Uniform quasi-multiplicativity of locally constant cocycles and applications

The paper’s Theorem 1.1 states precisely the claim under review and proves it by a clean contradiction argument built on the sequence Vk = Span{AI u : |I| = k}, a minimal-dimension choice, and a two-case dichotomy: a periodic case contradicting irreducibility of At, and a transversal case contradicting irreducibility of A∧m via eigenvector and wedge-space spanning arguments; see the theorem statement and proof in Section 3 (Theorem 1.1; Lemma 3.1; Case analysis; the Wk basis argument) . This establishes k-uniform spannability as defined in the preliminaries . By contrast, the model’s proof outline hinges on (i) a Burnside-type step that incorrectly deduces a scalar commutant and 𝔄 = M_d(ℝ) from the irreducibility of exterior powers, including a non-rigorous use of W2 = {v ∧ Jv}, and (ii) a false inference that an element commuting with all generators of S^{(t)} commutes with all of M_d(ℝ) simply because the two-sided ideal generated by S^{(t)} is all of M_d(ℝ). The key computation X(ZY) = (ZY)X is unjustified (it requires XZ = ZX), so Step 2 is invalid. Since later steps depend on this, the model’s argument does not go through.