Global Enclosure and Stop Rules: Even if this route is completely successful, what can it prove?

The document first firmly establishes the ceiling of the Phase 1 route: even if Banwait–Huang Algorithm 1 is completely successful, it only proves that "the base curve has an explicit, effectively enumerable infinite quadratic-twist subfamily whose members are guaranteed strong BSD by existing theorems"—it explicitly lists four things it does not prove: it is not that all twists of this base curve satisfy BSD, a base curve failing Algorithm 1 does not mean it has no strong-BSD twists, not all elliptic curves have such a family, and certainly not that full BSD holds for all $E/\mathbb Q$. Therefore, the correct positioning of this route is a Uniform infinite-family theorem, not $\forall E/\mathbb Q$. The document also explains that even without solving full BSD, Phase 1 can still generate cumulative value: theorem applicability atlas, curve family certificates, descent soundness audit, data/theorem separation, twist generator, and independent reproduction of external results. Finally, it defines a highly disciplined stopping rule: if for three consecutive rounds the work only amounts to "increasing the twist bound, listing more d's, recomputing the same batch of curves, or just adjusting runtime"—tasks with no new theorem predicate or certificate—the mainline must be frozen. Only a new theorem family, a new descent certificate, a new eligibility criterion, a discrepancy/bug with mathematical consequences, an independent global reproduction of the authors' results, or truly expanding family coverage to new types of curves can extend the mainline.