Chapter 39: The 3-Lemma Chain

the relevant chapter established the Idempotent Decomposition Lemma (II.L07): every τ-holomorphic map decomposes canonically as f = e_+ · f_+ + e_- · f_-. This chapter builds on that decomposition to prove the characterization theorem of Part VII: a function f : τ³ → H_τ is τ-holomorphic if and only if it is idempotent-supported.

The proof chains three lemmas. Lemma 1 (Branch Factorization, II.L08): every ω-germ transformer factors through the bipolar idempotents e_±. Lemma 2 (Prime-Split Support, II.L09): the e_+-component is supported on B-channel primes (γ-orbit), and the e_–component on C-channel primes (η-orbit), forced by Prime Polarity (I.T05, Book I). Lemma 3 (Polarity Symmetry, II.L10): the j-involution interchanges the two sectors: σ_j(G_+) = G_- and σ_j(G_-) = G_+, so the two channels carry symmetric information. Together, these three lemmas yield **the relevant theorem (II.T33): τ-holomorphic ⟺ idempotent-supported. This is the equivalence that unifies the analytic viewpoint (holomorphy) with the algebraic viewpoint (idempotent support structure), and it completes the regularity program initiated in the relevant chapter.