HEIGHTS AND MORPHISMS IN NUMBER FIELDS

The paper proves an asymptotic for N_{f^*H, P^m(K)}(X) with an explicit main constant c_K(f)=c_K(m)·c_{K,∞}(f)·c_{K,0}(f)/H(f)^{n(m+1)/d} and a power-saving error term; see Theorem 1.3 and its precise local factors and error bound, including when the log and C_f^∞ terms disappear . Crucially, the author explains that one cannot in general recast the problem as counting by an adelic Lipschitz height N_v(z)=(|F(z)|_v/|F|_v)^{1/d} and apply Widmer’s theorem “as-is,” because these local gauges may fail the ultrametric axiom (iv), explicitly demonstrated by a counterexample; hence the expression here is not a special case of Widmer’s adelic-Lipschitz framework . Instead, the paper develops a different route: it fibers over ideal classes and “excess divisors,” reduces nonarchimedean irregularities via periodicity modulo the resultant ideal, and then applies geometry-of-numbers (Masser–Vaaler volume, Widmer’s counting lemma for Lipschitz boundaries) in Minkowski space, carefully tracking Lipschitz parameters and unit-lattice geometry . The candidate solution reproduces the right main term and error-shape heuristically, but its core step—asserting that the N_v define an adelic Lipschitz system suitable for Widmer’s theorem—is false in general, directly contradicting the paper’s remark and example. Therefore, the model’s proof is invalid even though its final formula matches the paper’s result.