Icosahedral invariants from Dedekind zeta values of Q(ζ5)

This preprint provides an explicit arithmetic derivation of the classical icosahedral invariants from special values of Dedekind zeta and Dirichlet L-functions. All computations are exact and rely only on standard results from algebraic number theory. The work demonstrates that the integers 12,30,20,60,120 are not independent geometric data but spectral consequences of the arithmetic of the fifth cyclotomic field.The topological invariants of the icosahedron—its numbers of vertices, edges, and faces, together with the orders of its rotation group and its binary cover—are shown to arise as reciprocals of special values of the Dedekind zeta function of the fifth cyclotomic field Q(ζ₅) and its Dirichlet L‑factor decomposition. Exact evaluations at the negative integers s = −1 and s = −3 are obtained using generalized Bernoulli numbers. By a character parity mechanism, the full Dedekind zeta function vanishes at all negative integers, so that the icosahedral data reside entirely in the partial product corresponding to the real subfield Q(√5). The Euler characteristic of the sphere appears as a quadratic Dirichlet L‑value, and further identities relate these special values to regulator ratios, Casimir‑type constants, and the analytic class number formula. Together with earlier work connecting cyclotomic units to hyperbolic dynamics, this establishes a unified arithmetic–spectral picture in which the icosahedron emerges as a forced structure of the ring Z[ζ₅].