Universal Heat‑Kernel Duality: A Structural, Group‑Theoretic, and Geometric Compendium

This compendium presents a unified structural framework connecting the Selberg trace formula and the Migdal–Witten solution of two‑dimensional Yang–Mills theory through a shared mechanism built from heat kernels, Plancherel duality, and gluing along discrete combinatorial structures. Expanding on Anthony Perry’s foundational insight, the work generalises the correspondence across compact, non‑compact, and finite groups; develops a comprehensive extended dictionary linking spectral, geometric, and representation‑theoretic data; constructs a full numerical verification framework; and explores applications to BF theory, 2D Chern–Simons, random matrix models, and Hurwitz theory. The compendium remains strictly mathematical, avoiding physical speculation, and provides a modular, pedagogical, and computationally reproducible foundation for further research in solvable 2D theories and spectral geometry. Included a html visualiser app # KEYWORDSheat kernel, Plancherel duality, Selberg trace formula, Yang–Mills theory, representation theory, spectral geometry, hyperbolic geometry, Lie groups, BF theory, Chern–Simons theory, random matrix models, Hurwitz theory, geodesic spectrum, Casimir, spherical functions, conjugacy classes, numerical harmonic analysis, group theory, spectral decomposition, gluing rules # SUBJECTS- Mathematical Physics - Representation Theory - Spectral Geometry - Lie Groups and Harmonic Analysis - Topological Field Theory - Hyperbolic Geometry - Computational Mathematics - Gauge Theory (2D) - Automorphic Forms - Random Matrix Theory - Combinatorics of Finite Groups Contact: [email protected]