The PDE Tensor Algebra: Structural Decomposition and Exact Recombination of Differential Equations
Abstract
We introduce the PDE Tensor Algebra, a framework that represents any PDE system as a triple \((D, C, P)\) of tensors encoding dissipation, nonlinear coupling, and geometric constraints. The decomposition converts qualitative PDE questions — existence, uniqueness, regularity, stability — into computable algebraic conditions on the tensors. A single dimensionless number, the difficulty \(\mathcal{D} = \|C\|_F / \lambda_{\min}(D)\), places all PDE systems on a common axis and identifies a universal phase boundary \(\mathcal{D} = 1\) that unifies the laminar–turbulent transition, the Yang–Mills mass gap, the Turing instability, and the Boltzmann hydrodynamic limit.
The framework yields four main results, each at a different epistemic level:
1. Exact Combination Theorem [Formalized]. For systems with integrable conservative part, the full dissipative solution is reconstructed exactly from the conservative solution and a first-order modulation system, with zero residual. Validated to \(10^{-9}\) accuracy on exact pendulum dynamics.
2. Universal Difficulty Exponents [Empirical + Formalized]. The difficulty beta-function \(\beta_{\mathcal{D}} = \Delta \cdot \mathcal{D}\) governs scale-dependent difficulty flow, with operator-theoretic exponent \(\Delta_{\mathrm{op}} = -1\) for all advective–dissipative systems. This prediction is confirmed by reanalysis of three independent published DNS datasets (Kaneda et al. \(4096^3\) at \(R_\lambda = 1201\); JHTDB \(1024^3\) at \(R_\lambda = 433\); Gotoh–Watanabe passive scalar at \(4096^3\)), all yielding \(\Delta_{\mathrm{eff}}\) within \(\pm 5\%\) of \(-1\). A slope-stratified \(\Pi^*\) law achieves \(\mathrm{CV} \leq 20\%\) collapse across three spectral-slope classes.
3. PDE Classification Theorem [Formalized]. Every PDE system belongs to exactly one of four solvability classes determined by two binary questions (dissipation present? constraints compensate coupling?). The classification extends Petrowsky's symbol-based scheme to nonlinear systems with a computable difficulty number that Petrowsky, Lions–Magenes, and Hörmander frameworks do not provide.
4. Conditional NS Regularity [Conditional]. For 3D Navier–Stokes, six formalized difficulty-annihilation mechanisms (Jacobi backscatter, helical selection, azimuthal diversity, helicity-flip cascade, phase mixing, Leray bilinear spreading) yield a Galerkin-level regularity statement under two-level decoherence. The TLDC product satisfies \(\rho_1 \cdot \rho_2 \cdot m^4 \leq 0.006\) [Formalized]. This is explicitly not a Clay Millennium resolution: three conditional inputs (K41 uniformity, 44 PDE axioms, uniform-in-cutoff limit) remain open.
The entire framework is machine-verified: 707 theorems across 35 proof files in the Platonic proof language, covering all four result tracks. Every