Non-Semistable Family Theorem Schema

The document sets a self-limitation right at the beginning: "This document is not a theorem claim, but a list of complete proof obligations." The shape of the candidate theorem is: let $E/\mathbb Q$ be an optimal, analytic-rank-0 elliptic curve, not required to be semistable; let $\mathcal D(E)$ be a squarefree twist parameter family that satisfies a branch of Banwait–Huang Theorem 2.14; for $d\in\mathcal D(E)$, the candidate outputs $L(E_d,1)\ne0$ and $\operatorname{BSD}(E_d,2)$ are known; if we can further prove $\forall p>2,\ \operatorname{BSD}(E_d,p)$, then $\operatorname{BSD}(E_d)$ holds. The document lists five bridge hypotheses (FW-H1/H2/H3 hold for the base $E$, H1/H2 are preserved under twist, the splitting conditions of $d$ locally preserve the H3 witness, the period/Manin normalization of the FW Corollary is compatible with the Banwait BSD convention, and $L(E_d,1)\ne0$ can directly feed into the FW rank-zero corollary), all of which must hold to obtain $\forall d\in\mathcal D(E),\ \mathrm{BSD}(E_d)$. The document honestly calls out the two most dangerous current gaps: Gap A is the $\forall p>2$ repeatedly emphasized in documents 03 and 04, which has not yet been finite-ized; Gap B is new — the transition from the modular-form period to the Néron period in the Fouquet–Wan Corollary requires clean splicing for the Manin constant treatment at small primes $p$, and the two BSD conventions cannot be taken for granted as directly compatible. Finally, the document provides a pragmatic first-version strategy: use FW to handle "large/generic odd primes," retain Banwait's existing small-prime theorems to handle $3,5,7$, and retain the original Theorem 2.14 for $p=2$ — this kind of hybrid theorem might be easier to publish and verify than attempting to handle everything with a single framework.