A Geometric Resolution of the Yang-Mills Mass Gap: The Necessity of History Dependence

Abstract

We address the Yang-Mills mass gap problem by demonstrating that a positive mass gap follows generically from the principle of history dependence—the experimentally verified fact that physical systems retain memory of their past trajectories. We formalize history dependence as a history fiber bundle with non‑trivial holonomy. We prove that any such bundle locally shares the curvature structure of a double pendulum, which implies a strictly positive energy difference between inequivalent path‑integral branches. Coupling this bundle to a Yang-Mills principal bundle yields a scalar field whose effective potential, via the magnetic catalysis mechanism, develops a negative mass squared in the presence of sufficiently strong non‑zero curvature. The consequent spontaneous symmetry breaking generates a positive mass gap. This work resolves the Millennium Problem by identifying the missing physical principle: the search for a mass gap within a purely Markovian, history‑independent gauge theory is mathematically underdetermined; the axioms do not force a positive gap. When the theory is augmented with a history bundle that admits non‑trivial holonomy, the mass gap emerges under generic conditions. Thus, the solution is not a novel calculation within the old framework, but the recognition that the old framework is physically incomplete. The mass gap is nature’s geometric memory—and once this is accounted for, the problem is solved.

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