Structural Parallels Between Spectral Confinement on SU(2) and the Yang–Mills Mass Gap: A Conjecture Programme

Paper 15 in The Geometry of the Critical Line programme. We identify structural parallels between the spectral geometry of the compact group manifold SU(2) ≅ S³ and the Yang–Mills mass gap problem. The Laplace–Beltrami operator on S³ has a purely discrete spectrum with explicit eigenvalues λ_l = l(l+2)/R² and a positive spectral gap λ₁ = 3/R². In temporal gauge, the Yang–Mills Hamiltonian restricted to spatially constant gauge fields reduces to this Laplacian, so the geometric gap is a necessary condition for any spectral gap in the full quantum theory. The naive infinite-volume limit R → ∞ sends this gap to zero; the physical mass gap emerges from dimensional transmutation, not from the classical geometry alone. We identify a further structural correspondence: the Lambert spectrum from Paper 9 — a countable family of roots z_k(α) indexed by k ∈ ℤ — shares formal features with the topological winding-number classification of Yang–Mills vacuum sectors. Both are integer-indexed, both exhibit complex-conjugate pairing, and both have a distinguished principal pair. We state this as an explicit hypothesis and identify a scale-inversion paradox: the Lambert root magnitudes decay as 1/|k| while instanton actions grow as |k|. This paradox is the central open problem of the correspondence and must be resolved before any rigorous identification can be made. No claim is made that this constitutes a solution or partial solution to the Clay Millennium Problem. This paper is a conjecture programme: it states precise open problems connecting the SCT geometric framework to Yang–Mills physics, with every claim graded by rigour from proved theorem to speculative hypothesis. A six-point weight-limit section explicitly catalogues where the analogies break down. This is Paper 15 in a series spanning pure mathematics (Papers 0–10), nonlinear dynamics (Papers 11–12), warp drive phenomenology (Paper 13), computational PDE analysis (Paper 14), and now gauge theory.