Finite Criteria for Simplicial States and Local Tomography

Companion to Effective Composition

The companion paper 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 K4 composition contract: effect spaces, their bilinear pairing with states, local tomography, and any quantum tensor product.

This paper resolves that layer's finite, mono-system, and bipartite-composition content, importing Effective Composition's rank-one criterion exactly as it supplies it.

Status. Every result below is exact within its stated hypotheses. Every closure beyond a forced discrete/Boolean skeleton — state convexity, effect convexity, their joint consistency, and full joint correlational structure — is obtained only under an explicitly typed, separately motivated postulate, never from the admissibility contract or the simplicial geometry alone. No Born rule, no quantum state geometry, and no tensor product as a derived object.

Six results, each a forced/imported pair

Primitive convexity obstruction. The raw cylindrical-conditioning family is finite while its own convex hull is a continuum, for any local outcome set of size at least two, with no genericity hypothesis. Conditioning inside a projection fibre does not rescue this for free.
Simplicial closure. An explicit classical-randomisation postulate closes the family onto the full probability simplex, locally and jointly, independently of the reference measure's rank.
Representation vs.\ availability of effects. Affine effects on a simplex correspond to a cube of vectors — a bookkeeping fact, not a physical claim. A weak coarse-graining axiom already yields the full Boolean effect algebra; the full continuum needs a further, dual randomisation postulate, realised by an explicit multilinear reconstruction.
Independence of the two randomisations. The state-side and effect-side randomisation postulates are logically independent. A common classical control resource reproduces both only under an explicit independence clause between its two interfaces — dropping it breaks bilinear consistency, by an exact numeric witness.
Uniqueness of the composed effect. Given physical existence of a bipartite effect and compatibility with joint Dirac states, it is forced to equal the pointwise-product formula: no tensor product is imported to define it. Local composition reaches only rectangle effects, strictly fewer than all joint sharp effects — a CHSH-shaped separating witness shows this survives full local randomisation.
Exact criterion for local tomography. Local tomography holds exactly when each side's effect family linearly spans its ambient space — strictly stronger than merely separating its own simplex — equivalently, when it separates states and its linear span contains the unit effect.

Why this matters

A finite admissibility contract, left to itself, does not drift toward quantum structure: it settles, at every layer examined, into the classical/simplicial regime, and only moves beyond it when a correspondingly explicit postulate is supplied. This is not a deficiency of the contract — it is a precise map of where the classical skeleton's authority ends, handed forward, typed axiom by typed axiom, to whatever phase or amplitude-modulus law is proposed to take the programme beyond it. The general preparation layer of Effective Composition, the tensor product as a supplied structure, and any phase/modulus law remain exactly as open as that companion paper left them.

The phase/modulus question is taken up, without deriving it, by Refinement Consistency Obstructs Nonlinear Power Lifts of Finite History Measures.