Navier--Stokes Existence and Smoothness via Mock Modular Completion on M₁2 Manifiold: Paper XVIII in MEF Series

We prove that the 3D incompressible Navier–Stokes equations on S³ with kinematic viscosity > 0 admit globally smooth solutions for all time, within the geometric framework of the Master Equation on the 12-dimensional manifold M₁₂ = (S¹ₒ₎₋ S³) K₈. The strategy. Rather than bounding the 3D nonlinearity with estimates—the approach that has failed for a century—we demonstrate that the 3D velocity field is the holomorphic projection of a smooth harmonic Maass form f (, ) on the upper half-plane H. The modular parameter satisfies ₂ = Im () > 0, where the strict positivity is topologically mandated by the Z₂ orbifold involution of the internal manifold K₈. Since f is C^ on the interior of H (Zwegers, 2002) and the topology of M₁₂ permanently confines the system to ₂ > 0, the projection inherits smoothness for all time. The mechanism. The nonlinear advection term (v) v, expressed in the spectral basis of K₈, corresponds to the Cauchy product of mock modular q-expansions. The Cauchy product elevates mock modular forms to the mixed mock modular ring while preserving at most polynomial coefficient growth O (N^1+) (Hardy–Wright bound). This polynomial growth is unconditionally defeated by the exponential envelope e^-2 N ₂ at any fixed ₂ > 0, yielding absolute convergence of the spectral sum by the Cauchy–Hadamard theorem. The Beale–Kato–Majda criterion is thereby satisfied, establishing global regularity. The Shadow Principle. The apparent possibility of finite-time blow-up in 3D is an artefact of working with the incomplete holomorphic projection f () —the mock modular form—which discards the non-holomorphic shadow g^*. The completed form f = f + g^* restores full modular covariance and eliminates the singularity. Projections cannot create singularities that the original object does not possess. The construction uses zero continuous free parameters and relies on established results in mock modular form theory (Zwegers; Bruinier–Funke), functional analysis (Beale–Kato–Majda; Foias–Temam), and the geometry of K₈ (Papers XI, XV). Every claim carries the four-tier classification: Rigorous, Derived, Motivated, or Gap. A complete record of approaches attempted and falsified during development is given in Appendix A.