Phase 1 Agent Experiment Specifications: Certificate Atlas + Twist-Family Reproduction

The goal is not to "calculate a BSD ratio for millions of curves," but to generate theorem-applicability, certificate levels, and unclosed items for every isogeny class in a complete finite domain (N_E<500,000, full LMFDB coverage). The workflow is divided into five steps: Ingest (fetching LMFDB/Cremona label, a-invariants, rank, regulator, etc.), Evidence typing (tagging each field as exact/rigorous_computation/numerical/BSD_inferred/unknown), Theorem router (checking GZK, BSTW zeta-element, Banwait–Huang, main conjecture, p-converse, etc. item by item), Certificate level (tagging according to C0–C10), and Wall classification (restricting unclosed reasons to a controlled vocabulary). Testing is divided into three groups: R0 (rank 0, testing weak BSD known theorems and p-parts), R1 (rank 1, testing Heegner point/Kolyvagin applicability), and R2+ (rank≥2, not seeking comprehensive closure, only finding where theorem coverage suddenly drops). The first rank-2 sample is 389.a1 (LMFDB gives r_alg=r_an=2, numerical leading term matches), but the Agent's task is not to recalculate it, but to answer item by item the source of the certificates, whether generators are saturated, whether Sha_an=1 is an actual proof, and which p-parts are known. The first family reproduction is reproducing the Banwait–Huang 2026 algorithm, first running the authors' samples and then all conductor≤500,000, performing an adversarial audit on discrepancies. The success conditions explicitly do not require a new BSD theorem, only a complete schema, certificates for at least three classes of curves, algorithm reproducibility, the ability to distinguish evidence/theorem for each result, and finding the top three common bottlenecks for high rank.