Chapter 44: The BSD Coherence Theorem

Chapters 45 and 46 assembled the two sides of the BSD equation: the algebraic side (τ-rational points, rank function r(k), Mordell–Weil analogue) and the analytic side (proto-codes, the BSD functional BSD_T(k), spectral determinant L_T’(1,k)). This chapter proves that the two sides agree. The BSD Coherence Theorem states that for every τ-admissible elliptic datum—an E₁ object equipped with proto-code structure—the BSD functional stabilizes at finite primorial depth, and its stable value equals the rank of the τ-rational point group. The proof rests on three ingredients, isolated first as Proposition [prop:bsd-three-ingredient]: rank stabilization from the Mordell–Weil analogue (Proposition [prop:mordell-weil-analogue], Ch. 45), L-value stabilization from spectral determinant tower coherence (the relevant definition, the relevant theorem), and equality from the E₁ Mutual Determination instance (the relevant definition, Ch. 44). The chapter then draws a sharp scope boundary between what τ-BSD proves and what the Clay BSD prize requires, and closes with export contracts specifying what downstream chapters inherit.