Global signature overhead is target-set collision excess

Exact statement

For a fixed flat memoryless Boolean-product expression, let $T_\sigma$ be the useful canonical product states entered by event signature $\sigma$ . Then $\Delta(E)=\sum_\sigma|T_\sigma|-|\bigcup_\sigma T_\sigma|$ . Inclusion–exclusion needs only compatibility cliques; with one support per letter these are exactly support matchings. The pair-intersection bound is exact iff no state receives three or more signatures.

StatusSelf-contained double-counting and inclusion–exclusion proof; complete finite audit passes

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

Context

The local clique number controls one fiber. Total overhead instead counts repeated coverage of useful product states by signature target sets.

Definitions

$T_\sigma=\{q\ne q_0:\text{a useful edge of signature }\sigma\text{ enters }q\}$ .

Proof or evidence

Double-count signature–target incidences, subtract the target-set union, and apply inclusion–exclusion. Every nonempty intersection is a compatibility clique; in a one-support-per-letter policy it is a support matching.

Global collision calculus →

Verification notes

All 65,536 four-set families on a four-point universe and all 28,856 audited one-support signature intersections pass the exact identities and equality boundary.

Limitations

The set-theoretic identity is classical inclusion–exclusion; the contribution is its exact automata interpretation.
It does not by itself optimize target-set sizes from component state counts.
Novelty of the automata-specific formulation is plausible but unconfirmed.

Open work

Optimize the matching-indexed target intersections for overlapping one-support policies.