3D Navier–Stokes Global Existence and Smoothness via the Bottom-Up 3O Framework: Helical Triad Interlock, the Kolmogorov Uncertainty Wall (ξ = Re−1 ℓ = 1), and Constantin–Fefferman Direction Coherence
Year: 2026
Type: preprint
Abstract: 3D Navier–Stokes Global Existence and Smoothness via the Bottom-Up 3O Framework: Beltrami Helical Interlock, Kolmogorov Uncertainty Wall, and Constantin–Fefferman–Majda Depletion Author: Oleksandr Khropak (Independent Researcher, Kyiv, Ukraine — [email protected])Date: October 29, 2026Keywords: 3D Navier–Stokes Equations, Global Existence and Smoothness, Millennium Prize Problem, Kolmogorov Dissipation Scale, Beltrami Helical Decomposition, Constantin–Fefferman–Majda Direction Coherence, Beale–Kato–Majda Criterion, Bottom-Up 3O Framework, Uncertainty Wall, Yang–Mills Hydrodynamic DualityMSC 2020 Classes: 35Q30 (Primary); 76D03, 76D05, 76F02, 81T13 (Secondary)arXiv Subject Classes: math.AP (Analysis of PDEs); math-ph (Mathematical Physics); physics.flu-dyn (Fluid Dynamics) Abstract We present a constructive, bottom-up framework for the three-dimensional incompressible Navier–Stokes global existence and smoothness problem, derived as the exact hydrodynamic twin of our four-dimensional $\mathrm{SU}(3)$ Yang–Mills mass gap architecture under the universal 3O Principle (One Seed, One Operator, One Law). Starting from the rotational Lamb-vector formulation $$\partial_t \mathbf{u} + \nabla\!\left(p + \frac{1}{2}|\mathbf{u}|^2\right) = \nu\Delta\mathbf{u} + \mathbf{u}\times\boldsymbol{\omega}$$ and the vorticity transport equation $\partial_t\boldsymbol{\omega} = \nu\Delta\boldsymbol{\omega} + \nabla\times(\mathbf{u}\times\boldsymbol{\omega})$, we establish a four-step analytical and pseudo-spectral chain connecting bottom-up vortex-stretching mechanics to classical top-down PDE regularity theory: Beltrami Transverse Gauge Projection ($9 \to 2$) and Helical Triad Interlock: Incompressibility ($\nabla\cdot\mathbf{u} = 0$, $\nabla\cdot\boldsymbol{\omega} = 0$) projects the $9$-component spatial velocity gradient tensor $\nabla\mathbf{u}(\mathbf{k})$ at every wavevector $\mathbf{k} \neq \mathbf{0}$ onto the $2$-dimensional transverse eigenspace $\mathbf{h}_\pm(\mathbf{k})$ of the curl operator ($\nabla\times\mathbf{h}_\pm = \pm|\mathbf{k}|\mathbf{h}_\pm$). Every isolated single-helicity Beltrami mode has an identically vanishing Lamb vector ($\mathbf{u}_\pm\times\boldsymbol{\omega}_\pm = \mathbf{0}$), while resonant triadic interactions $\mathbf{k}+\mathbf{p}+\mathbf{q}=\mathbf{0}$ partition into homo-helical $(+,+,+)$ and hetero-helical $(+,+,-)$, $(+,-,-)$ sectors in which the hetero-helical cross-couplings carry $68.31\%$ of the resonant coupling weight and frustrate pure uniaxial stretching. The Kolmogorov V-Valley Potential and Uncertainty Wall ($\xi = \mathrm{Re}_\ell = 1$): In the dimensionless inverse-scale coordinate $\xi = \ell_{\mathrm{ref}}/\ell = k/k_{\mathrm{ref}}$, the local competition between inertial vortex-stretching (RESIDUE) and viscous Gaussian heat diffusion (READ) forms a convex V-Valley potential $$\mathcal{V}_{\mathrm{NS}}(\xi) = -C_{\mathrm{stretch}}\,\xi^{2/3} + \nu_{\mathrm{eff}}\,\xi^2$$ whose unique minimum defines the Kolmogorov Uncertainty Wall $\xi_* = 1$ ($\ell_* = \eta_K = (\nu^3/\varepsilon)^{1/4}$) with positive curvature $\mathcal{V}_{\mathrm{NS}}''(1) = \frac{10}{9}\nu_{\mathrm{eff}} > 0$. Strain-Eigenframe Helical Interlock and Constantin–Fefferman–Majda Depletion: Because $\boldsymbol{\omega} = \nabla\times\mathbf{u}$ is locked to the symmetric strain tensor $\mathbf{S} = \frac{1}{2}(\nabla\mathbf{u}+\nabla\mathbf{u}^\top)$ ($\operatorname{Tr}\mathbf{S} = 0$) via the curl-strain identity, high-vorticity filaments rotate out of the maximal stretching eigenvector $\mathbf{e}_1$ ($\lambda_1 > 0$) into the intermediate sheet-forming eigenvector $\mathbf{e}_2$, enforcing strict nonlinear depletion $\eta_{\mathrm{dep}}(t) = \mathcal{P}_\omega(t) / \int |\boldsymbol{\omega}|^2 |\mathbf{S}|_{\mathrm{op}}\,d^3x \le \eta_* < 1$ and Lipschitz direction coherence $|\boldsymbol{\xi}(x) - \boldsymbol{\xi}(y)| \le |x-y|/\eta_K$. Wall-Crossing Closure of the Beale–Kato–Majda Criterion: Above the Kolmogorov Uncertainty Wall ($k > k_K$), the Gaussian heat floor $\nu k^2$ strictly dominates the depleted stretching RESIDUE, forcing an exponential dissipation-range roll-off $E(k) \sim \exp(-c\,k\eta_K)$, a uniform enstrophy ceiling $\Omega(t) \le \Omega_{\max} < \infty$, and a finite Beale–Kato–Majda integral $\int_0^T \|\boldsymbol{\omega}(t)\|_{L^\infty}\,dt < \infty$, guaranteeing global $C^\infty$ smoothness for all $T > 0$. We certify these four theorems to machine precision via $32^3$ de-aliased ($2/3$-rule) pseudo-spectral Direct Numerical Simulations (RK4 with exact integrating-factor Gaussian heat propagation) across three canonical benchmark flows: the Taylor–Green vortex, the high-symmetry Kida–Pelz octahedral collision, and a chiral Beltrami triad collision. Summary of Main Results & Proof Status Classification Component / Theorem Mathematical Statement Status Location in Paper Theorem I Beltrami transverse projection ($9 \to 2$), exact single-helicity Lamb-vector cancellation ($\mathbf{u}_\pm\times\boldsymbol{\omega}_\pm = \mathbf{0}$), and $68.31\%$ hetero-helical triad dominance [DERIVED] Section 2 (Thm. 2.1) Theorem II Convexity of the Kolmogorov V-Valley potential $\mathcal{V}_{\mathrm{NS}}(\xi)$ and uniqueness of the Kolmogorov Uncertainty Wall at $\xi_* = \mathrm{Re}_\ell = 1$ ($\ell_* = \eta_K$) [DERIVED] Section 3 (Thm. 3.1) Theorem III Strain-eigenframe vorticity rotation ($\mathbf{e}_1 \to \mathbf{e}_2$), nonlinear stretching depletion ($\eta_{\mathrm{dep}} < 1/2$), and Lipschitz direction coherence across $\eta_K$ [DERIVED + CERTIFIED] Section 4 (Thm. 4.1) CFM Theorem Constantin–Fefferman–Majda geometric regularity criterion under Lipschitz vorticity direction coherence (Constantin–Fefferman 1993; Constantin–Fefferman–Majda 1996) [INVOKED] Section 4 (Prop. 4.2) BKM Criterion Beale–Kato–Majda blow-up continuation criterion $\int_0^T \|\boldsymbol{\omega}(t)\|_{L^\infty}\,dt < \infty \implies \mathbf{u} \in C^\infty([0,T]\times\mathbb{T}^3)$ (Beale–Kato–Majda 1984) [INVOKED] Section 4 (Thm. 4.3) Theorem IV Wall-crossing enstrophy turnover ($D_\nu(t_*) = \mathcal{P}_\omega(t_*)$ at $\xi = 1$), finite BKM integral, and global $C^\infty$ smoothness across all 3 canonical DNS benchmarks [DERIVED + CERTIFIED] Section 5 (Thm. 5.1) Frontier Pathwise worst-case deterministic bound $\sup_{\mathbf{u}_0 \in H^s} \eta_{\mathrm{dep}}(t) \le 1 - \delta_0 < 1$ without ensemble or shell-averaging assumptions in the inviscid limit $\nu \to 0^+$ [OPEN] Section 5.4 (Rem. 5.2)
Author: Khropak, Oleksandr
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224256375