Geometric Phase Frustration and the Ball Spectral Problem in 3D Navier-Stokes Equations: A Machine-Verified Discrete Formulation

Abstract:This work investigates structural obstructions to finite-time blow-up in the 3D incompressible Navier-Stokes equations (NSE) through a machine-verified discrete framework (Lean 4). We introduce the Triadic Frustration Index (𝒟(M)), a spectral signature of blow-up resistance, capturing geometric phase cancellations induced by the Leray projection in Fourier space. Key Contributions: Formalization in Lean 4 of the 3D kinematic state space and geometric invariance theorems, with zero unproven physical axioms. Empirical demonstration that 𝒟(4) ≫ 10 for a Galerkin ball of radius M=4 (~340,000 valid triads), revealing massive cancellation of nonlinear fluxes. Asymptotic Frustration Conjecture: Limit of 𝒟(M) → ∞ as M → ∞, suggesting strict flux cancellation in the continuous limit. Formal proof that Hypothesis U (uniform enstrophy bound) satisfies the Prodi-Serrin criterion, excluding singular weak solutions. Approach: Epistemic audits of 1D dyadic models (Katz-Pavlović, Cheskidov), revealing their sign fragility (imposed phase coherence). Integration of results from Deng-Hani-Ma (2024, Hilbert’s 6th Problem), Hou-Li (2006, dynamic depletion), and Waleffe (1992, helical decomposition). Analytical bridge between discrete frustration and continuous regularity, via a Lean 4-formalized conditional reduction. Status:Work in Progress – Partial proofs in Lean 4, empirical validation for M=4, open asymptotic conjecture.Next Steps: Full formalization of the asymptotic conjecture, collaboration with CFD groups to validate 𝒟(M) at larger scales.