Known Theorem Closure Map

Not a literature review, but a layer-by-layer answer to "which component of BSD does each existing theory actually close." The Modularity Theorem has already provided the analytic continuation and functional equation of L(E,s), so "whether the L-function exists" is not a frontline problem. Gross–Zagier and Kolyvagin, combined with the Modularity Theorem, provide the core closure for weak BSD when the analytic rank is 0 or 1; this is the most important low-rank result for BSD, but it cannot be extrapolated to analytic rank ≥2. In recent years, Iwasawa theory, Euler systems, Kato classes, Heegner points, and p-adic L-functions have proven p-part BSD, the main conjecture, p-converse, and strong BSD for infinite twist families of non-CM curves under a large number of conditions for rank 0/1, but each result usually has explicit conditions (semistable?, does p divide the conductor?, ordinary/supersingular?, residual representation?). For high rank (rank≥2), there have been structural advances such as generalized Kato classes and higher Gross-Zagier formulas, but they cannot currently be organized into a general closure and remain the main wall for BSD-W. Keller–Stoll type work shows that full strong BSD can be unconditionally and precisely verified on explicit finite sets; the lesson is that "a complete BSD certificate is engineerable, but every component must be independently closed." Arithmetic statistics (Selmer group average, rank distribution) can predict, find families, and determine Agent budgets, but positive density/average results do not equal ∀E; they are research routing, not global closure.