Finite-Size Effects in the Delaunay Heuristics for Tate-Shafarevich Groups: A Systematic Verification Across Ranks

The Cohen-Lenstra-Delaunay heuristics predict the asymptotic distribution of the Tate-Shafarevich group Sha (E/Q) for elliptic curves of rank u, ordered by conductor. Using the complete Cremona database (conductor 1|rank=0) /P (|Sha|>1|rank=2) is ~1600 in the data versus ~20 predicted, an 80-fold discrepancy from rank-dependent finite-size effects. (4) For rank 0, P (Sha2>0) proportional to N⁰. 22 implies convergence requires conductor N >> 10⁸. (5) Nontrivial Sha for rank-2 curves first appears at conductor 194, 040; all 21 such cases have |Sha|=4. These results provide the first empirical constraints on the distance to asymptotics for the Delaunay heuristics. Data: Cremona database via SageMath 10. 8. Computations in Python 3. 11/SQLite. The 21 rank-2 curves with nontrivial Sha are listed in full in Table 9. This investigation originated as an auxiliary study during AI-assisted scientific discovery research (companion paper tao2026c).