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.
Tao Rui (Fri,) studied this question.