The topological necessity of the mass gap

The Yang–Mills mass-gap problem asks why the lowest excitation of a gauge field has strictly positive energy. In this paper, I present a mechanical derivation of the mass gap based on the geometry of the 137-voxel Möbius loop, the fine-structure constant, and the energetic cost of time. Previous work established that the inverse fine-structure constant arises from a torsional secant-stretch of the 137-unit matter loop. This stretch produces a geometric overshoot, which forces a minimal protrusion into the surrounding Zero-Point Energy (ZPE) matrix. The ZPE medium has a mechanically derived local density, making any protrusion-displacement energetically costly. Recent results in thermodynamic timekeeping demonstrate that time itself has a minimal energetic cost. Because the protrusion > 0 is topologically unavoidable for any 137-loop, the production of time — and therefore energy — is unavoidable.