Support and Preparation Criteria for Independent Effective Subsystems

The composition gap, and the K4 contract

Laboratory physics is organised around independent subsystems, yet the Cosmochrony substrate is not supplied with any factorisation $\Omega \simeq \Omega_A \times \Omega_B$. The paper names the missing piece as a transversal contract,

K4: Admissibility ⟶ Composable effective subsystems,

and works on it in the finite case, from two effective projections $\Pi_A$, $\Pi_B$ on one undivided substrate and cylindrical admissibility conditions combined by intersection.

Status, layer by layer. The support layer is completely characterised under the proposed contract. The preparation layer is resolved for two reference classes — the full simplex, and the strictly positive cylindrical conditionings — and remains open in general. The states-and-effects layer of K4 (effects, local tomography, any quantum tensor product) is untouched. The theorems are exact within the contract; the contract itself is proposed, not derived from the programme's axioms or from an operational theory.

The two layers, and their exact criteria

Support layer. The pairs of local outcome sets on which joint interrogation is margin-stable and jointly rectangular — the independence blocks — are exactly the nonempty rectangles (bicliques) $S_A \times S_B \subseteq R$ of the projected support relation.
Benchmark. On a block, if every probability measure on the underlying support is an admissible preparation, every joint law — in particular every product — lifts to the substrate. The absence of a supplied factorisation alone obstructs nothing.
Preparation layer (two classes resolved; general case open). For the normalised cylindrical conditionings of a strictly positive weight matrix $W$, closure under products of independently locally conditioned states holds if and only if $\operatorname{rank} W = 1$. Other admissible preparation families need their own criteria.
Separation. The counting weights $N_3 = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$ give a complete rectangular support whose induced preparations are correlated: support rectangularity does not imply preparation independence.
Type discipline. States of one side obtained by conditioning on the other are steered conditional states, not local preparations; admitting them manufactures spurious witnesses.

Why this matters

The absence of a supplied factorisation does not forbid independent effective subsystems — it relocates their independence, from an inherited property of the substrate to an emergent property of the projected support (which outcome pairs are jointly possible) and of the preparation weights (which mixtures are jointly realisable). What a laboratory calls “two systems” is, in this structural reading, a regime where both layers cooperate: a biclique carrying a rank-one weight structure. Corpus constructions that use a bipartite composite consume a supplied composition structure; this paper states exactly what that supply must provide.