FB(S³)R: Geometric Infrared Scale of Yang–Mills Theory on a Compact Three-Manifold: Discreteness, Spectral Threshold, and Fibonacci Hierarchy

In this work, we revisit the long-standing Yang–Mills mass gap problem from the perspective of spectral geometry on compact spatial manifolds. Instead of modifying the gauge sector or introducing phenomenological parameters, we explore a geometrically minimal setting: SU(N) Yang–Mills theory defined on the compact, simply-connected three-manifold S³. The key idea is conceptually simple but physically powerful: compact topology enforces spectral discreteness. On a compact manifold, elliptic operators such as the transverse vector Laplacian possess a purely discrete spectrum. Discreteness implies the existence of a lowest admissible mode frequency — a natural infrared threshold. For the three-sphere of radius R, the lowest vector eigenvalue yields the geometric spectral threshold: Δ = √3 / R which defines the minimal excitation scale of the gauge field in the linearised theory on S³(R) × ℝ. Identifying the compactification scale with the confinement scale through asymptotic freedom, R = 1 / ΛQCD leads to a characteristic energy: Δ ≈ √3 ΛQCD ≈ 500 MeV remarkably close to the effective gluon mass scale obtained from lattice QCD simulations and Dyson–Schwinger analyses. A central conceptual clarification introduced in this work is the Principle of Spectral Separation: the mass gap is determined by the lowest eigenfrequency of the gauge-field Hamiltonian; the structure of higher excitations is governed by an independent spectral hierarchy; hierarchical scaling must not be conflated with the origin of the gap itself. Within this framework we demonstrate that the Fibonacci spectral stratification constitutes the unique minimally redundant, gapless, self-similar coarse-graining of the Yang–Mills mode spectrum on S³. The Fibonacci organisation does not generate the gap, but canonically structures the tower of excitations above it. In the high-energy limit, ratios of successive glueball masses asymptotically approach the golden ratio: φ = (1 + √5) / 2 providing a scale-free signature that can be tested against future lattice calculations at higher spin. Conceptually, the work contributes to a broader programme exploring how global geometric structure constrains quantum field spectra. In general relativity, curvature determines gravitational dynamics; in compact quantum gauge theory, topology constrains the admissible eigenmodes of the field. From this viewpoint: compact topology → discrete spectrum → positive geometric threshold while the Fibonacci hierarchy provides the canonical organisation of excitations above that threshold. Although the persistence of the gap in the infinite-volume limit ℝ³ × ℝ remains an open problem — and is directly related to the Clay Millennium Problem formulated by Jaffe and Witten — the compact formulation separates geometric and dynamical contributions in a mathematically controlled way. The results suggest that aspects of confinement may be understood not only as consequences of nonlinear gauge dynamics, but also as manifestations of global spectral constraints imposed by spatial geometry. In this sense, the Yang–Mills mass scale may be viewed as emerging at the intersection of topology, spectral theory, and quantum gauge dynamics.