UDCT v2.6.4: Higgs Hierarchy Plaquette Susceptibility Technical Note with Reproducible Codes and Gauge Invariance Tests in Geometric Back-Reaction

This technical note presents UDCT v2.6.4, which performs a systematic finite-size scaling study of plaquette susceptibility in compact U(1) lattice gauge theory as part of the ongoing UDCT program on Higgs hierarchy stabilization via memory kernel in geometric back-reaction. Using an improved Metropolis Monte Carlo algorithm with staple summation and automatic proposal-width tuning, we investigate lattices with linear sizes L = 4, 6, and 8. As the lattice volume increases, finite-size effects are progressively suppressed, resulting in a clear sharpening of the susceptibility peak near β ≈ 1.0 and a substantial increase in its maximum value (approximately 72% increase from L = 4 to L = 8). These results provide direct numerical evidence that the essential features of the confinement-deconfinement crossover become more pronounced on larger volumes and establish a reliable computational foundation for future Higgs hierarchy studies. This upload includes the full technical note together with two self-contained, reproducible C++ software packages: 1. Reproducible Monte Carlo Code Package Complete C++ implementation used to generate all data, tables, and figures presented in the technical note. The code features an improved Metropolis algorithm with staple summation, automatic proposal-width (ε) tuning (target acceptance rate 40–60%), and plaquette susceptibility measurement with proper finite-volume correction. It supports multiple lattice sizes (L = 4, 6, 8) and is fully documented for independent reproduction. 2. U(1) Gauge Field Implementation and Gauge Invariance Test Code Package Supporting software package for introducing U(1) gauge fields via link variables and performing quantitative numerical verification of gauge invariance preservation. This package includes a complete U(1) link variable implementation, gauge transformation generators (random and smooth), and gauge invariance test routines with defined tolerance criteria. It serves as a foundational testbed for integrating gauge fields into the memory kernel framework and represents an important step toward extending the Higgs hierarchy stabilization approach to gauge theories. A detailed design document outlining the roadmap to non-Abelian extensions (planned for v2.6.5) is also included. Both code packages are written in standard C++17 with no external dependencies and are accompanied by comprehensive documentation to ensure full reproducibility and facilitate further development within the UDCT Higgs hierarchy program. This work builds directly upon previous UDCT technical notes (v2.6.1–v2.6.3) on Higgs hierarchy stabilization via memory kernel in geometric back-reaction. Keywords: Lattice gauge theory, Compact U(1), Monte Carlo simulation, Finite-size scaling, Plaquette susceptibility, Gauge invariance, Higgs hierarchy, Memory kernel, Reproducible software