Finite Criteria for Simplicial States and Local Tomography toward Emergent Operational Composition

Abstract

The companion paper on effective composition characterises, for a finite substrate without a supplied subsystem factorisation, the possibilistic support layer completely and the preparation layer for two reference classes, and explicitly leaves open the states-and-effects layer of the resulting composition contract. We resolve that layer's finite, mono-system, and bipartite-composition questions, keeping the general preparation layer and the tensor product exactly as open as the companion paper left them. The primitive family arising from cylindrical conditioning of a strictly positive, rank-one reference measure is not itself convex: it is finite while its own convex hull is a continuum, unconditionally, for any local outcome set of size at least two. An explicit classical-randomisation postulate closes it onto the full classical probability simplex, independently of the reference measure's rank; rank governs composition of the primitive family, not the shape of its closure once randomisation is added. Simplicial geometry fixes the mathematical representation of affine effects as vectors in a cube, but not their physical availability as measurements; a weak coarse-graining axiom already yields the entire Boolean skeleton of sharp effects for free, while the full continuum of effects requires a further, dual randomisation postulate on the measurement side. Under two further, distinct, and logically independent axioms — one for a common classical control resource, one for the physical existence of a composed bipartite effect — the pointwise product of two local effects is proved unique, not postulated, and local tomography holds exactly when the local effect families linearly span their ambient functional spaces, equivalently when they separate their local states and contain the unit effect in their linear span. All results are finite, exact, and reproduced by committed scripts using rational arithmetic. Interpretive reading (structural, not a result): the finite hypotheses examined here force, successively and for free, a classical, simplicial operational skeleton — convex states once mixing is supplied, a Boolean effect algebra once coarse-graining is supplied, and exact local tomography once spanning local families are supplied — under precisely identified imports, and produce neither a quantum state geometry nor the Born rule. Locating exactly how much operational structure a finite admissibility contract yields for free, and where every further richness must instead be imported, is the intended service of this result to the non-injective foundations programme: it marks the boundary this classical skeleton cannot cross on its own, and hands that boundary, precisely typed, to whatever module or phase law is proposed next.