Shift Equivalence Implies Flow Equivalence for Shifts of Finite Type

The paper gives a complete, correct proof that shift equivalence over Z+ implies flow equivalence for shifts of finite type via an explicit polynomial shift equivalence (PSE) over Z[t], a partitioned refinement, and the known flow-equivalence classification stated as Theorem 5.5; see Theorem 1.1 and the proof in Section 6, including equations (6.2)–(6.3) and the positivity-on-cycles check . By contrast, the model’s Step 1 constructs a purported explicit homeomorphism H: Y_T → Y_{T^ℓ} together with an inverse G([x,s]) = [x,s/ℓ]; this inverse is incorrect and does not yield G ∘ H = id in the mapping torus (the quotient identifications are mishandled), so the central reduction in the model’s argument is invalid as written. Although the high-level strategy (powers of a system have flow-equivalent suspensions; A^ℓ and B^ℓ are ESSE and hence conjugate) is a known workable route, the model’s proof fails at the level of details it provides. The paper’s proof remains correct and self-contained.