Geometric Phase Frustration and the Ball Spectral Problem in 3D Navier-Stokes Equations: A Machine-Verified Discrete Formulation
Year: 2026
Type: article
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 (๐(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 ๐(4) โซ 10 for a Galerkin ball of radius M=4 (~340,000 valid triads), revealing massive cancellation of nonlinear fluxes. Asymptotic Frustration Conjecture: Limit of ๐(M) โ โ as M โ โ, 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-Pavloviฤ, Cheskidov), revealing their sign fragility (imposed phase coherence). Integration of results from Deng-Hani-Ma (2024, Hilbertโs 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 โ 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 ๐(M) at larger scales.
Author: Callens, Xavier
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204229531
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