Exact one-shot Boolean-product overhead is #P-complete

Exact statement

Computing the one-shot overhead $\Delta(G)=\sum_M(|M|-1)_+$ is #P-complete under polynomial-time Turing reductions. If $Z(G)$ counts all graph matchings, then $Z(G)=1+\Delta(G\sqcup K_2)-2\Delta(G)$ . Hardness therefore holds with size-two supports and depth-one component expressions.

StatusSelf-contained membership and isolated-edge reduction; exact finite audit passes

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

Context

Dancing Links gives the desired product-free exact enumerator, but exact evaluation cannot generally collapse to polynomial input time.

Proof or evidence

A size-k matching has k-1 countable overhead witnesses. Adding one isolated edge transforms the overhead into twice the old overhead plus the number of nonempty matchings.

Counting-hardness boundary →

Verification notes

The isolated-edge recovery identity passes on every one of the 1,355 audited graphs.

Limitations

The theorem is under polynomial-time Turing reductions and does not give a fine-grained lower bound.
It does not rank practical XCC solvers or prove Dancing Links optimal.
Valiant's counting theorem is classical; novelty is only plausible for this automata application.

Open work

Study sparse, bounded-width, and structurally restricted policy classes where XCC enumeration is effective.