Chapter 57: BSD Bridge: Proto-Rationality in Split-Complex Regime

This chapter bridges from Book II’s Central Theorem to one of the seven Millennium Problems: the Birch and Swinnerton-Dyer (BSD) Conjecture. Book II does not prove BSD — that requires the spectral force framework of Book III (enrichment layer E₂). What this chapter establishes is the structural prerequisite: proto-rationality (the relevant definition, II.D66), the condition that spectral coefficients of τ-L-functions lie in the image of the canonical basis B_τ (the relevant definition, II.D45). The BSD Bridge (Remark [rem:bsd-bridge], II.R18) identifies what Book II provides (the Central Theorem, the spectral algebra, calibration) and what Book III must add (L-function machinery, analytic continuation in H_τ, force-theoretic rank computation). The split-complex regime fundamentally alters the analytic rank: orders of vanishing split into bipolar components ord_+ and ord_- via the idempotent decomposition (Lemma [lem:idempotent-decomposition], II.L07). This is a forward pointer, not a proof; the chapter is honest about the distance from proto-rationality to BSD.