Compatibility clique number is the exact universal local split

Exact statement

Let $C_\psi$ have the policy signatures $(a,S)$ as vertices, with distinct vertices adjacent exactly when they share a letter or have disjoint supports. Over every tuple of ordinary component expressions, the largest attainable number of incoming signatures at one useful canonical product state is exactly $\omega(C_\psi)$ . Hence $|\operatorname{Sig}(q)|\le\omega(C_\psi)$ and $\Delta(E)\le(\omega(C_\psi)-1)(|Q(P)|-1)$ .

StatusSelf-contained upper bound and exact starred realization; finite audit passes

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

Context

The preceding expansion theorem gives an exact expression-level overhead formula. This theorem extracts the strongest possible one-state multiplicity from the support policy alone.

Definitions

Two signatures $(a,S)$ and $(b,T)$ are compatible when $a=b$ or $S\cap T=\varnothing$ .
$\Delta(E)=\sum_{q\ne q_0}(|\operatorname{Sig}(q)|-1)$ .

Proof or evidence

Every incoming-signature family is a clique because overlapping supports cannot impose two distinct incoming letters on one homogeneous component target. Conversely, a maximum clique is realized at one useful product state by choosing one-letter stars in every used component.

Compatibility-graph theorem →

Verification notes

All 16,452 two-letter policies through arity three agree with exhaustive homogeneous starred-state assignments; 114,884 policy-signature incidences are exercised.

Limitations

The theorem gives an exact local multiplicity and a global upper bound, not an exact maximum total overhead for prescribed component sizes.
Nested and stateful Boolean products are outside the statement.
Novelty is plausible but unconfirmed; no external review is recorded.

Open work

Use FPRD-SH-T17 and FPRD-SH-T18 for the exact global collision receipt and disjoint-support count.