Isogeny-Dependence of Non-Trivial Sha: A Rank-Graded Concentration Phenomenon for Elliptic Curves over Q

We report a systematic computational study of the relationship between rational isogenies and the Tate-Shafarevich group III(E/Q) for elliptic curves over Q, using the Cremona database (2,483,649 curves with conductor <= 400,000). We discover a Sha-isogeny concentration gradient: the conditional probability that a curve with non-trivial III[p] admits a rational p-isogeny increases monotonically with the Mordell-Weil rank, reaching 100% for p=2 at rank 2 (enrichment 5.9x over baseline, p < 10⁻¹⁶). For p=3, the enrichment reaches 7.9x at rank 1, and at rank 2 none of the 6,694 curves with 3-isogeny has non-trivial III[3]. We further show that all 21 rank-2 curves with |III|>1 possess a partial-split 2-isogeny (exactly one rational 2-torsion point, p < 10⁻¹¹), and that their anomalously small Tamagawa products arise from a preponderance of I_1 Kodaira types (60.3% vs 25.0% in controls). Within each isogeny class, the BSD quotient is constant, making |III|=4 an algebraic consequence of Tamagawa redistribution. A BSD budget analysis shows that rank-2 regulators inflate the BSD denominator 33x relative to rank-0, while L-values grow only 3x, leaving zero budget for |III|=9 among all 6,694 rank-2 curves with 3-isogeny. We propose a Sha-Isogeny Concentration Conjecture: P(p-isogeny | III[p] != 0) -> 1 as rank -> infinity, and provide an isogeny-corrected Delaunay heuristic with empirical parameters. Data: Cremona elliptic curve database (conductor <= 400,000) via SageMath 10.8 / SQLite. All verification scripts are included in the supplementary materials. This paper is a companion to arXiv:2604.xxxxx (Finite-Size Effects in the Delaunay Heuristics, Tao 2026). Key findings: (1) Sha-isogeny concentration gradient with monotonic rank dependence; (2) Partial-split 2-isogeny as necessary condition for non-trivial Sha at rank 2; (3) Four-layer filtration mechanism (isogeny -> split type -> Kodaira types -> period); (4) BSD redistribution within isogeny classes; (5) BSD budget constraint explaining absence of Sha[3] at rank 2.