Pattern Complexity of Aperiodic Substitutive Subshifts

The paper proves the quadratic lower bound via a single-scale injection (Lemma 7) and a maximal desubstitution depth k(n), yielding |L_{n,n}(c)| ≥ n^2(1 + 1/⌈n/M^{k(n)}⌉^2), and then uses Solomyak’s recognizability (Theorem 2) plus Lemma 8 to promote this to a uniform constant K = 1 + 1/(ρ+1)^2 independent of n and c; this is correct and complete in the text (Theorem 9, proof) . The candidate solution, however, asserts a stronger multi-scale injection that sums contributions across all powers t (a geometric series). That summation is not established in the paper and is not justified: it would require proving disjointness of images across different desubstitution levels, which is not provided and is generally false without additional arguments. Consequently, the model’s key Step 4 is flawed, and the final constant Kσ = 1 + 1/(Mℓσ)^2 is not derived by the given reasoning. The paper’s use of Kari–Moutot’s per-configuration bound (Corollary 1) is consistent and correctly applied; the model’s use of it is fine per se but attached to an invalid additive injection across levels .