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Theorem · exact sparse constraint compilation

FPRD-SH-T19

Policy-signature compatibility has an exact XCC compilation

Exact statement

Create one primary selector item for every policy signature σ=(a,S)\sigma=(a,S), with an off option and an on option containing each participant in SS as a nonprimary item colored aa. XCC solutions are then in bijection with cliques of the signature-compatibility graph. With one support per letter, they are exactly support-hypergraph matchings.

StatusSelf-contained bijection proof; exhaustive color-consistency audit passes
External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

Color agreement is exactly the Boolean-policy rule that overlapping signatures may coexist only when they carry the same letter.

Proof or evidence

Every selector is covered exactly once, choosing its signature on or off. Knuth's color purification accepts the chosen on-options exactly when all overlaps agree in letter.

Exact-cover-with-colors compilation →

Verification notes

All 16,384 subsets of the fourteen signatures formed by two letters and all nonempty supports on three participants satisfy XCC consistency iff they are pairwise policy-compatible.

Limitations

  • The compilation enumerates compatible signature families; expression-specific target intersections may still be empty.
  • It does not claim polynomial total runtime when the solution family is exponential.
  • Novelty is plausible but unconfirmed; no external review is recorded.

Open work

Use FPRD-SH-T20 and FPRD-SH-T21 for the exact one-shot algorithm and its counting boundary.