A Noncommutative Geometric Realization of Prime Spectral Triples via Modular Holonomy

We propose a noncommutative geometric framework in which primes are realized as spectral data of a modular holonomy system. The construction is based on a lattice representation of PSL (2, ℤ), where non-abelian holonomy induces a decomposition into two irreducible sectors associated with Fibonacci and Pell eigen-directions. Within this structure, we define a family of operators TpeffTₚ^effTpeff acting on a Hilbert space HHH, and associate to each prime ppp a spectral triple (Tpeff, λp, Yp∗), (Tₚ^eff, ₚ, Yₚ^*), (Tpeff, λp, Yp∗), where Yp∗Yₚ^*Yp∗ is a non-decomposable eigenmode invariant under the Hecke action up to scaling. The interaction between the two sectors generates an effective coupling μeff=ηpΞN, ₄₅₅ = ₚ N, μeff=ηpΞN, which governs the transition between Poisson and GUE statistics. In this setting, primes correspond to those spectral modes that remain irreducible under the full noncommutative channel algebra and persist under higher-order (non-abelian) transport. This perspective suggests that the statistical behavior of zeta zeros arises from an underlying noncommutative geometry, where primes are not elementary objects but invariant spectral carriers encoding the global structure of the system.