Part XV: Global Hartogs

Part Overview

Every tool in this book has been forged for a single purpose: to prove that the limit determines the stages. In classical complex analysis, Hartogs’ extension theorem says that a holomorphic function defined outside a thin set extends uniquely to the whole domain. The τ-analog is stronger: every τ-holomorphic function on 𝕃 ∖ K, with K primordially thin, extends uniquely to all of 𝕃.

The proof uses everything earned so far: spectral coefficients from the character decomposition (Part X), the CRT extension lemma from the primorial coverage (Part IX), tower coherence from the holomorphic transformer (Part XII), and the Identity Theorem for uniqueness (Part XII). The result is the Global Hartogs Extension Theorem — the crown jewel and climax of Book I.

As a corollary, omega-tail data on the algebraic lemniscate 𝕃 uniquely determines all finite-stage values on τ³. This is the canonical passage to Book II, where the Central Theorem O(τ³) ≅ A_spec(𝕃) sharpens this algebraic principle into the geometric slogan boundary determines interior.

Chapters

• Chapter 65: Spectral Coefficients and the Restriction Map
• Chapter 66: Thinness and the Removable Singularity Criterion
• Chapter 67: The Global Hartogs Extension Theorem
• Chapter 68: The Limit-Stage Passage