Part VII: Regularity and Mutual Determination

Part Overview

Part VII executes the regularity program, proving that holomorphic = idempotent-supported. Five chapters build a three-lemma chain leading to this equivalence.

the relevant chapter establishes the Idempotent Decomposition Lemma (II.L07): the key structural result. Every holomorphic map decomposes: f = e_+ · f_+ + e_- · f_-. Decomposition is canonical, functorial, and complete.

the relevant chapter strings three lemmas: (II.L08) Branch Factorization: ω-germs factor through bipolar idempotents; (II.L09) Prime-Split Support: Factorization supported on B/C prime partition; (II.L10) Polarity Symmetry: Sectors interchanged by j-involution. Together: Theorem II.T29: Holomorphic ⇔ idempotent-supported.

the relevant chapter shifts from negative regularity (absence of singularity) to positive regularity (existence of stabilized ω-germ). A map is regular if and only if it has a canonical extension to the interior. No “removable singularity” pathology: either regular or genuinely singular.

the relevant chapter shows the embedding Hol_τ ↪ d(τ³): holomorphic functions are both transformers and germs. Self-referential structure. This is a preview of Part VIII’s full Yoneda theorem.

the relevant chapter constructs the bijection: boundary streams (codes) ↔ interior extensions (decodes). The diagonal-free discipline (I.X05) blocks zero-divisor pathology; e_± exist and decompose, but arbitrary projections are not earned. This is why split-complex algebra doesn’t collapse.

Part VII closes with the regularity theory established. Holomorphic functions are rigorously characterized as idempotent-supported ω-germ transformers.

Chapters

• Chapter 38: Idempotent Decomposition Lemma
• Chapter 39: The 3-Lemma Chain
• Chapter 40: Regularity as Positive Structure
• Chapter 41: Pre-Yoneda Embedding
• Chapter 42: Code/Decode and Diagonal Protection