An Equivalent Toroidal-Helix Construction of an Elliptic Curve: From the Weierstrass Equation to Toroidal-Helix Dynamics — The Equivalence Theorem and the Geometric Form of BSD

An equivalent toroidal-helix construction of an elliptic curve, in strict duality with the spherical-helix equivalence of the ζ function (V14). The core is the equivalence theorem: an elliptic curve is isometrically embedded as a flat torus (Clifford torus ⊂ S³ ⊂ R⁴), where pO ⊥ tangent plane and the group law P+Q+R=O manifests as central symmetry of the tangent-plane section. Torsion points correspond to rational-angle helices (closing), infinite-order points to irrational-angle helices (never closing, densely filling by Kronecker–Weyl); the Mordell–Weil rank equals the number of independent irrational-angle helix directions. The Hasse–Weil local factor 1−a_p p^{-s}+p^{1-2s} is the closure factor of the φ-helix on the discrete torus mod p, with a_p the winding deficit and the Hasse bound |a_p|≤2√p its boundedness on S³. The equivalence theorem (four propositions) is rigorously proved; the BSD substance (geometric rank = analytic order of vanishing at s=1) is explicitly marked as a target statement built upon the equivalence theorem and not proved here, with the gap located at the integer quantization of the continuous projection angle and the weaving of local windings into the global order of vanishing. This is the geometric foundation on which the companion paper (BSD V1, DOI 10.5281/zenodo.20647368) proves the rank identity via the isosceles-triangle construction. Bilingual (Chinese and English PDFs).