Finite-Dimensional Spectral Closure and a Relative Forced Singular Branch for Navier–Stokes on the Round Three-Sphere

This study develops a unified mathematical framework connecting finite-time singular dynamics, exact spectral closure, and geometric structure in the forced incompressible Navier–Stokes equations on the round three-sphere S³_R. Building on five explicitly stated assumptions about a Euclidean forced-blowup construction, the paper formulates a conditional transfer of a bounded-energy singular branch to compact, boundaryless geometry. The geometric mechanism rests on the exact identity √det(g_R) = r, which preserves the cylindrical incompressibility operator while introducing controlled curvature corrections into the momentum equations. The construction is conditional on the external source assumptions and their solvability, matching, and localization estimates; it does not independently prove the Euclidean source theorem. The independent mathematical contribution establishes a coordinate-free criterion for exact finite-dimensional spectral closure. With the symmetric polarization Q(u,v) = ½[B(u,v) + B(v,u)], a finite modal space W is an exact reducing space precisely when AW ⊂ W and Q(W,W) ⊂ W. The equivalent zero-defect test 𝔏_Q(W) = 0 is basis-independent; an analogous representation-theoretic criterion applies to finite unions of complete signed curl shells. Killing fields, maximal-torus reductions, and Beltrami eigenspaces furnish exact invariant solution families. A finite-sector obstruction shows that no fixed finite-dimensional exact reducing space can support bounded-energy L∞ blowup under time-integrable L² forcing. The central structural consequence is infinite-dimensional spectral escape. On S³_R, the exact curl-band rank is N_K = (K + 1)(K + 2)(2K + 9)/3. Sharp projector estimates imply that any bounded-energy singular branch must escape every fixed finite spectral band: capturing a nonzero fraction of its increasing velocity peak requires an unbounded number of modes. Localization of the singular core strengthens this to a concentration-dependent rank bound. These findings connect physical-space concentration, nonlinear modal closure, and growing spectral complexity, and provide a general test for exact reductions of quadratic evolution equations.