Geometrical representation of subshifts for primitive substitutions

The paper’s Theorem 1.2 precisely asserts that for a proper primitive pseudo‑unimodular substitution with Pisot Perron eigenvalue β of degree d+1, and with the sum‑zero condition on generalized eigenvectors for every other eigenvalue with modulus ≥ 1, the subshift (Ωσ, S) is a finite extension of a minimal translation on Td; this is proved via an explicit construction of d independent eigenvalues, a torus factor map φV′ built from a summation scheme on the abelianized prefix–suffix paths, and a piecewise‑translation bridge ψ from a proper Rauzy fractal domain exchange to the torus that is finite‑to‑one almost everywhere (see Theorem 1.2 and Section 8 for the proof outline; details for the factor map and domain exchange appear in Sections 3–5 ). By contrast, the model’s solution relies on unproven or incorrect steps: (i) it asserts that properness implies strong coincidence and uses that to deduce finiteness—whereas the paper explicitly avoids assuming strong coincidence by working with proper substitutions and an explicit conjugacy to a domain exchange (Theorem 5.2 and discussion) ; (ii) it claims pseudo‑unimodularity “eliminates p‑adic/solenoidal components,” guaranteeing a full‑rank lattice Λ in the contracting space and hence a torus quotient, a claim not established in the paper’s argument (the paper instead constructs the torus factor directly from eigenvalues via V′ and reduction modulo ZA) ; and (iii) it never demonstrates the existence of d rationally independent eigenvalues needed for minimality of the torus translation, which the paper obtains by a careful linear‑algebraic argument (Lemma 8.2 and subsequent steps) . These gaps and a few other unverified identifications (e.g., the independence of the increment α and the finiteness almost everywhere without the ψ bridge) make the model’s proof outline incomplete and in parts incorrect, whereas the paper’s proof is self‑contained and correct under the stated hypotheses.