[kumagi]

The actual PR3851 boundary solver factors round_P(S), then stores rounded pivot reciprocals. This package makes both roundings explicit and proves the conditional original-Schur error of the unchanged scalar BandedSolve:

T x-r = e_f + Lhat e_b, with Ueff_ii=1/round_P(1/U_ii), ||T^-1||<=32p, and ||x-S^-1 r||<2^17 p^5 eta max(1,||r||) for the stated production premises. The constant is already present in the retained RaD report; the contribution is the actual-caller bridge and discriminating residual evidence.

It executes a small actual old-H Axis/Window preparation and two unchanged scalar solver calls, reconstructs the Schur system independently with exact rationals, and separates initial-Schur, factor and online residual errors. Controls distinguish omitted initial rounding, a substituted rounded diagonal and premature upper-residual rounding. The frozen source and selected frontier are unchanged.

Validation: source/metadata pins, complete registered replay, optimized replay, publication-byte/source-history preflight and a committed Git-free export replay. RESULT.json records the finite case and every observed residual. These are author checks with additional AI review, not independent human or formal verification.

Limits: the global old-H gap, matching setup, factor caps, primitive arithmetic and storage widths remain explicit premises. Reduced p=16,P=24 is a conformance fixture, not production accuracy evidence. Complete interior/Core/locality/tensor/CRT error and cost admission remain open.

Canonical scoped claim/result. Fresh main base ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision7; narrower #5081 OQ1, distinct from the separately completed conditional downstream tensor join. Credit: Jose Manuel/jmbarrancoml #3851, Chafik Boukhalfa #489 and physical-word/RaD lineage, James Reed's factor-margin audit, JoyboyBrian #5081/#5059, and retained notices. Prepared by Hiroki KUMAZAKI with substantial OpenAI Codex assistance; Apache-2.0.