Cartesian support tensors induce strong graph powers

Exact statement

Let $J_\psi$ join distinct signatures whose nonempty supports overlap. Define $\psi^{\otimes k}$ on the participant universe $U^k$ by $S_{(a_1,\ldots,a_k)}=S_{a_1}\times\cdots\times S_{a_k}$ . Then $J_{\psi^{\otimes k}}=J_\psi^{\boxtimes k}$ .

StatusSelf-contained all-length proof; exhaustive finite tensor audit passes

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

The overlap graph is the complement of the earlier compatibility graph: its independent sets are mutually disjoint support families.

Proof or evidence

Two Cartesian-product supports intersect exactly when every coordinate support intersects; equality in a coordinate supplies the reflexive part of strong-product adjacency.

Support-tensor theorem →

Verification notes

All 39,693 audited tensor vertex pairs agree with strong-product adjacency, including tensor representations of every graph through four vertices.

Limitations

The theorem concerns the policy-level local-split invariant; expression reachability can impose additional restrictions on realized global states.
Novelty of the automata formulation is unconfirmed and no external review is recorded.

Open work

Use FPRD-SH-T26 as a policy-level capacity dictionary; keep expression reachability as a separate proof obligation.