Yang-Mills–Ω: Entropic Gauge Spectra and Mass-Gap Structure Euclidean Construction, Spectral Relations, and the UEFT–Ω Ecosystem

This article develops a mathematical formulation of non-Abelian Yang–Mills gauge dynamics and mass-gap structure within the Unified Entropic Field Theory framework (UEFT–Ω). Starting from the probabilistic entropy field (S(x,t)), the study develops entropy-weighted Euclidean gauge measures, reflection-positivity conditions, transfer-operator spectral relations, gauge-invariant correlation functions, confinement-related estimates, and renormalization-group structures. The mathematical investigation includes twelve lemmas, associated theorems, and detailed derivation chains addressing the theoretical construction and its conditional implications. Three complementary mass-gap formulations are examined: spectral eigenvalue separation, exponential decay of gauge-invariant correlations, and entropy-curvature scaling. The framework integrates UCAT–Ω/UCAT-GP, ECO–Ω, EWC–Ω, and SCOUT–Ω for spectral analysis, robust estimation, uncertainty assessment, and consistency verification. Analytical numerical estimates, exploratory computational implementations, and a proposed 36-setting lattice simulation protocol are also presented, with proposed simulations distinguished from completed numerical calculations. Five-Article UEFT–Ω Research Series This publication is Article I of a coordinated five-article research programme, developed between 16 October 2025 and 9 March 2026 to investigate entropy-based formulations of gauge dynamics, particle masses, and gravity: Article I — Yang–Mills: Entropic Gauge Spectra and Mass-Gap Structure. Article II — Higgs: Entropy-Dependent Scalar Potential, Mass and Vacuum Structure. Article III — Neutrinos: Entropic Mass Matrices, Hierarchies and Mixing. Article IV — Unified Mass: Sectoral Entropy-Curvature Architecture and Unified Mass Generation. Article V — Gravity and QEP–Ω: Mass, Gravity and Branchwise Quantum Equivalence. The articles investigate distinct but interconnected physical sectors under a common UEFT–Ω mathematical foundation. The present article develops the Yang–Mills contribution, while the companion articles address scalar and neutrino mass generation, the unified mass architecture, and gravitational and quantum-equivalence relationships.