Decidability of the Isomorphism Problem Between Multidimensional Substitutive Subshifts

The paper’s Theorem 6.2 proves exactly the two claims at issue—after a global shift Sj, (i) the factor can be taken to have radius at most 2 r̄ + R_{ζ2} + 1, and (ii) there exist n and f ∈ F_n with S^f ψ ζ_1^n = ζ_2^n ψ—via a careful construction using Proposition 6.1, the renormalized maps ϕ_n defined by S^{f_n(ϕ)} ϕ ζ_1^n = ζ_2^n ϕ_n, and a finiteness/pigeonhole argument that yields stabilization and the desired intertwining (eqs. (9) and (14) in the paper) . By contrast, the model’s proof outline hinges on two unproven steps: (A) a phase-alignment claim that requires the L(ℤ^d)-subaction to be minimal to force a constant phase J∘ϕ, which the paper neither assumes nor needs and which is nontrivial as stated; and (B) a local ‘no-carry’/interior-ball argument to deduce S^f ψ ζ_1^n = ζ_2^n ψ from patch comparisons, which tacitly assumes image supertile coherence that the paper instead secures by Proposition 6.1 and the ϕ_n-renormalization. The model reaches the right conclusion but does not justify these key steps, whereas the paper provides a complete, rigorous route to the same result.