# Verification and limitations

The five-page manuscript proves a signed simplex decomposition with coefficients +1 or -1 under weak nondegeneracy. All simplices can share a prescribed allowed apex. It also proves the bounded-facet-jump extension for integer-valued polyhedral functions.

The written proof was checked for zero entries in the affine relation, full-dimensional cross-pair intersections, all four gauge-interval inequalities, integer coefficients, facet vertex inheritance, hyperplanes through the chosen apex, and duplicate simplices. The known ABR integer-decomposition and coning theorems are explicit dependencies.

The exact checker passed identically with and without Python optimization: 6,267,982 small coefficient configurations, 69,426 admissible cases, 23 rational geometric configurations, 107 strict intersection witnesses, 958 triangulation tests, and 2,394 pointwise coning tests plus four moment identities. No floating-point checks are used. These finite tests supplement the written proof and do not certify the general theorem.

The source question is interpreted with its 2013 weak nondegeneracy hypothesis and unit-density convention. The later arbitrary-configuration conjecture remains unaddressed. The broad frozen record is not counted as an unqualified closure.

This is a self-audited, AI-assisted, unrefereed preprint. No independent expert review, proof-assistant verification or exhaustive priority determination is asserted. Sources and a portable checker are included by citation or original code; third-party PDFs are not redistributed.