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 fold4 · 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
- claySolvedByThisFold
- 0
- physicalFtlClaim
- 0
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ flow MODEL — doubleTorusSurface counter-oriented lobes finite and L≠R distinct: a finite genus-2 sample of a flow geometry, NOT the PDE itself - ✓
f₂ frontier honesty — the verified-partials fold computes, holding this row at modeled-partial (no regularity or blow-up claim)
- 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