Auerbach bases, projection constants, and the joint spectral radius of principal submatrices

The paper proves (via extremal norms and Auerbach bases) that there exists a similarity T placing all entries of all matrices in the set within the joint spectral radius ρ(M) (Proposition 3.3), and shows that no analogous uniform bound holds for higher-dimensional principal submatrices by constructing finite sets whose principal-submatrix JSRs are uniformly larger than ρ(M) across the entire similarity orbit (Section 4; Example with ρ((T^{-1}MT)_{J,J}) ≥ 1.01 for all T and |J|=2 in d=3). These statements are clearly presented and their proofs are coherent and complete in the provided excerpts. By contrast, the model’s Part 1 hinges on an unproven and in fact incorrect inequality Q_{p+q} ≥ Q_p ⊗ Q_q to deduce ρ_⊗(P) ≤ ρ(M), and it assumes attainment of a max-times Collatz–Wielandt minimizer without addressing reducibility. Part 2 gives only the trivial singleton family {C I_d}, which does not capture the paper’s obstruction (it yields equality, not a strict lower bound). Therefore, the paper’s results stand, while the model’s proof has fundamental gaps and does not address the stronger negative statement for principal submatrices. See the paper’s main statements and proofs in Proposition 3.3 and the negative results for principal submatrices in Section 4 and its example constructions .