Prime Field Pathfinder v5.0 — Identification of the Effective Hamiltonian and Growth Modes in Modular Prime Dynamics

This work introduces Prime Field Pathfinder v5.0, a theoretical framework for operator-level identification of phase transitions in modular prime residue dynamics. Building on empirically established energy thresholding (Ecrit ≈ 0.17) and localized spectral rigidity (RI ≈ 10) in the Crystal-9 subspace (v4.1–v4.2), we formulate the inverse problem for the effective Hamiltonian governing laminar dynamics. We show that the deviation field evolution is consistent with a linear instability regime (effective negative-viscosity growth mode) containing a structured unstable eigenmode constrained by modular symmetry (Δa ≡ 0 mod 9). This growth mode drives systematic accumulation of structural stress, followed by nonlinear stabilization manifested as GUE-type spectral freezing. The framework provides a falsifiable Hamiltonian-learning protocol and a scalable roadmap for multi-resolution refinement (Crystal(9^k)) and future quantum simulation of arithmetic dynamical structures.