Chapter 38: Idempotent Decomposition Lemma

Part VI established the sheaf O_τ and its coherence properties. Part VII now asks: what is the internal structure of a τ-holomorphic function? The answer begins with the observation that the bipolar idempotents e_± = (1 ± j)/2 (I.D21, Book I) decompose not only scalars but also functions. If f : τ³ → H_τ is τ-holomorphic, then f_+ := e_+ · f and f_- := e_- · f are each τ-holomorphic, and f = f_+ + f_-. This is the Idempotent Decomposition Lemma (II.L07): the algebraic idempotent splitting of H_τ lifts to a functional splitting of the holomorphic function space. The decomposition is canonical (no choices are made), functorial (it respects composition: (g ∘ f)± = g± ∘ f_±, the channels do not mix), and complete (f_+ and f_- together determine f uniquely). the relevant definition (II.D48) formalizes the canonical decomposition. **Proposition [prop:decomposition-functoriality] (II.P09) establishes functoriality. This lemma is the engine for the three-lemma chain in the next chapter: it converts the algebraic decomposition of the codomain H_τ into a functional decomposition of the holomorphic function space O_τ(τ³).