Millennium Problems Remain Unsolved; No New Breakthroughs in 2025–2026 — E8 Intelligence Research

FINDING: The search results are dominated by popular expositions of the seven Clay Millennium Problems, with one arXiv preprint claiming solutions to P vs NP (3rd) and Navier–Stokes (4th). No new 2025–2026 breakthroughs are evident; the core unsolved status remains. MATH: - Millennium list: P vs NP, Hodge conjecture, Poincaré conjecture (solved 2003, Perelman), Riemann hypothesis, Yang–Mills mass gap, Navier–Stokes existence/smoothness, Birch–Swinnerton-Dyer. - The arXiv preprint (1902.07265v23) asserts solutions but has not been peer-accepted; no equations extracted from the snippet. - No new constants, ratios, or symmetry groups appear in the provided text. CONNECTION: - Poincaré conjecture (solved) relates to the 3-sphere \( S^3 \) — a compact, simply-connected 3-manifold — which is the boundary of the 4-ball, and its geometry is tied to the Hopf fibration \( S^1 \to S^3 \to S^2 \), a root of quaternionic symmetry (SU(2) double cover of SO(3)). - Yang–Mills mass gap Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com