Results Glossary Entry Canonical mathematics The τ-Ultrametric Distance (II.D13) is the τ-categorical metric on the boundary, satisfying the strong triangle inequality d(x,z) ≤ max(d(x,y), d(y,z)) — the ultrametric property. The distance is induced by the K1 strict order and the K2 bo…
Results · Mathematics Glossary · Definition MathG-D16-ultrametric-distance d_τ Canonical

τ-Ultrametric Distance

The τ-Ultrametric Distance (II.D13) is the τ-categorical metric on the boundary, satisfying the strong triangle inequality d(x,z) ≤ max(d(x,y), d(y,z)) — the ultrametric property. The distance is induced by the K1 strict order and the K2 boundary topology. With 14 incoming edges, it underlies Book II's domains and convergence machinery.

Mathematics Glossary Primary: II.D13

τ-Definition

The τ-Ultrametric Distance (II.D13) is the τ-categorical metric on the boundary, satisfying the strong triangle inequality d(x,z) ≤ max(d(x,y), d(y,z)) — the ultrametric property. The distance is induced by the K1 strict order and the K2 boundary topology. With 14 incoming edges, it underlies Book II's domains and convergence machinery.

Categorical invariant. A τ-categorical metric d_τ : B_τ × B_τ → ℝ_≥0 satisfying the ultrametric inequality.

Primary registry anchor: II.D13

Supporting items: I.D19, II.D14

τ-Derivation Chain

  1. I.K0 — Universe Postulate
  2. I.D19 — Boundary Ring
  3. II.D14 — τ-Stone Space (compact totally-disconnected)
  4. II.D13 — τ-Ultrametric Distance — strong triangle inequality on the boundary

Lean modules referenced: TauLib.BookII.Domains.Ultrametric

Mathematical content

Definition d_τ
Definition

d_τ : B_τ × B_τ → ℝ_≥0 satisfies: (a) d_τ(x, y) = 0 iff x = y; (b) d_τ(x, y) = d_τ(y, x); (c) d_τ(x, z) ≤ max(d_τ(x, y), d_τ(y, z)) — the ultrametric (strong) triangle inequality; (d) compatibility with K1 strict order and K2 topology.

Why ultrametric. The K2 labelled boundary is totally disconnected (D14 τ-Stone Space). On totally-disconnected spaces, the natural metric is ultrametric — every triangle has at least two equal sides. The τ-Ultrametric Distance is the canonical such metric on B_τ.

Consequences:

  • All convergence in the τ-categorical framework is ultrametric (each Cauchy sequence's terms eventually become indistinguishable in the strong sense).
  • Holomorphy tower (S02) levels are ultrametric balls.
  • p-adic-style behavior of B_τ — many convergence results have classical p-adic-analysis analogues.

Lean Coverage

Status: Formalized

Module: TauLib.BookII.Domains.Ultrametric

Lean kind: def

Lean symbol: Tau.BookII.Domains.ultrametricDistance

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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