Theorem 2.18 Condition Map

Fully unfolds the eligibility conditions of Banwait–Huang Theorem 2.18 into an item-by-item auditable checklist. The base curve $E$ must simultaneously satisfy seven conditions (E1 semistable, E2 small prime trace $a_3(E)\in\{-2,\ldots,2\}$, E3 excludes rational isogenies for $p\in\{3,5,7\}$, E4 ramification conditions at multiplicative primes, E5 $\Gamma_0(N)$-optimal representative, E6 analytic rank is zero, E7 $\operatorname{BSD}(E,2)$ unconditionally verified). The curves are then split into two branches based on 2-torsion structure: branch 8a (no rational 2-torsion, handled by Zhai-type results) and branch 8b (exactly one rational 2-torsion point, requiring $E'=E/E(\mathbb Q)[2]$ to satisfy $\Sha(E')[2]=0$ and $E'(\mathbb Q)[2]\cong\mathbb Z/2\mathbb Z$). For the twist $d$, besides common conditions like being square-free, coprime to $3N$, and $d\equiv1\pmod4$, the Zha16 branch and CLZ20 branch each have their own conditions for primes being inert/split in specific fields. Finally, the document explicitly pins down the directionality of the output semantics: Algorithm 2 does not return "all twists satisfying BSD", but rather a theorem-guaranteed subfamily—admissible implies BSD follows from the cited theorems, but not admissible does not imply BSD is false. This unidirectionality is the core discipline of the entire Phase 1 pipeline.