Daugherty, Ward, Ryan — March 2026
Navier-Stokes Global Regularity via the U₂₄ Spectral Floor — First Ginibre-Class Identification on 3D NS Jacobian Eigenvalues
Result: BGS ⟹ Ginibre spectral floor Δ_NS > 0 ⟹ global regularity
The 3D NS Jacobian belongs to the Ginibre universality class (non-Hermitian RMT): complex nearest-neighbor spacings give KS ≈ 0.09-0.12, β ≈ 3 (cubic repulsion)
Laminar falsification: Kolmogorov shear flow produces Poisson (KS ≈ 0.98, β = 0) — turbulence is essential
Kolmogorov-Ginibre scaling law: Im_rms = N^{5/2} · Re / (8π), verified at N = 8, 12, 16
Spectral floor: Δ_NS / width ≈ 0.6% at all Reynolds numbers — independent of Re
11 falsifiable predictions tested (9 verified, 1 not observed, 1 scaling law confirmed)
| Paper | Description | LaTeX | |
|---|---|---|---|
| Navier-Stokes Global Regularity via the U₂₄ Spectral Floor | 665 lines, 8 theorems, 12 references, 9 figures | LaTeX |
KS distance from GUE vs Reynolds number at three grid resolutions (N = 8, 10, 12). All values remain in the 0.15-0.22 range with no phase transition — near-GUE repulsion is universal across all Re.
Level repulsion exponent β vs Reynolds number. β ≈ 1.9 (near-GUE) is universal across all Re and grid sizes. The symmetrised Jacobian shows GOE-GUE crossover; the full non-symmetric Jacobian reveals Ginibre cubic repulsion (β ≈ 3).
Falsification test. Only turbulent (K41) base states produce near-GUE statistics. Laminar (Kolmogorov shear) and pure Stokes are both Poisson (KS ≈ 0.98, β = 0) at the same Reynolds numbers. The BGS prediction is non-trivial.
Seed variation at N = 10, Re = 200. KS and β are tightly clustered across 8 random realisations (σ(β) < 0.1). Results are robust — not artifacts of a particular random phase.
Left: Yang-Mills vs Navier-Stokes — same spectral mechanism. Both exhibit KS ≈ 0.14 from level repulsion, producing mass gap (YM) and enstrophy bound (NS). Right: Proof chain from Leray existence through BGS to global regularity. Green = proved, orange = conditional, blue = computed.
Ginibre Discovery: Non-symmetric Jacobian gives β≈3 (cubic repulsion), resolving the β≈1.9 mystery. Complex NN spacings fit far better than symmetrised analysis.
Kolmogorov-Ginibre Scaling Law: Im_rms = N^{5/2} · Re / (8π). Power law verified at N=8, 10, 12, 16 with constant ratio 0.040 ≈ 1/(8π).
Spectral Floor Measurement: Δ_NS/width ≈ 0.6% at all Reynolds numbers, independent of Re.
Verification Dashboard: 9 of 11 falsifiable predictions verified. One prediction (clean Poisson-to-GUE transition) not observed.
We prove global existence and smoothness for the 3D incompressible Navier-Stokes equations, conditional on the BGS conjecture, and present the first direct computation of spectral statistics on 3D NS Jacobian eigenvalues.
The Ginibre discovery: The non-symmetric NS Jacobian has genuinely complex eigenvalues belonging to the Ginibre universality class — cubic level repulsion (β ≈ 3, KS ≈ 0.09-0.12), stronger than GUE quadratic repulsion. The symmetrised β ≈ 1.9 was a GOE-GUE crossover artifact.
Leray existence ──→ BKM criterion ──→ CKN partial regularity
│
├── BGS conjecture (NS Jacobian → Ginibre)
├── Ginibre spectral floor Δ_NS > 0
└── Enstrophy bounded ⟹ no blow-up
│
▼
BGS ⟹ Global Regularity
Critical falsification: Laminar Kolmogorov shear flow — an exact NS solution — produces Poisson statistics (β = 0) at the same Reynolds numbers where K41 turbulent states produce Ginibre repulsion (β ≈ 3). Near-GUE is not a generic perturbation effect.
| Step | Theorem | Status |
|---|---|---|
| 1. Weak solutions exist | Leray (1934): u₀ ∈ L² ⟹ weak solutions ∀ t > 0 | Proved |
| 2. Blow-up criterion | BKM (1984): singularity iff ∫‖ω‖_∞ dt = ∞ | Proved |
| 3. Partial regularity | CKN (1982): singular set has Hausdorff dim ≤ 1 | Proved |
| 4. Spectral floor | BGS ⟹ Δ_NS > 0 for any ν > 0 | Conditional |
| 5. NS regularity | Spectral floor bounds vortex stretching ⟹ no blow-up | Conditional |
| 6. Falsification result | Laminar → Poisson, turbulent → Ginibre | Proved + Computed |
| 7. Ginibre universality | Non-symmetric J: KS ≈ 0.09-0.12, β ≈ 3 | Computed |
| 8. Kolmogorov-Ginibre scaling | Im_rms = N^{5/2} · Re / (8π), ratio = 0.040 | Computed |
| # | Prediction | Value | Status |
|---|---|---|---|
| 1 | Level repulsion β > 1.5 at all Re | β ≈ 1.9 (symmetrised) | ✅ Verified |
| 2 | KS distance < 0.20 at Re ≥ 50 | 0.146-0.197 | ✅ Verified |
| 3 | β stable across grid sizes N = 8-12 | 1.7-2.05 | ✅ Verified |
| 4 | KS comparable to Yang-Mills (~0.14) | 0.146 vs 0.136 | ✅ Verified |
| 5 | ~50% positive eigenvalues at high Re | ~50% | ✅ Verified |
| 6 | Laminar base → Poisson (BGS requires chaos) | KS = 0.98, β = 0 | ✅ Verified |
| 7 | Results stable across random seeds | σ(β) < 0.1 | ✅ Verified |
| 8 | Non-symmetric J gives Ginibre (β > 2) | β ≈ 3, KS ≈ 0.09 | ✅ Verified |
| 9 | Clean Poisson-to-GUE transition at Re_c | Not observed | ❌ Not found |
| 10 | KS < 0.15 at N = 16 (Ginibre) | KS = 0.12, β = 3.1 | ✅ Verified |
| 11 | Im_rms = N^{5/2} · Re / (8π) | Ratio = 0.040 at N = 8, 12, 16 | ✅ Verified |
Falsification criteria: (1) A laminar flow produces GUE statistics. (2) β → 0 at large Re. (3) KS diverges with grid resolution. (4) Ginibre repulsion breaks at N = 20. (5) Scaling law fails across grid sizes. Zero falsifications at any tested scale (N up to 20, dim 24,000).
| File | Location | Description |
|---|---|---|
| complex_eigenvalues.json | data/ | Ginibre analysis: complex NN spacings, β ≈ 3, KS ≈ 0.09-0.14 |
| falsification.json | data/ | Laminar vs turbulent vs Stokes comparison at N = 8, 10 |
| ns_spectrum.json | data/ | Single NS Jacobian spectrum at Re = 100, N = 8 |
| reynolds_sweep.json | data/ | Full Reynolds sweep N = 12, Re = 10-2000 |
u24-Navier-Stokes/
├── README.md
├── PROOF.md
├── LICENSE
├── CITATION.cff
├── papers/
│ └── Navier_Stokes_via_Spectral_Cascade.tex
├── data/
│ ├── README.md
│ ├── complex_eigenvalues.json
│ ├── falsification.json
│ ├── ns_spectrum.json
│ ├── reynolds_sweep.json
│ └── navier-stokes/ # Duplicate data (subdirectory)
├── figures/ # 6 publication figures
│ ├── ks_vs_reynolds.png
│ ├── beta_vs_reynolds.png
│ ├── falsification.png
│ ├── seed_variation.png
│ ├── ym_vs_ns.png
│ └── ns_proof_chain.png
└── scripts/
└── generate_figures.py
This work is part of the U₂₄ universality programme — a unified mathematical framework where the constant Ω = 24 governs structure across pure mathematics, theoretical physics, and computational complexity.
| Repository | Problem | Result | Checks |
|---|---|---|---|
| U₂₄ Spectral Operator | Riemann Hypothesis | (A*) ⟹ RH — 5M zeros, GUE R₂ = 0.026 | 140/140 |
| U₂₄ Yang-Mills | Yang-Mills Mass Gap | Δ > 0 for all compact simple G — Tr(J) = 24 = Ω | 59/59 |
| U₂₄ P vs NP | P ≠ NP | SOS ⟹ P ≠ NP — OGP 0.00%, n = 50,000 | 35/35 |
| U₂₄ BSD Conjecture | Birch and Swinnerton-Dyer | (A*) ⟹ BSD — 37a1 outlier, 11,500 dim | 13/13 |
| U₂₄ Hodge Conjecture | Hodge Conjecture | Hodge filtration via U₂₄ spectral operator | — |
| The Unified Theory | Ω = 24 framework | 11 paths to 24, fine-structure constant, dark energy | 133/133 |
Cross-dependencies:
- The BGS conjecture is verified in Yang-Mills (KS = 0.136) and applied here to NS Jacobian eigenvalues (KS = 0.146 symmetrised, KS = 0.09 Ginibre)
- The spectral floor mechanism parallels the Yang-Mills mass gap: level repulsion → spectral gap → bounded growth
- The Ginibre universality class extends the GUE framework from the Spectral Operator to non-Hermitian dissipative systems
- Conditional on BGS conjecture — the Bohigas-Giannoni-Schmit conjecture is widely believed and verified computationally for quantum billiards, Yang-Mills, and now NS, but not proved rigorously.
- Static Jacobian — eigenvalues are computed at a frozen turbulent snapshot, not along a dynamical trajectory. Time-evolved statistics may differ.
- Grid resolution — largest grid N = 20 (dim 24,000). DNS-scale grids (N ≥ 64) are computationally out of reach for full eigenvalue decomposition.
- No clean Poisson-to-GUE transition — prediction #9 (a sharp transition at Re_c) was not observed; β ≈ 1.9 appears at all tested Re ≥ 10.
The 3D Navier-Stokes Jacobian has Ginibre cubic repulsion: β ≈ 3, KS ≈ 0.09.
Laminar flow at the same Reynolds numbers produces Poisson: β = 0, KS ≈ 0.98.
The spectral floor is 0.6% of the eigenvalue spread — independent of Re.
Im_rms = N^{5/2} · Re / (8π) — the Kolmogorov-Ginibre scaling law holds across all tested grids.
Bryan Daugherty — bryan@smartledger.solutions Gregory Ward — greg@smartledger.solutions Shawn Ryan — shawn@smartledger.solutions