Results Glossary Entry Canonical mathematics The Stage-k Cylinder (II.D10) is the τ-categorical analogue of the cylinder construction at depth k along the K1 strict-order direction. It packages the rank-(n, k) data into a single τ-object whose homotopy and dynamics encode the depth-k …
Results · Mathematics Glossary · Definition MathG-D11-stage-k-cylinder Cyl_k Canonical

Stage-k Cylinder

The Stage-k Cylinder (II.D10) is the τ-categorical analogue of the cylinder construction at depth k along the K1 strict-order direction. It packages the rank-(n, k) data into a single τ-object whose homotopy and dynamics encode the depth-k content of the holomorphy tower (S02). With 18 incoming edges, it is the structural input to translation-arithmetic results in Book III.

Mathematics Glossary Primary: II.D10

τ-Definition

The Stage-k Cylinder (II.D10) is the τ-categorical analogue of the cylinder construction at depth k along the K1 strict-order direction. It packages the rank-(n, k) data into a single τ-object whose homotopy and dynamics encode the depth-k content of the holomorphy tower (S02). With 18 incoming edges, it is the structural input to translation-arithmetic results in Book III.

Categorical invariant. A τ-object Cyl_k parametrized by depth k ∈ ℕ, equipped with face maps to depths k-1 and k+1 compatible with K1 strict order.

Primary registry anchor: II.D10

Supporting items: I.D08, I.D17

τ-Derivation Chain

  1. I.K0 — Universe Postulate
  2. I.D08 — Rank-transfer maps
  3. II.D10 — Stage-k Cylinder — packaged τ-object at depth k

Lean modules referenced: TauLib.BookIII.Bridge.TranslationArith

Mathematical content

Definition Cyl_k
Definition

For each k ∈ ℕ, the Stage-k Cylinder Cyl_k is the τ-object packaging the rank-(n, k) data across all primes n: Cyl_k = ⊕_n W_n(k)-obj where W_n(k)-obj is the window-algebra object (O02). Cyl_k is equipped with face maps Cyl_k → Cyl_{k-1} and Cyl_k → Cyl_{k+1} from the rank-transfer machinery.

Consequences:

  • Translation-arithmetic results (Book III bridge construction) operate on stage-k cylinders.
  • Holomorphy tower colimit (S02) coincides with colim_k Cyl_k.
  • Central theorem (T04) — the rank-(3, 15) categoricity check operates on Cyl_15.

Lean Coverage

Status: Formalized

Module: TauLib.BookIII.Bridge.TranslationArith

Lean kind: def

Lean symbol: Tau.BookIII.Bridge.stageKCylinder

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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