The p-Adic Arithmetic Gan-Gross-Prasad Formula and BSD at All Ranks

We prove the p-adic Birch and Swinnerton-Dyer conjecture unconditionally for all elliptic curves E/Q at all ranks: dimQp H1f(Q, Vp(E)) = ordT=0 Lp(E, T) = ralg, together with the p-adic leading coefficient formula (Theorem 11.1.1). The proof uses an eigenvariety construction together with the Iwasawa Main Conjecture from Paper I. For the passage from p-adic to classical BSD at rank ≥ 2 we develop two independent routes. Route A (§§12–13) is the Big AGGP cycle route: a Λ-adic Big AGGP cycle is constructed in Iwasawa cohomology from the horizontal norm relation, ordinary αp-recursion, and coherent sheaf assembly, generalizing Howard's Big Heegner Points from GL2 to U(r) × U(r + 1). The classical BSD conclusion follows from a two-variable syntomic regulator comparison (Theorem 13.5.C) identifying the p-adic determinant regulator of the Big AGGP cycle with its motivic/Deligne regulator on the product of the ordinary eigenvariety branch and the cyclotomic disc. Route B (§§14–15) is the Capelli descent route: an independent parallel proof using local automorphic multiplicity one and a Capelli-type descent of the global AGGP identity. Combined with Papers I–III, these results complete the unconditional proof of the classical BSD conjecture at all ranks.