Chapter 7: Bipolar Decomposition of Interior Points

Book I earned the bipolar idempotents e_± = (1 ± j)/2 as boundary structure: the algebraic lemniscate 𝕃 decomposes into two sectors via the split-complex algebra H_τ. This chapter extends the bipolar decomposition from the boundary to the interior. Each τ-admissible point (A, B, C, D) inherits a canonical sector assignment from its fiber coordinates (B, C): the γ/η dominance determines which sector the point belongs to. Every interior point thus carries two independent information channels—a B-channel governed by e_+ and a C-channel governed by e_-. At the ω-limit, the sector assignment degenerates to the two lobes of 𝕃, and the crossing point where B ≈ C maps to the node.