Global Hartogs Extension: Boundary Determines Interior
Any τ-holomorphic function on the boundary L extends uniquely to the interior of τ³ — the Global Hartogs Extension Theorem.
Overview
I.T31 (Global Hartogs Extension Theorem) proves that any τ-holomorphic function defined on the lemniscate boundary L = S¹ ∨ S¹ extends uniquely to a τ-holomorphic function on all of τ³. This is the climax of Book I: the foundational result that shows the boundary encodes the interior, preparing for the Central Theorem (II.T40) in Book II.
Detail
The classical Hartogs extension theorem in complex analysis states that a function holomorphic on the boundary of a domain of holomorphy extends holomorphically to the domain. I.T31 is the τ-analogue: a τ-holomorphic function on the boundary L extends uniquely to all of τ³. The proof in Book I uses the bipolar prime structure established by the Prime Polarity Theorem (I.T05) and the ABCD coordinate chart (I.T04): the boundary data, decomposed into γ-even and η-odd components, propagates inward via the tower-graded structure of τ³. Uniqueness follows from the Tau-Identity Theorem (I.T21): if two holomorphic functions agree on the boundary, they agree everywhere. I.T31 is the Book I climax because it proves the holographic principle at the foundational level, before the full machinery of Book II is needed.
Result Statement
I.T31: Any τ-holomorphic function defined on the lemniscate boundary L extends uniquely to a τ-holomorphic function on all of τ³. Boundary determines interior.