Prime Multiplication as a Physical Organizing Principle: Journal-Aligned Computational Dataset and Reproducibility Package

This dataset and reproducibility package supports the manuscript “Prime Multiplication as a Physical Organizing Principle: Topological Coarse-Graining, Thermal Fixed Points, and Finite-Size Scaling.” The record contains journal-aligned machine-readable data corresponding directly to the current manuscript and Supplemental Material, including complete congruence Wilson kernels for k = 2,...,10; exact thermal endpoint values; finite-size contrast tables; 5,000 seeded global-label permutation controls at selected cutoffs; a consecutive coprime-integer control; direct-versus-compressed validation at X = 500 and s = 1.5; residue-preserving permutation diagnostics; thermal-flow data; and the complete prime-mode catalogue through X = 1,000,000. The largest register contains 78,498 prime modes, representing 3,080,928,753 implicit unordered pairs without explicit pair enumeration. No external empirical dataset is used. All numerical outputs are deterministic and derive directly from the mathematical definitions and computational protocol reported in the manuscript. Additional exploratory calculations not required for the submitted claims are separated in the extended directory. This Zenodo record is independent of earlier repositories and is intended as the persistent computational record for the present study. The Hamiltonian considered in the work is an engineered programmable spectrum. The dataset does not claim that prime numbers are particles, fields, or microscopic constituents of matter.