Chapter 30: The Bipolar Spectral Algebra

The polarized omega-germs of the relevant chapter split the boundary data of τ into three families: B-polarized germs, C-polarized germs, and the unique crossing-point germ. This chapter asks: what is the natural scalar algebra for functions on this boundary? The answer is forced by the bipolar structure: split-complex scalars satisfying j² = +1 (not elliptic i² = -1), organized into a bipolar spectral algebra H_τ = A_τ^(B) × A_τ^(C) with canonical idempotents e_± = (1 ± j)/2. Holomorphic functions are primitive in the τ-framework: they are omega-germ transformers valued in H_τ, defined before topology, continuity, or geometry. The bipolar spectral algebra, together with the crossing-point germ as identity and the polarity involution σ as structural flip, constitutes the algebraic lemniscate — the pre-geometric, pre-topological boundary of τ. The geometric form S¹ ∨ S¹ emerges in Book II when topology is earned.