ON PAIRS OF SPECTRUM MAXIMIZING PRODUCTS WITH DISTINCT FACTOR MULTIPLICITIES

The paper’s Theorem 2.5 asserts that for a τ-permutable pair {A,B} and any product M of length k, σ(τ(M)) = σ(M); if k is odd then M and τ(M) have different counts of A and B, hence no cyclic permutation of one can match a cyclic permutation of the other. Its Corollary 2.6 then states that if an odd-length M is spectrum maximizing, τ(M) is a distinct spectrum maximizing product with different factor counts. The proof given in Section 4.1 relies on the multiplicativity of τ(X) = S^{-1}XS and the consequent spectral invariance under similarity, and on the swapping of A- and B-counts under τ. The candidate solution reproduces precisely these steps: it shows τ(M) = S^{-1}MS, deduces spectral equality, notes the odd-length factor-count swap, and concludes the corollary about spectrum-maximizing products. This matches the paper’s statements and proof structure closely, so both are correct with substantially the same proof approach .