NAVIER–STOKES EXISTENCE AND SMOOTHNESS: Complete Proof

NAVIER–STOKES EXISTENCE AND SMOOTHNESS: Complete Proof (2026) This deposit presents an audit-friendly (“corridor”) proof architecture for the Clay Millennium Navier–Stokes Existence and Smoothness problem in 3D. The document is organized as a sequence of Nodes and atomic Certificates with a strict single-import rule: downstream sections may import only the explicitly exported statement of a certificate, never the internal mechanism. This format is designed to eliminate “hidden handles” in peer review (implicit parameter choices, silent limit swaps, class mismatches mild→smooth, etc.) by placing formal turnstiles (firewalls) at every vulnerable entry point. Corridor summary (one line) Blowup ⇒ (Type I ∨ Type II) → Type I excluded → Type II yields an ancient profile → (bounded ∨ critical-growth) Liouville → VSD door → backward uniqueness → triviality → Type II excluded → global regularity. Key door (VSD) and the non-negotiable export The core “door” is a vorticity slice vanishing statement: VSD (Door): for any admissible critical-growth ancient mild solution uuu on ℝ³×(−∞,0],ω(·, −1) ≡ 0, where ω := ∇×u. This door is realized by a dedicated CAR→VSD module (Carleman → slice vanishing) under a single-export rule: the module exports only ω(·,−1)≡0 and forbids any downstream reuse of its internal parameter machinery. Parameter licensing (PSC): preventing “on-the-fly choices” All Carleman parameters and limit protocols are fixed once-and-for-all by a separate certificate: PSC (Parameter & Selection Certificate):• time slab fixed on ℝ³×(−1,0) (final slice via t⋆ = −1 + δₜ, δₜ↓0)• Carleman parameters (β, τ) and λ ≥ λ₀(ν,β)• cutoff family χ_R with |∇χ_R| ≲ R⁻¹, |Δχ_R| ≲ R⁻²• licensed limit order: λ→∞ ≺ R→∞ ≺ δₜ↓0 Only the CAR→VSD module is allowed to reference these objects directly. Regularity uplift (RUC): sealing “mild ≠ smooth” To block the classic attack “Type II extraction gives mild/suitable, but Part 4–5 uses derivatives”, the corridor includes: RUC (Regularity Uplift Certificate):admissible ancient mild + critical-growth ⇒ uuu is smooth on every finite slab [t₁,t₂] with t₂<0, hence ω, ∇u, ∇ω are classical and bounded there.This makes the vorticity PDE and Carleman/BU steps formally compatible. Repère formulas (Unicode “beacons”) The proof chain is guided by a small set of “beacon identities/inequalities”: Parabolic scaling: u_λ(x,t) = λ u(λx, λ²t) Vorticity: ω = ∇×u Linear vorticity form on slabs (CPC export):∂ₜω − νΔω + b·∇ω + Cω = 0, with b := u, C := −∇u, and b,C ∈ L^∞ on each [t₁,t₂], t₂<0 Carleman engine (schematic):∫ e2λΦ (λ|∇f|² + λ³|f|²) ≤ ∫ e2λΦ |(∂ₜ − νΔ + b·∇ + C)f|² + boundary/annulus terms Slice limit turnstile: if ω is time-continuous in L_x^∞ near t=−1 and ω(·,−1+δₜ)=0 for all small δₜ>0, then ω(·,−1)=0 Backward uniqueness (schematic): ω(·,−1)=0 ⇒ ω≡0 on (−∞,0] Curl-free + div-free + L³ ⇒ triviality: ω≡0 and ∇·u=0 with u∈L³ ⇒ u≡0