Arithmetic spectroscopy of elliptic curves. Part VI: The supersingular extension and the full spectral BSD formula

The Birch and Swinnerton-Dyer conjecture asserts that the order of vanishing of the L-function L (E, s) of an elliptic curve E/Q at s=1 equals the Mordell–Weil rank and gives an explicit formula for the leading coefficient in terms of the real period, Néron–Tate regulator, order of the Tate–Shafarevich group, and product of Tamagawa numbers. This six-part series develops a spectral programme in which a one-parameter family of Schrödinger operators Hₗambda, whose potential is built from the Néron–Tate height pairing on the Mordell–Weil lattice, encodes the central arithmetic invariants of E. The present sixth part supplies the supersingular extension. We introduce a bipartite monad constructed from Wach modules and Frobenius idempotents, define signed spectral zeta functions, and construct a recombination operator that cancels the supersingular singularities at the critical point. We prove that the residual structure of the full spectral zeta function zetaH (s) at s=1 coincides with that of L (E, s) via the Kato Euler system. The spectral trace identity holds to relative error better than 1. 3 × 10^-8 on all tested curves (ranks 0 to 4, including supersingular reduction at p=2, 3, 5, 7). All theoretical results are conditional on the vanishing of the mu-invariant and the Iwasawa main conjecture (Sprung 2012 for supersingular primes; Skinner–Urban 2014 for ordinary primes). These conditions appear precisely where the spectral geometry isolates the remaining classical arithmetic input. Numerical verification across eight curves of conductor at most 10⁴ confirms that the Witten index equals the algebraic rank independently of reduction type, the spectral regulator agrees with the Néron–Tate regulator to relative error better than 2 × 10^-5, and the three-sector thermal trace recovers the BSD leading term.