Explorations for alternating FPU-chains with large mass

Both the paper and the candidate solution reduce the alternating-mass α–FPU chain on a symmetric invariant manifold to q¨ + B q = α N(q), diagonalize B to obtain quasi-harmonic coordinates, and explicitly exhibit nonzero quadratic “square” forcing terms that couple acoustic and optical groups for N=6 and N=10; both then use computer-assisted evidence to verify such coupling for all odd primes p ≤ 47 and invoke the embedding argument to extend to suitable composite sizes. The paper documents the reduction and the observed cross-group square terms with explicit systems (e.g., eqs. (6)–(7), (10), (12)) and describes a Mathematica pipeline used to confirm the pattern up to N ≤ 100, while leaving a general theorem as an open problem; the model reproduces this path, adds closed-form small‑a expansions for p=3, and performs analogous numerical checks for primes. Minor discrepancies arise from normalization/scale choices in the quasi-harmonic coordinates (not fixed in either account) and from reliance on computation for general p, but there is no substantive conflict in claims of existence of cross-group square forcing. See the paper’s reduction and invariant-manifold setup (q̈ + Bq = αN(q)) and explicit forms (2)–(3) for p prime, and the N=6,10 quasi-harmonic systems documenting nonzero x_opt^2 and x_ac^2 terms (e.g., (6)–(7), (10), (12)) . The computational survey and observed square-term pattern for primes up to 47 are summarized in Section 6 and 6.1 , and the embedding argument is stated in Section 2 and tied to [2] .