Navier-Stokes Problem Remains Unsolved: No Verified Breakthrough Found — E8 Intelligence Research

Abstract

FINDING: The Navier-Stokes existence and smoothness problem remains open; the search results are educational expositions and one preprint claiming a new method, but no verified breakthrough or novel mathematical constant emerges. | MATH: The core system is the incompressible Navier-Stokes equation: ∂u/∂t + (u·∇)u = −∇p + ν∇²u, ∇·u = 0, with u(x,t) ∈ ℝ³, ν > 0 (kinematic viscosity). The Clay problem asks: for smooth, divergence-free initial data u₀ with finite energy, do smooth global solutions exist for all t > 0, with bounded energy? The critical scaling is u_λ(x,t) = λ u(λx, λ²t), which leaves the equation invariant — this yields the scaling dimension of the solution, and the critical Sobolev space is H^{1/2}(ℝ³). No new constants, ratios, or exact solutions are presented in the search results. | CONNECTION: The scaling symmetry λ → λ² in time vs. λ in space is a parabolic anisotropy — not a golden-ratio or base-60 harmonic. However, the criticality at H^{1/2} relates to the Fourier Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com