Navier–Stokes Millennium Problem Still Open: No Verified Solution — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: No verified solution exists; the Navier–Stokes Millennium Problem remains open, with only unconfirmed AI claims and non-peer-reviewed preprints in the search results. | MATH: 3D incompressible N–S: ∂ₜu + (u·∇)u = −∇p/ρ + ν∇²u, ∇·u = 0. Millennium statement: for smooth divergence-free initial data u₀ ∈ C^∞ with finite energy, prove either (A) global smooth solution exists for all t>0, or (B) finite-time blow-up occurs (‖∇u‖_L∞ → ∞ as t→T*). Critical scaling: u_λ(x,t)=λu(λx,λ²t) leaves equations invariant; ‖u₀‖_L³ and ‖u₀‖_Ḣ^{1/2} are scale-invariant (critical) norms. Energy: d/dt ‖u‖²_L² = −2ν‖∇u‖²_L². Prodi–Serrin: regularity if u ∈ L^p_t L^q_x with 2/p+3/q ≤ 1, q>3. | CONNECTION: No geometric-harmony constants (0.382, 0.618, 0.786, 1.618, 2.618) appear in the N–S literature or in these sources. The only structural ratios are the scaling exponents (2,1) in the symmetry group and the Prodi–Serrin exponent relation 2/p+3/q=1 — a dimensional-balance condition, not a golden-ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224994230