BASS NOTE SPECTRA OF BINARY FORMS

The paper proves that for every R-isotropic real binary form P of degree n ≥ 3 with nonzero discriminant, the spectrum Spec(P) is exactly the interval [0, MP], with MP > 0, by combining Mahler-type compactness, a no-isolated-maximum result near extremal lattices, a full-measure filling of [0, MP], and a final “fixed point perturbations” step to pass from full measure to a full interval (Theorem 1.6; cf. Sections 5–8) . The model’s solution correctly notes upper semicontinuity and the attainment of MP (consistent with Mahler’s compactness/upper semicontinuity, cf. Proposition 2.1) , but its key Step (4) is invalid: from upper semicontinuity alone one cannot conclude that every boundary point of A_s = {Λ : m(Λ) ≥ s} has m(Λ) = s. This requires at least lower semicontinuity or continuity of m, which the paper does not assert (indeed, only upper semicontinuity is established) . The paper overcomes this gap with substantial Diophantine and perturbative arguments rather than the model’s topological boundary argument.