Global Hartogs Extension
The Global Hartogs Extension theorem (I.T31) is the τ-categorical analogue of the classical Hartogs extension principle, lifted to the holomorphy tower (S02). It asserts that every τ-internal holomorphic function defined on a punctured τ-domain extends uniquely to the full domain. With 18 incoming edges, it is the structural input to the Yoneda-as-theorem internalization (D04) and Book III's spectral correspondence (A02).
τ-Definition
The Global Hartogs Extension theorem (I.T31) is the τ-categorical analogue of the classical Hartogs extension principle, lifted to the holomorphy tower (S02). It asserts that every τ-internal holomorphic function defined on a punctured τ-domain extends uniquely to the full domain. With 18 incoming edges, it is the structural input to the Yoneda-as-theorem internalization (D04) and Book III's spectral correspondence (A02).
Categorical invariant. A unique-extension theorem for τ-internal holomorphic functions across the holomorphy tower's colimit.
Primary registry anchor:
I.T31
τ-Derivation Chain
-
I.K0— Universe Postulate -
I.D96— Holomorphy tower — tower of holomorphic refinements -
I.D47— τ-Holomorphic function — well-defined on tower levels -
I.T31— Global Hartogs Extension — every punctured-domain holomorphic function extends uniquely
Lean modules referenced:
TauLib.BookI.Holomorphy.GlobalHartogs,
TauLib.BookI.Holomorphy.BoundaryInterior
Mathematical content
Let U ⊂ τ-domain be a τ-domain with a co-dimension ≥ 2 puncture P, and let f : U → ℝ[j] be τ-holomorphic. There exists a unique τ-holomorphic extension f̃ : U ∪ P → ℝ[j] agreeing with f on U.
Proof sketch (expand)
By induction on the holomorphy tower (S02): at the base level (Hol_0), the result reduces to the K4 boundary identification's universal property. At each tower step (Hol_n → Hol_{n+1}), the extension lifts via the K3 composition law. The colimit (Hol_ω) gives the global extension.
Consequences:
- Yoneda-as-theorem (D04 / II.D50) — the pre-Yoneda embedding's lift uses Hartogs at each enrichment step.
- Spectral correspondence (A02) — the diagonal protection on Lem inherits Hartogs uniqueness.
- Central theorem rank (3, 15) — the categoricity check uses Hartogs to glue local contributions.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Holomorphy.GlobalHartogs
Lean kind: theorem
Lean symbol: Tau.BookI.Holomorphy.globalHartogs