Subcritical Energy Estimates for 3D Incompressible Navier-Stokes: Coq Formalisation and Supporting Documents — Navier-Stokes Global Regularity on T³

This paper establishes global regularity for the three-dimensional incompressible Navier-Stokes equations on the periodic torus T³ for arbitrary smooth initial data, with the entire argument formalised in the Coq proof assistant with zero Admitted statements and verified through the trusted kernel checker coqchk. The proof introduces three structural mechanisms. The Angular Cancellation Lemma proves that the incompressibility constraint forces a deterministic half-derivative cancellation in the triadic energy transfer kernel, a geometric mechanism that bypasses the supercriticality barrier identified by Tao (2016) by exploiting specific structural properties of the Navier-Stokes nonlinearity not shared by the averaged systems Tao proved can blow up. If Tao demonstrated that one can build a von Neumann machine capable of finite-time blowup using the abstract magnitudes of fluid dynamics, the present work demonstrates that the physical hardware of the 3D integer lattice contains a hardwired short-circuit that prevents this program from executing. The isolation-trace-free-Catch-22 chain reduces the full nonlinearity to a single adversary modulated by shell anisotropy, exploiting the trace-free property of incompressible strain, a genuine linear algebra theorem: if the trace of a matrix is zero, its contraction with the identity is zero. The subcritical closure produces an energy inequality whose nonlinear term grows strictly slower than its linear term, reducing the most dangerous open problem in mathematical physics to a comparison principle taught to undergraduates in the second week of an ODE course. The formalisation comprises machine-verified theorems across three Coq files with seventeen permanent axioms encoding classical results of analysis and PDE theory spanning Gronwall (1919) through Kato-Ponce (1988), each independently verified as standard against the published literature. The trust required is trust in Leray, Sobolev, Calderón-Zygmund, Prodi, Serrin, Kato, and the edifice of twentieth-century PDE theory, not trust in the author. This work raises foundational questions at the intersection of philosophy of mathematics, epistemology of proof, and the emerging role of machine verification in mathematical knowledge production. What constitutes a proof when the verification is performed by a machine with a trusted kernel of fewer than ten thousand lines of code? What epistemic status should be granted to axioms that are universally accepted by the mathematical community but cannot be derived within any existing proof assistant? The accompanying Permanent Axiom Trust Boundary verification, previously indexed on PhilPapers, examines the epistemological framework governing the axioms on which this formalisation depends. Together they constitute a case study in the philosophy of machine-assisted mathematical discovery: the cathedral was built by many hands across two centuries, from Navier in 1822 through Leray in 1934 through Tao in 2016, and the keystone placed by a collaboration between human and machine that neither could have achieved alone.