Abstract: We introduce a “period removal” and “mirror selection” framework for elliptic curves over ℚ. The method separates global structure from local algebra by (i) canonically normalizing local data in a compact, period-removed space and (ii) selecting representatives via a deterministic mirror principle (maximization on compact sets with tie-breaking). This yields a universal residual compactum and a uniform depth bound. From this bound we derive a prime-cut mechanism and an exponent bound for Sha(E/ℚ), implying finiteness of Sha. Combined with a standard analytic bridge Z(s)=u(s)Λ(E,s) with u(1)≠0, a no-ghost decomposition shows that the Mordell–Weil continuous factor is the only non-compact direction. Poisson/Mellin analysis on the Mordell–Weil lattice then gives ordₛ₌₁ L(E,s)=rank E(ℚ), and a canonical local-height normalization of the kernel yields the full leading-term formula in the strong Birch–Swinnerton-Dyer conjecture. An explicit dependency map and referee appendix are included to support line-by-line auditability.
Carlos Alberto Terencio de Bastos (Sun,) studied this question.