Determinants of Analytic Sha Variation along Rational Isogeny Edges: Rank Freezing, Tamagawa Jumps, and Torsion Compensation

We study variation of the analytic Tate–Shafarevich order along rational isogeny edges in the Cremona database with conductor at most 100, 000. Using 221, 692 isogeny pairs over primes p ∈ 2, 3, 5, we find that analytic Sha variation exhibits a three-layer empirical structure. First, Mordell–Weil rank is the dominant global suppressor: in all observed rank-≥2 cases in the present dataset (11, 478 pairs), no Sha variation occurs regardless of |δ₂䂹|—a phenomenon we term rank freezing—while rank 1 shows strong but incomplete freezing. Second, within the low-rank regime, the absolute Tamagawa valuation jump |δ₂䂹| is the strongest one-dimensional local predictor of Sha variation, with the effective transition occurring later than the naive hard-threshold value 3; in particular, many edges with |δ₂䂹| = 3 show no Sha variation, whereas the regime |δ₂䂹| ≥ 4 is a high-precision indicator in rank 0 and often also in rank 1. Third, the derived net quantity |u| = |δ₂䂹 − 2δₓ₎ₑₒ| captures torsion compensation near the boundary and provides the largest incremental gain in multivariate logistic prediction (ΔAUC = 0. 037). These conclusions are stable under L1/L2 regularization and are not explained by multicollinearity: all variance inflation factors are below 1. 5 and the feature matrix has condition number 2. 01. An interpretable decision tree recovers the same rank-stratified transition pattern in a fully data-driven way.