Spectral Cocycle for Substitution Tilings

The paper’s Theorem 4.2 establishes the exact lower local-dimension bounds via the spectral cocycle and a boundary-sensitive decomposition of twisted ergodic integrals; the proof is complete and correct in Section 7.4, relying on Lemma 7.1 and the two-regime analysis that produces the “min{…,1}” switch (see the statement and proof of Theorem 4.2 and Lemma 7.1) . The candidate solution recreates the main outline (renormalization via M(z,n), choice of the resonance scale, and the PF-based zero-frequency bound), but contains a critical misestimate: it drops the θ^{n(1−d)} factor in the remainder, leading to a term of order r e^{(χ_+ + ε) n} and then incorrectly simplifies it to r^2, which does not logically yield the boundary-driven “+ r^2” term nor the min-switch. This gap means the model’s derivation does not actually prove the claimed bound, even though the final statements match the paper.