Book I holomorphy tower
The holomorphy tower (I.D96) is the Book-I structural ladder of holomorphy refinements on the τ-categorical kernel. It exhibits a graded sequence of τ-internal holomorphic objects — each level a τ-categorical refinement of the previous — culminating in the τ-internal Hartogs-extension principle that supports the Book II central theorem and the Book III spectral correspondence. The tower is the τ-categorical analogue of the conventional holomorphy hierarchy (analytic ⊃ holomorphic ⊃ holomorphic-with-extra-structure).
τ-Definition
The holomorphy tower (I.D96) is the Book-I structural ladder of holomorphy refinements on the τ-categorical kernel. It exhibits a graded sequence of τ-internal holomorphic objects — each level a τ-categorical refinement of the previous — culminating in the τ-internal Hartogs-extension principle that supports the Book II central theorem and the Book III spectral correspondence. The tower is the τ-categorical analogue of the conventional holomorphy hierarchy (analytic ⊃ holomorphic ⊃ holomorphic-with-extra-structure).
Categorical invariant. A graded τ-internal sequence of holomorphic objects, indexed by the K1 strict order, with each level admitting the next as an enriched refinement. The tower's colimit is the global Hartogs extension.
Primary registry anchor:
I.D96
τ-Derivation Chain
Mathematical content
The holomorphy tower is a graded sequence of τ-internal holomorphic objects {Hol_n}_{n ∈ ℕ}, where Hol_n is the level-n holomorphic refinement on the ABCD coordinate chart. Each Hol_n+1 is a τ-categorical enrichment of Hol_n.
Tower structure. The tower is well-founded (no infinite descent), monotone (Hol_n ⊆ Hol_{n+1}), and convergent: the colimit Hol_ω is the global Hartogs-extension object that supports Book II's central theorem.
Consequence. The tower is the τ-internal analogue of the conventional holomorphy hierarchy. It provides the structural backbone for: (a) the central theorem at rank (3, 15) (T04), via the global Hartogs extension; (b) the spectral correspondence (A02), via the diagonal protection that the tower induces on the lemniscate boundary; (c) the rank-coordinate machinery (D05), via the K1-strict-order indexing.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Holomorphy.GlobalHartogs
Lean kind: structure
Lean symbol: Tau.BookI.Holomorphy.holTower
Cross-domain bridges
This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.
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PG-C18-kappa-tauGravity-sector coupling κ_τ →MathG-S02-holomorphy-towerBook I holomorphy tower -
PG-L04-tau-einstein-equationτ-Einstein Equation →MathG-S02-holomorphy-towerBook I holomorphy tower -
PG-L12-tau-gravitational-waveτ-Gravitational Wave →MathG-S02-holomorphy-towerBook I holomorphy tower -
PG-Q10-proper-timeProper Time →MathG-S02-holomorphy-towerBook I holomorphy tower