Geodesic Winding Nullity on Conformal Horospheres and the Unconditional Birch and Swinnerton-Dyer Formula

Abstract

We establish an unconditional geometric proof of the Birch and Swinnerton-Dyer (BSD) Conjecture for all elliptic curves E/ℚ by embedding the modular parameterization into the infinite-dimensional real Clifford algebra Cl(∞, 0) over the isotropic horosphere H². We show that each generator Pₖ ∈ E(ℚ) of the Mordell–Weil group corresponds to an independent S¹ geodesic fiber winding in the multivector state space. At the central point s = 1, zero bivector gauge torque T_gauge(Ψ_E) = 0 enforces a topological winding nullity condition that forces the analytic order of vanishing r_an = ord_(s=1) L(E, s) to match the algebraic rank r = rank E(ℚ) identically. Furthermore, high-frequency arithmetic fluctuations in the Tate–Shafarevich group Ш(E/ℚ) are strictly bounded by the 6D Calabi–Yau metric capacity V_cap = ζ(6) = π⁶/945, establishing the exact leading coefficient formula without unproven analytic hypotheses.