Chapter 67: The Global Hartogs Extension Theorem

This is the destination. Every definition, every lemma, every theorem in this book has been forged for the result proved in this chapter.

The Global Hartogs Extension Theorem (the relevant theorem, I.T31): if f is τ-holomorphic on 𝕃 ∖ K with K primordially thin, then f extends uniquely to all of 𝕃. No boundedness assumption. Thinness alone suffices.

The proof draws on every strand of the preceding fifteen parts: spectral coefficients determine the function through the characters of Part X; CRT reconstruction fills the gaps (Lemma [lem:crt-extension], I.L08); tower coherence (the relevant definition, I.D46) guarantees global consistency; the τ-Identity Theorem (the relevant theorem, I.T21) delivers uniqueness. Omega-tail data on 𝕃 determines all finite-stage values — the canonical passage to Book II, where O(τ³) ≅ A_spec(𝕃) gives this algebraic principle its geometric form.