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Theorem · sharp universal policy classification

FPRD-SH-T14

Signature rigidity is the sharp universal Boolean-policy criterion

Exact statement

A memoryless Boolean policy makes the natural prefix automaton isomorphic to the useful canonical component-prefix product for every tuple of ordinary component expressions if and only if every active letter aa has one allowed support SaS_a and Sa∩Sb≠∅S_a\cap S_b\ne\varnothing for every two distinct active letters. In the positive case the quotient is the identity-block quotient; every policy violation has a starred-expression witness whose natural automaton is strictly larger than the product.

StatusSelf-contained necessity and sufficiency proof with independent complete small-policy audit
External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

The policy criterion is uniform over all ordinary component expressions and is strictly broader than strong full synchronization.

Hypotheses and scope

  • Allowed event supports are nonempty and depend only on the current letter.
  • The Boolean-product expression is flat.

Proof or evidence

Unique per-letter support removes same-letter ambiguity. Pairwise support intersection makes distinct-letter ambiguity impossible because one homogeneous component target would need both labels. Two supports for one letter are separated by all-a-star components; disjoint distinct-letter supports are separated by a-star and b-star components.

Signature-rigidity classification →

Verification notes

The source checker covers 16,448 policies at arities two and three. An independent implementation covers all 16,452 two-letter policies at arities one through three, separates all 16,382 non-rigid policies, and checks 172,128 same-letter support-pair occurrences at the unmarked boundary.

Limitations

  • The cited papers supply the component constructions; no literature-priority claim is made for this policy classification.
  • Stateful weak synchronization is outside the memoryless policy theorem.

Open work

Use FPRD-SH-T15 and FPRD-SH-T16 for the exact local bound and its policy-only complexity boundary.