Part IX: The Central Theorem and Categoricity

Part Overview

Part IX is the climax: the Central Theorem and categoricity. Six chapters assemble boundary-to-interior correspondence and prove uniqueness.

the relevant chapter restates the boundary ring ℤ_τ spectral characters (I.D19, I.D22–I.D23) as idempotent-supported objects. Each character decomposes: χ = e_+ · χ_+ + e_- · χ_- (B-channel vs. C-channel).

the relevant chapter proves Theorem II.T37: each idempotent-supported character extends uniquely to the interior. The extension lives in split-complex codomain H_τ (not classical ℂ). Uniqueness follows from bipolar channel independence.

the relevant chapter proves Theorem II.T38: Hartogs extensions are ω-germ transformers. Stagewise naturality carries forward the boundary character structure. This is the boundary → interior bridge.

the relevant chapter applies the Yoneda theorem (II.T36) to τ³ and proves Theorem II.T39: ω-germs are holomorphic functions. Probe naturality = ω-germ naturality = holomorphy. The loop closes: characters → extensions → ω-germs → holomorphic functions.

the relevant chapter assembles the main result. Theorem II.T40: The Central Theorem

Boundary determines interior; interior encodes boundary. Exact holographic principle. Spectral coefficients are numerically calibrated via ι_τ (Part V).

the relevant chapter resolves the classical Liouville theorem in the τ setting. Theorem II.T41: Wave-type PDEs (not elliptic) permit non-constant bounded solutions, dodging Liouville. Theorem II.T42: Categoricity. The six axioms (I.K0–I.K5) force τ³ uniquely. Moduli space = {pt} (no parameters). τ³ is discovered, not constructed.

Part IX closes with the Central Theorem proven and categoricity established. The interior is completely determined by boundary character algebra.

Chapters

• Chapter 48: Boundary Characters via Idempotent Support
• Chapter 49: Hartogs Extension in H_τ
• Chapter 50: Extensions Are ω-Germ Transformers
• Chapter 51: Yoneda Applied — ω-Germs Are Holomorphic Functions
• Chapter 52: The Central Theorem
• Chapter 53: Liouville Categorical Dodge and Categoricity