Nonlinear Calculus and Solving Three-Dimensional Navier–Stokes

The paper develops an author-originated matrix-of-error framework called nonlinear calculus and applies that through one shared derivation-and-testing process across distinct, omni-dimensional dynamical systems. It recovers established cross-system results, derives and validates the full three-dimensional Navier–Stokes representation, and connects that representation to a smooth-continuation interpretation through established regularity mathematics. Abstract: Nonlinear calculus proposes the Universal Matrix-of-Error Theory: one dimension-agnostic derivation-and-testing map can evaluate the operative correction structure of different dynamical systems once each system’s proper variables and established mathematical inputs are supplied. The associated omni-dimensional-equilibrium hypothesis identifies zero-error cases as the non error, textbook equilibria against which operative corrections can be resolved. Section 1 tests the proposal through one shared weak-form process that independently recovers established governing structures in Mercury’s perihelion dynamics, viscous Burgers dynamics, and two- and three-dimensional Navier–Stokes vorticity. Section 2 formalizes the common map, relates it directly to Magnus, SINDy, PDE-FIND, WSINDy, and SPIDER, and derives the full three-dimensional Navier–Stokes Matrix of Error ENS =[Sω | ν∆ω]. Under an exact two-plus-one decomposition, the complete coefficient structure is (B,b,s,C2,C1). Section 3 compares that derived representation with the earlier naive pre-derivation form (B,C2,C1). The full representation passes its decomposition, rotational, diffusion-block, and independent enstrophy-rate checks at approximately machine precision. The naive form fails in all 48 predetermined flow/frame/split evaluations, and a matched 32^3 trajectory reproduces the result. Viscous dissipation exceeds vortex stretching at all 204 primary sampled flow-time states, with B/C ≤0.955849, and the sampled enstrophy values decrease along all four primary trajectories. Section 4 applies the completed matrix without changing its terms. Exact amplitude scaling constructs strict-zero, dissipation-dominant, nonzero aggregate-balance, and stretching-dominant cases from one reference field. Under one fixed evolution, the below and boundary cases decay; the above case reaches 1.034606 times its initial enstrophy, crosses into dissipation dominance at t = 0.458898, and ends at 1.034245. The complete five-block sum closes at approximately machine precision, and every path remains below its corresponding smooth-continuation envelope. Together, Parts I–III close the derivation-and-testing program with a positive result and support the Universal Matrix-of-Error Theory over the completed range. Part IV makes the range of behavior produced by that result physically legible without adding another proof condition. The full matrix is advanced as the theory’s three-dimensional representation for the Navier–Stokes existence-and-smoothness problem. The observed dissipative inequality is tied directly to the Section 2 enstrophy identity, and its proposed general form supplies the stated smooth-continuation implication. Section 5 assembles the full visible chain from the derived matrix and exact enstrophy identity to the computed remainder, the selected blockwise majorant, and the finite-integral smooth-continuation criterion. It compares this criterion with Chae–Choe and Miller, proves the conditional implication, and explains how either initial-data estimates or rigorous solution dependent bounds can supply its integrability hypothesis. The universal smooth-continuation interpretation is stated as the theory’s claim; the exact identities, conditional result, and completed numerical observations give its explicitly identified mathematical and empirical basis.