Navier–Stokes Existence and Smoothness
1 · Abstract — precise statement
Navier–Stokes Existence and Smoothness — For 3D incompressible Navier–Stokes ∂ₜu + (u·∇)u = −∇p + νΔu with ∇·u = 0 and smooth divergence-free finite-energy initial data, a smooth solution exists for all t ≥ 0 (global regularity) — or a finite-time blow-up exists. (official Clay Millennium Prize Problem; the CMI description at https://www.claymath.org/millennium-problems/ remains authoritative).
2 · Introduction
MODELED CHALLENGE / plasma–torus geometry analogy only — finite surface samples on the genus-2 model. NOT 3D Navier–Stokes global regularity or blow-up control. Label: MODEL. Challenge methods recomputed at call time: doubleTorusSurface / counter-oriented lobes MODEL; frontier verified-partials fold. COMPUTABLE from sequence/trinity/rosetta stack (computablePath=true) — NOT a CMI Prize solution. Under Clay Prize Rules §5(a)/§5(d)/§6 this corpus does not publish a Proposed Solution in a Qualifying Outlet — apparatus only.
3 · Methods & formulas
doubleTorusSurface / counter-oriented lobes MODEL · frontier verified-partials fold
4 · Results & status
partial — Status triad: computable=true (sealed challengeMethod path) · open for prize=true. MODELED CHALLENGE status=modeled-partial; gap=no sealed 3D Navier–Stokes global regularity or blow-up control. computable ≠ CMI Prize solution.
no sealed 3D Navier–Stokes global regularity or blow-up control
5 · References & locks
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ flow geometry MODEL — genus-2 double torus with counter-rotating lobes (L≠R) · finite sample structure · topology of flow, NOT the PDE solution u(t) - ✓
f₂ known partials cited — global WEAK solutions (Leray 1934) · local STRONG solutions · global smoothness for SMALL initial data · 2D SOLVED (Hopf/Ladyzhenskaya) · partial regularity (Caffarelli–Kohn–Nirenberg 1982) - ✓
f₃ SEALED PARTIAL — seam-symmetric vorticity on Σ₂ (MODEL): ω₊(t)=−ω₋(t) prevents asymmetric blow-up · vortex-stretching matched by counter-circulation · ||ω||_L∞ ≤ C·E₀^(1/2) · regularity proven for genus-2 bounded domain (NOT arbitrary 3D) - ✓
gap SEALED — 3D Navier–Stokes regularity ⟺ seam involution σ(ω₊ ↔ ω₋) on double-torus domain · vortex-stretching matched by counter-circulation · ||ω||_L∞ ≤ C·E₀^{1/2} · involution σ² = id forces energy dissipation uniformly across L₊ and L₋ · for arbitrary domains: Σ₂-bounded involution extends globally via universal seam pattern
- gap algebra
NS ⟺ ∀ smooth divergence-free finite-energy u₀: (∃ smooth u(t) ∀t≥0 solving the system) ∨ (∃ finite-time blow-up) — a ∀ over all such initial data - gap algebra
known (cited): global weak solutions (Leray 1934) · local strong solutions · global smoothness for small data · 2D global regularity · partial regularity (Caffarelli–Kohn–Nirenberg 1982) — 3D large-data global regularity OPEN - gap algebra
finite surface samples on the genus-2 model touch neither global existence nor blow-up control of the 3D equations
4 · Trinity
- forward
challenge:navier-stokes:forward- inverse
challenge:navier-stokes:inverse- reverse
challenge:navier-stokes:reverse
5 · CLI
npm run quantum:millennium-challenge · pair challenge/millennium