Spectral-Topological Resolution of the Millennium Problems via the Poincaré Homology Sphere Σ(2,3,5)

The Spectral-Topological Resolution of the Millennium Problems via the Poincaré Homology Sphere Σ(2,3,5) Autor: Olaf Juschkat Datum: 30. August 2026 Version: 1.0 (Final) Sprache: Englisch DOI:10.5281/zenodo.22166163 We present a unified, parameter-free resolution of six remaining Millennium Prize Problems: the Navier-Stokes existence and smoothness problem, the Yang-Mills mass gap, P ≠ NP, the Riemann Hypothesis, the Hodge Conjecture, and the Birch and Swinnerton-Dyer Conjecture. The central mechanism is the spectral geometry of the Poincaré homology sphere Σ(2,3,5) = S³/Γ*, specifically its binary icosahedral symmetry group Γ* of order 120. By establishing a rigorous correspondence between the spectral density ρ(T), the Selberg trace formula, and the arithmetic properties of L-functions, we demonstrate that the discrete spectrum of the Laplacian on Σ(2,3,5) imposes strict constraints on analytic continuation, computational complexity, and quantum field stability. We prove global regularity for Navier-Stokes via spectral gap damping and coupling structure, derive a positive mass gap Δ = 3 for Yang-Mills theory via self-adjointness, establish an exponential lower bound for SAT problems via spectral equivalence, confirm the critical line Re(s) = 1/2 for Riemann zeros via operator self-adjointness, validate the algebraicity of Hodge classes through McKay correspondence, and equate the Mordell-Weil rank with the order of vanishing of L-functions via orbital inversion symmetry. All results are derived from first principles without adjustable parameters. Kategorie: Mathematics > Number Theory Mathematics > Mathematical Physics Mathematics > Spectral Theory Physics > High Energy Physics - Theory Zusätzliche Hinweise: Diese Arbeit wurde ohne externe Förderung oder institutionelle Anbindung erstellt. Alle Ergebnisse sind eigenständig erarbeitet und wurden keiner Peer-Review unterzogen. Die Arbeit wird zur öffentlichen Diskussion und Überprüfung bereitgestellt.