Release v1.0.1 — Four-Branch Formal Architecture for 3D Navier–Stokes Regularity

What's new in v1.0.1

This release extends the formal architecture to cover all four standard regularity branches of the 3D incompressible Navier–Stokes problem, on both the torus T³ and the Euclidean domain R³, under a single irreducible PDE assumption ω₀ ∈ L².

Summary of changes from v1.0.0

• T³ global regularity branch (B) completed as an end-to-end named theorem surface.
• T³ breakdown counterexample branch (D) completed as an end-to-end named theorem surface.
• R³ breakdown counterexample branch (C) completed with actual analytic realization (Euclidean Littlewood–Paley multipliers, Bernstein inequality, PDE-facing interface).
• R³ global regularity branch (A) completed via structural transfer of the T³(B) architecture.
• All four branches now share the same upper-layer theorems (closure, bootstrap, continuation, K_max-uniform limit passage) in a backend-agnostic manner.

Formal Verification Statistics

Four Named Final Theorems

All four terminal theorems share the same irreducible PDE hypothesis ω₀ ∈ L²:

The authors make no claim regarding the resolution of the Clay Millennium Problem. The scope and interpretation of such claims rests with the Clay Mathematics Institute.

Architectural Principle

The proof introduces no new mathematical constructs. All estimates are derived from standard tools — Cauchy–Schwarz, Gronwall's inequality, Arzelà–Ascoli, and the k⁴ dissipative scaling inherent in the viscous term — connected into typed return-value chains over existing Mathlib primitives.

The architecture is backend-agnostic: the same upper-layer theorems (closure, bootstrap, continuation, limit passage) apply to both T³ (discrete Fourier modes) and R³ (continuous frequency multipliers), once the backend-specific shell projector family is supplied. The R³ implementation treats L²(R³) as the frequency-side Hilbert space and realizes dyadic shell projectors as pointwise multiplication by indicator functions, avoiding the full Fourier transform machinery while retaining contraction, orthogonality, and Parseval-type decomposition.

Irreducible PDE Frontier

The sole external assumption of the entire architecture remains:

omega_mem : MemLp (omega k) 2    — initial vorticity ω₀ ∈ L²(T³)

R³-side initial data is packaged internally via R3SmoothInitialData and is not an additional external hypothesis at the architectural level.

All other assumptions that might appear external — Sop_ae_bound, sigma_mem, stretch_mem, Galerkin ODE existence, Arzelà–Ascoli extraction — are internalized as formally verified theorems.

Proof Architecture (unchanged from v1.0.0, extended to four branches)

• Theorem I (Closure): Conditional discrete Gronwall envelope from triad geometry, finite Fourier–Bernstein bounds, and shell decomposition.
• Theorem II (Realization): True Navier–Stokes field data discharges the hypotheses of Theorem I via Galerkin ODE realization.
• Theorem III (No-Blowup): Bootstrap strain regeneration and inductive continuation exclude any minimal blowup scenario.
• K_max-Uniform Limit Passage: Arzelà–Ascoli extraction and limit-closure stability pass the closure mechanism to the limit.

Core Idea (unchanged)

In the enstrophy budget dΩ/dt = P − D, the dissipation term D scales as νk⁴ in Fourier space, while nonlinear production P scales at most as k². For any finite initial energy, the k⁴ barrier prevents the stretching-dominated spectral band from advancing indefinitely. The closure factor F_low acts on low-shell energy (an L² quantity), not on enstrophy (H¹); combined with the Navier–Stokes energy identity (monotone L² decrease), this linearizes the naive dE/dt ≤ A·E^{3/2} inequality into a step bound compatible with discrete Gronwall.

The physical content of the second law of thermodynamics (monotone energy dissipation of viscous flow) becomes the structural fuel for the mathematical linearization.

Repository Contents

• /lean4/ — Complete Lean 4 formalization (159 files, lake build for full verification)
• /lean4/NSBarrier/THEOREM_MAP_ANNOTATED.md — Layered guide to the proof architecture
• /lean4/NSBarrier/FULL_THEOREM_REGISTRY.md — Complete registry of all 1,007 verified statements
• /solver/ — Pseudo-spectral DNS code (JAX/GPU) for the companion JFM paper
• /runs/ — Simulation scripts for all paper figures
• /figures/ — Figure generation code

Reproducing the Verification

git clone https://github.com/miosync-masa/Scale-resolved-competition.git
cd Scale-resolved-competition/lean4/
lake update
lake build
# Expected: Build completed successfully. (0 errors)

Key Mathlib Connections (all sorry-free)

• IsPicardLindelof — Cauchy–Lipschitz ODE existence
• BoundedContinuousFunction.arzela_ascoli — compactness extraction
• IsCompact.tendsto_subseq — subsequence extraction
• Convex.norm_image_sub_le_of_norm_hasDerivWithin_le — MVT for Lipschitz
• integral_eq_sub_of_hasDerivAt — FTC for shellwise identity
• MeasureTheory.Measure.addHaar_ball — Euclidean ball volume (R³ branch)

Acknowledgments

This proof introduces no new mathematical constructs. Every tool used — from Cauchy–Schwarz to Arzelà–Ascoli — was forged by generations of mathematicians and formalized by the Lean 4 and Mathlib communities. This work merely connects them where they were needed. The author's confidence in the correctness of this work does not rest on the author's own mathematical judgment. It rests on the Lean 4 theorem prover and the Mathlib4 library — infrastructure built by a community of researchers, most of whom the author has never met, across many years of careful and open collaboration.

Author

Masamichi Iizumi (Miosync, Inc.) ORCID: 0009-0007-0755-403X

License

MIT

Previous version

See v1.0.0 for the original T³-focused release (DOI: 10.5281/zenodo.19592265).