An effective criterion for prescribed block-code extensions

Abstract

For a specified onto block self-map f of a nonempty two-sided irreducible sofic shift T, this note gives a complete finite computable criterion for extending that same map into T from a containing shift of finite type. It answers the specified-map question in Boyle's Problem 16.3, including its mixing case. For any total completion F of the local rule, let L be the allowed finite-word language and B the regular two-sided ideal of input words whose finite output is forbidden. Extension is equivalent to locally testable separation of L from B. An irreducibility-based saturation proof converts the credited Place–van Rooijen–Zeitoun2014 Theorem4.2 bound into the single canonical test L(T[N]) intersect B empty, with N=max(2,ell,4(|M|+1)) for a common recognizing monoid M. Success gives F(T[N])=T; failure excludes all other radii, completions and containing alphabets. Earlier steadiness terminology and Dykstra's2007 broader sofic-homomorphism question, Barth–Dykstra2007 terminology as credited by Ballier, and Ballier's2013/2015 canonical approximation normalization and restricted closing/continuing results are credited. The general LT-separation theorem is published prior machinery, used rather than independently reproved. The result does not solve the separate existence-over-all-maps or cellular-automaton stability questions, or Dykstra's full unrestricted reducible-sofic problem. It claims no new general separation theorem, practical complexity improvement or implemented arbitrary-input solver. Bounded primary-source and citation reviews identified no full predecessor in inspected statements; originality, historical firstness and current worldwide open status remain unestablished. Inaccessible bodies, versions and indexing limits are explicit in the companion archive. The supporting source archive preserves all48 frozen candidate files, two independent mathematical families, two independent priority reviews and a later peer challenge, exact source hashes and locators, finite automata/symbolic checks and root reproduction receipts. Optimization-safe integrity checks preserve the distinction between example computation and general proof. Software does not formally certify the theorem, its imported ingredient or exhaustive priority. AI tools were used extensively in development, source investigation, writing, computation, reproduction and adversarial review. This is an unrefereed preprint without external human peer review or proof-assistant certification. The note and authored companion material are CC BY4.0; no third-party full source bodies are redistributed.