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 χR with |∇χR| ≲ R⁻¹, |ΔχR| ≲ 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 t₂0, 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