A BF-Theoretic Spectral Origin for the Yang–Mills Mass Gap: Internal Poincare Coercivity from a Universal Curvature Framework

Abstract

The Yang-Mills mass gap problem remains one of the most important unresolved questions in mathematical physics. Existing approaches include lattice gauge theory, Schwinger-Dyson equations, functional renormalization group methods, geometric gauge theory, instanton methods, topological approaches, and constructive quantum field theory. Despite overwhelming numerical and physical evidence for a positive mass gap, a rigorous analytical derivation remains absent. In this work, we propose a novel perspective inspired by a previously developed Unified Curvature Conjecture based on BF theory, formulating a tripartite conceptual framework that bridges top-down, inverse, and lateral approaches. Rather than searching directly for a mass gap in the standard Yang-Mills sector, we investigate whether a spectral gap may emerge from a more primitive BF variational structure defined on a universal Spin(10) bundle. A central result of the present proposal is the introduction of an internal spectral coercivity condition analogous to the classical Poincare inequality. We show formally that the complement sector of the Lie algebra decomposition may induce an effective spectral lower bound in the Yang-Mills Hessian through a Schur-complement mechanism. Under suitable assumptions, the effective Yang-Mills operator acquires a positive spectral gap generated by the geometry of the internal symmetry space itself. This top-down mechanism is further validated through an inverse spectral analysis in the infrared limit and cross-examined via a lateral approach connecting discretized lattice spin foams and Hooft confinement dualities. The resulting framework suggests a mathematically coherent bridge between topological BF theory, Lie-algebra decompositions, spectral coercivity, and the emergence of non-perturbative mass scales in Yang-Mills theories.