An Explicit and General Disproof of the Birch and Swinnerton-Dyer and Tate Conjecture via Plastic Number Descent and Transcendental Curvature Operators

We construct the first explicit rational counterexample to the Birch and Swinnerton-Dyer (BSD) Conjecture using a recursive curvature descent operator seeded by the plastic number . The resulting elliptic curve exhibits analytic rank one and arithmetic rank zero, thereby violating BSD. We generalize this to an infinite class of rational elliptic curves indexed by transcendental phase constants and rational phase denominators , each exhibiting the same rank divergence. The paper introduces a new class of anti-Diophantine numbers and defines a transcendental curvature operator whose recursive fixed points collapse cohomological, geometric, and arithmetic correspondences—yielding a falsification of the Tate Conjecture as well. This work establishes the field of transcendental arithmetic geometry and constructs an infinite family of Schrödinger curves that exhibit curvature-locked rational traces with no algebraic origin. All results are constructive, verifiable, and algorithmically reproducible.