Projection-Enforced Computation, Solenoidal Turbulence Regularity, and Manifold-Locked Architecture

This work establishes the mathematical and physical foundations of the Deterministic Spectral Manifold framework, resolving numerical instability, generative hallucinations, and three-dimensional fluid regularity through geometric admissibility. By intercepting continuous non-linear updates with an orthogonal projection engine and algebraic integer quantization on an invariant discrete manifold, computational trajectories are strictly confined to valid states, preventing error compounding across arbitrary network depth. The underlying scaffold is constructed on the eight-hundred-sixty-one-node Johnson association scheme derived from the prime conductor of Euler's polynomial. This structure possesses a regular vertex valency of eighty and an invariant integer spectral gap. Arithmetic rigidity is anchored at the terminal class-number-one Heegner discriminant, ensuring unique algebraic factorization and eliminating floating-point mantissa drift. The six-dimensional rate-of-strain deformation space packs into the seventy-two kissing vectors of the exceptional Lie root lattice, yielding an invariant minus one-twenty-fourth commensurability defect that arrests phase diffusion and defines a universal topological boundary residue. In fluid dynamics, discrete exterior calculus on the triangular clique complex enforces exact coboundary nilpotency, guaranteeing bit-level divergence-free velocity and vorticity fields. Global three-dimensional Navier–Stokes regularity is established via uniform core exclusion: parallel vorton dipole pairs are mutually self-excluding due to singular Hamiltonian repulsion, while antiparallel pairs activate localized induction reconnection at the Kolmogorov dissipation scale, forcing cores apart and preventing finite-time vorticity blow-up for all time. The architecture is physically realized as a manifold-locked silicon processor comprising a spectral projection engine, a Heegner-rigid arithmetic unit, and a high-bandwidth synchronous crossbar fabric. The entire processor state vector fits within primary cache memory. Invalid computational candidate branches are shunted to ground at the analog crossbar before switching downstream digital logic gates, eliminating generative hallucinations while keeping operational energy dissipation bounded by thermodynamic Landauer limits.