A COMPLETE INVARIANT FOR SHIFT EQUIVALENCE FOR BOOLEAN MATRICES AND FINITE RELATIONS

The paper proves that for any finite relation R, the triple (R≥, p, [ξ])—period, poset of strongly connected recurrent components, and the Lp-valued cocycle class—is a complete invariant of shift equivalence, via a canonical-form reduction and a precise cocycle/isomorphism calculus. See the statement of the main theorem and its construction of the invariant and canonical form, and the corollary extending completeness beyond canonical form (Theorem and introduction; Theorem 2.4; Theorems 3.10, 3.13, 3.15 and Corollary 3.16) . By contrast, the model’s direct construction contains two critical errors: (i) it asserts that A(x) ⊆ R′q(A(x)) forces every x′ ∈ A(x) to be recurrent for R′; this does not follow for general relations from S ⊆ R′q(S) for a finite set S, and requires additional argument the model does not provide; (ii) in the backward direction it claims that a union ⋃s∈ξ(a,a) R2L+q+s collapses to a single power RL+q by modular periodicity when L ≡ 0 (mod p). This is false in general, since periodicity occurs mod p while s ranges over the subgroup ξ(a,a) ≤ Z/pZ (not necessarily {0}), so the exponents 2L+q+s need not be congruent mod p; the union cannot be replaced by a single power without additional structure. The paper’s method avoids these traps by passing to canonical form and proving isomorphism classification for the induced cocycle class, with cohomology taken in the Lp-monoid sense (not group-valued H1), thereby yielding a correct and complete classification .