Prime Field Pathfinder v4.2 — Operator Blueprint and Multi-Scale Refinement for Modular Prime Dynamics

This record provides the Prime Field Pathfinder v4.2 “Operator Blueprint” and test-suite requirements for modeling phase transitions in modular prime dynamics. Scope: We study prime residue dynamics in the Crystal-9 branch (residues ≡ 8 mod 9) and its multi-scale refinement to Crystal(9^k). The goal of v4.2 is not prime enumeration on quantum hardware, but operator-level system identification: an effective Hamiltonian whose dynamics reproduces the observed phase structure. Key points: Data-to-model contract (must reproduce): • Energy accumulation prior to the rigid island (Ecrit ≈ 0.17) • Localized spectral freezing with GUE-type level repulsion (RI ≈ 10) in the 85M–90M window • Post-transition dissipation / relaxation Operator blueprint: Heff(Z) = 1/2 · Zᵀ K Z + λ ||Z||⁴, where Z is the normalized deviation field in the refined branch and K is a coupling matrix to be learned from laminar data. Multi-scale refinement: Crystal(9^k) = {8 + 9t mod 9^k}, enabling resolution increase from k=2 (mod 81) toward k=3,4 classically, and k≥5 as a candidate target for quantum/HPC simulation. Related works: • Prime Field Pathfinder v4.1 — Energy Diagnostics (Ecrit and phase-contrast table) • Prime Field Pathfinder v4.0 — Energy Accumulation and Spectral Freezing • Prime Field Pathfinder v3.0 — Executive Summary + Appendix A (Crystal-9 window tables, 70M–100M) • v1.2 baseline preprint (HMM perspective) Author: Andrey Kuznetsov (ORCID: 0009-0000-7232-3018)