Unit-Coefficient Simplex Decompositions under Weak Nondegeneracy

Let P be a finite union of compact convex polytopes in real d-space, and let S contain its algebraic vertices. Suppose that every d+2 points of S affinely span the ambient space. We prove that the unit-density measure on P is a sum of distinct simplex measures with coefficients +1 or -1 and all simplex vertices in S. All simplices may share any prescribed point of S as a vertex. The proof combines the signed-section and coning results of Akopyan, Barany and Robins with an elementary bounded-coefficient lemma for a configuration having at most one affine dependence. This answers the coefficient question in the weakly nondegenerate setting of Pasechnik's 2013 Oberwolfach report; it does not resolve the later conjecture for arbitrary configurations. A more general bound by the largest facet jump holds for integer-valued polyhedral functions. Exact regression scripts accompany the written proof. This AI-assisted preprint is self-audited and unrefereed; independent review, formal certification and exhaustive priority determination are not claimed.