Higgs–Ω: Entropy-Dependent Scalar Potential, Mass and Vacuum Structure Symmetry Breaking, Mass Relations, Renormalization and Scalar Phenomenology

This article develops a mathematical formulation of the entropy-dependent Higgs scalar potential, vacuum structure, and mass-generation relations within the Unified Entropic Field Theory framework (UEFT–Ω). Starting from the probabilistic entropy field (S(x,t)), the study constructs entropy-dressed Higgs kinetic and potential terms and derives the associated vacuum stationarity conditions, scalar-potential curvature, canonical field normalization, and physical mass relations. The mathematical investigation includes fourteen lemmas, six principal H–Ω theorems, four propositions, and detailed derivation chains connecting the foundational effective action to scalar mass relations, parameter fitting, renormalization dynamics, vacuum stability, and phenomenological observables. The analysis develops entropy-dependent Higgs parameter-fit manifolds and distinguishes potential-coordinate quantities from canonically normalized physical variables. Renormalization-group relations are examined through coupled entropy-dependent flow equations, analytical solutions, explicitly calculated numerical trajectories, and stability conditions. Moderate-dressing parameter scans and phase diagrams establish the mathematical boundaries associated with kinetic normalization, scalar curvature, and quartic-potential stability within the specified model. Higgs self-interactions, trilinear and quartic couplings, and illustrative collider-related benchmarks are also investigated. The derivations distinguish exact algebraic identities, conditional results of the reduced dynamical model, and phenomenological estimates from independently established experimental predictions. The framework integrates UCAT–Ω, ECO–Ω, EWC–Ω, and SCOUT–Ω for parameter-response analysis, mathematical consistency, uncertainty assessment, and stability evaluation. Analytical numerical calculations, reproducible Python implementations, scalar-response estimates, and computational phase diagrams accompany the theoretical development. The article presents an interconnected mathematical construction of Higgs mass and vacuum behavior within UEFT–Ω, while retaining the distinction between internal model consistency and independent experimental verification. Five-Article UEFT–Ω Research Series This publication is Article II 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. Article I develops the Yang–Mills gauge-sector and mass-gap construction. The present Article II develops the Higgs scalar-sector contribution, including entropy-dependent symmetry breaking, vacuum curvature, mass relations, renormalization dynamics, and phenomenological implications. The companion articles address neutrino masses and mixing, unified mass generation, and gravitational and quantum-equivalence relationships. Version 1.0 — First edition: 6 November 2026.