A Reduction Attempt of the Mass Gap Problem via Topological Defects in Q8 Discrete Geometry: A Nonperturbative Conjecture in a Scalar Field Model

The mass gap is one of the central challenges in quantum field theory. This paper does not attempt to directly solve the problem, but rather to reduce it to a more fundamental mathematical proposition. Specifically, we construct a scalar field topological defect model based on the discrete geometry of the Q8 hypercube graph, and prove that if the spectral gap of the Q8 discrete Laplacian can constrain the amplitude of topological defects (i.e., if Conjecture 1 holds), then the global minimum of the energy functional for any topological defect configuration with nonzero winding number is strictly greater than zero — this is precisely the geometric counterpart of the mass gap. In other words, we reduce the proposition “the mass gap exists” to the more specific and more tractable mathematical conjecture that “the Q8 spectral gap constrains the topological defect amplitude.” The reduction itself is rigorous; Conjecture 1, on which the reduction depends, has not yet been proven and is stated as an independent mathematical conjecture. In addition, we note a heuristic numerical coincidence: χ_top^{1/4}/Λ_QCD is numerically close to sin(π/8), and we present this as a target for future lattice QCD tests. The contribution of this paper is not to solve the mass gap problem, but to transform it into a mathematically well-defined and tractable conjecture, thereby providing a clear direction for subsequent research.