Unified Mass–Ω: Sectoral Entropy–Curvature Architecture and Unified Mass Generation Gauge Spectral Gaps, Scalar Vacuum Curvature, and Neutrino Mass Operators

This article develops the sectoral entropy–curvature architecture of Unified Entropic Field Theory (UEFT–Ω), establishing a common mathematical framework for investigating mass generation across non-Abelian gauge fields, the Higgs sector, and neutrino flavor dynamics. Building upon the mathematical formulations established in Articles I–III, the study retains the original probabilistic entropy field (S(x,t)) as the common theoretical foundation while preserving the distinct physical and spectral definitions of mass in each sector. The Yang–Mills mass is represented through the physical Hamiltonian spectral threshold, the Higgs mass through canonically normalized vacuum curvature, and neutrino masses through entropy-dependent Majorana mass matrices and Takagi diagonalization. The investigation derives entropy-dressed quadratic actions, generalized mass operators, canonical field normalizations, vacuum stationarity conditions, spectral response equations, and the mathematical relationships connecting Higgs vacuum structure to effective neutrino mass generation. Explicit calculations establish the relevant normalization identities and demonstrate how distinct mass observables can depend on a common entropy background without requiring their underlying mathematical operators to be identical. The article develops five principal mathematical results: Theorem U–Ω–1 establishes the conditional unified mass architecture connecting the Yang–Mills spectral mass, Higgs vacuum curvature, and neutrino mass-matrix structures. Theorem U–Ω–2 develops the cross-sector entropy-response formulation, deriving differential relationships governing scalar mass variations, neutrino spectral responses, and entropy-dependent gauge-sector stiffness. Theorem U–Ω–3 constructs the quantitative mass-hierarchy framework through three benchmark families: flavor-first, scalar-first, and combined scalar–flavor response. These constructions preserve the original entropy-dependent parameterizations and incorporate exact matrix diagonalization, physical normalization, and spectral calculations. Theorem U–Ω–4 establishes the mathematical composition of the UEFT–Ω computational and certification framework, retaining the canonical sequence UCAT–Ω → ECO–Ω → EWC–Ω → SCOUT–Ω and the Definition → Certificate → Execution → Closure mathematical structure. Theorem U–Ω–5 formulates the interface between sectoral mass generation and gravitational response, identifying the mathematical conditions required to connect entropy-dependent mass structures with total stress–energy, weak gravitational responses, and branchwise quantum-equivalence investigations. The manuscript contains expanded lemmas, propositions, and theorem proofs, with intermediate algebraic substitutions, matrix transformations, spectral identities, normalization calculations, differential relations, and theorem dependencies developed explicitly. The calculations preserve the established mathematical findings of the preceding three articles and distinguish exact identities, conditional theoretical relations, and numerical verification from independently demonstrated physical results. The quantitative investigation examines the original UEFT–Ω benchmark families A, B, and C, including Higgs-sector evolution, Majorana mass spectra, neutrino flavor mixing, spectral thresholds, and cross-sector parameter sensitivities. A reproducibility appendix provides executable Python verification code and numerical calculations using the specified theoretical inputs. The resulting architecture presents unified mass generation as a coordinated mathematical description of distinct entropy-dependent physical sectors. It provides a foundation for further investigation of mass hierarchies, vacuum response, spectral stability, and relationships between particle masses and gravitation, while retaining the independent proof obligations associated with continuum Yang–Mills theory, experimental neutrino phenomenology, and gravitational equivalence. Five-Article UEFT–Ω Research Series This publication is Article IV of a coordinated five-article research programme, developed between 16 October 2025 and 9 March 2026, examining entropy-dependent structures in fundamental physics. Article I — Yang–Mills: Entropic Gauge Spectra and Mass-Gap Structure develops the non-Abelian gauge-sector spectral formulation. Article II — Higgs: Entropy-Dependent Scalar Potential, Mass and Vacuum Structure investigates entropy-dependent scalar potentials, vacuum configurations, and physical Higgs mass relations. Article III — Neutrinos: Entropic Mass Matrices, Hierarchies and Mixing develops effective neutrino mass operators, Majorana spectra, flavor hierarchies, and mixing structures. Article IV — Unified Mass: Sectoral Entropy–Curvature Architecture and Unified Mass Generation integrates the preceding sectoral mathematical formulations into a common entropy-dependent mass architecture. Article V — Gravity and QEP–Ω: Mass, Gravity and Branchwise Quantum Equivalence extends the investigation toward gravitational response and branchwise quantum equivalence. Together, the five articles constitute a connected research programme investigating the mathematical relationships among entropy, field dynamics, mass generation, and gravitation within UEFT–Ω. Author: Nazım ArıResearch position: Independent Researcher, Freiburg im Breisgau, GermanyOriginal research period: 16 October 2025–9 March 2026Version: 1.0 — First edition prepared: 10 November 2026DOI: https://doi.org/10.5281/zenodo.24094559ORCID: https://orcid.org/0009-0004-2131-5930 Copyright: © 2026 Nazım Arı. All rights reserved.