Spectral Geometry of the Helical Triadic Tensor and Global Regularity for the 3D Navier–Stokes System: Constructive Formalization in Lean 4 and Deterministic Verification in Rust

This monograph establishes a self-contained spectral framework for the global regularity analysis of the three-dimensional incompressible Navier–Stokes equations on \(R^3\) and \(T^3\). In the helical polarization basis, the non-linear convective tensor admits an exact chiral parity decomposition governed by a structural asymmetry between homochiral and heterochiral triadic channels. This geometric structure induces a dissipative Riccati flow on the unit Sobolev sphere that bounds vortex alignment through a negative-definite homochiral damping mechanism. We prove uniform saturation bounds that remain stable under the continuum limit \(Kₘₐₓ → ∞\), ensuring the global existence of unique smooth solutions for both homogeneous and forced Cauchy regimes. The complete deductive architecture is mechanically certified in the Lean 4 interactive theorem prover (ProtocolNS_Bivariant) across 2,472 targets with zero unverified axioms (0 sorry). In parallel, the quantitative invariants, multi-scale continuum calibrations up to \(Kₘₐₓ=1024\), and IEEE 754 floating-point backward error stability (\(≤ 1.8 × 10⁻¹⁴\)) are deterministically audited by a dedicated, zero-allocation Rust computational verification engine (stokes_scaling) executing on consumer silicon in less than 25 MB RAM. Technological Disclosure and Declaration of Open Heritage Notice on Prior Art and Public Domain Stewardship: The analytical architecture, spectral decompositions, and computational verification algorithms established throughout this work—specifically the exact chiral parity decoupling of the non-linear convective tensor, the negative-definite homochiral Riccati damping, the invariant coherence barrier \(χ(t) ≤ 1 - θ_* < 1\), and the collapse of the turbulent dynamics to a finite-dimensional inertial manifold (\(dKY ≤ 6.00\))—describe intrinsic, immutable mathematical properties of physical continuum mechanics. Under 35 U.S.C. § 101 and Article 52(2)(a) of the European Patent Convention (EPC), discoveries of natural order, physical phenomena, and pure mathematical methods constitute non-patentable subject matter. Deposited into permanent open repositories with immutable cryptographic timestamping, this release establishes an incontestable record of Prior Art under 35 U.S.C. § 102 and Article 54 EPC to prevent any corporate, private, or state enclosure of the topological resolution of fluid turbulence. By demonstrating that rigorous continuum verification executes on consumer silicon with less than 25 MB RAM, this framework dismantles the economic monopoly of multi-million-dollar computing clusters, granting computational sovereignty to independent scholars, public universities, and environmental/climate researchers globally.