling 9 p^2.
• Place a core row past the agreement radius p^26 (no new κ) #4491 supplies ||P - H_sec^{-1}||_2 < ε against each section's own polynomial, under that section's Lower the within-cell halo exponents to 11, 24, 26, 26 and 6 (no new κ) #4452 gauge. HALO makes the section polynomial match the global polynomial only on rows at distance A = ceil(p^{48}).
• The matrix with rows (3, 0) and (0, 9) has first-row squared length 9 < 8^2, while M e_2 has squared length 81 > 8^2. A small row does not force the operator 2-norm below 8.
• The matrix with rows (0, 0) and (0, 2) shows that a zero row does not force ||P_B - P_C||_2 = 0. The triangle then keeps an extra term, and 3 + 2 + 3 = 8 is not strictly below 2 · 4.

This does not assert that the Gaussian section polynomials equal the displayed matrices. The locality audit's sharp row at distance (2J - 1) W, and its 2 p^{10} exp(-p^2) inverse split, are different statements. Both ideal-gap gauges stay hypotheses. No new κ. The selected record remains 23/10000.

This answers the open question of #4512. A bound on ||P_sec - P_global||_2 off the rows at distance A is left open.

Test plan

• python3.12 -B scripts/research_package.py --package research/gaussian-operator-lift
• Prints PASS_GAUSSIAN_OPERATOR_LIFT and outcome=completed
• Maintainer replay was not run

Base f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc, priority revision 7. Upstream fetched at 2026-10-22T03:07:14Z was that commit, and the pre-submit fetch at 2026-10-22T03:10:56Z was unchanged. κ remains 23/10000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.