Physics and Computation of Short-Range Fundamental Interactions: A Topological–Historical Gauge-Spectral Reconstruction of Quantum Chromodynamics, Electroweak Dynamics, Finite Particle Cores, Causal Response, and Unified Non-Lattice Computation

This monograph develops a continuous mathematical framework for the strong and electroweak short-distance sectors while keeping their distinct infrared meanings explicit. The color connection is a massless non-Abelian gauge connection at the level of the local action; observable short-range strong behavior is instead tied to color-singlet asymptotic states, confinement, a physical mass gap, and string breaking. The weak interaction is short ranged because the charged and neutral weak response directions possess positive pole masses, whereas the electromagnetic direction remains exactly massless. The construction combines finite topological particle cores, a conditional faithful internal gauge group, BRST reduction, causal historical auxiliary systems, covariant operator spectra, and representation-space harmonic analysis. Its computational realization, called the Continuous Topological–Historical Gauge-Spectral Method for Short-Range Interactions (CTHG-SRI), uses no regular spacetime lattice as a defining structure. Every nonstandard correction is separated from ordinary propagation, self-energy, renormalization-group running, bound-state retardation, environmental decoherence, and detector memory. Results are classified as definitions, representations, conditional theorems, controlled limits, computations, experimental inputs, or rejection criteria; only three physical axioms are inherited. The present opening module fixes the logical types, ownership rules, complete short-range state, conditional gauge-group import, and the first exact electroweak stiffness benchmark. It proves that a gauge-invariant rank-three stiffness on \(SU(2)_L\times U(1)_Y\) leaves one and only one massless photon direction, while producing positive tree-level \(W^\pm\) and \(Z\) masses. The theorem is accompanied by a reproducible computed eigensystem, dimensional checks, perturbation bounds, and the standard recovery limit. Keywords Quantum chromodynamics; electroweak theory; BRST cohomology; finite topological cores; causal memory; Peter–Weyl expansion; covariant spectra; non-lattice computation; confinement; weak interaction.