τ-categorical structure
The τ-categorical structure is the categorical content of the τ-kernel: τ as a category in its own right, equipped with the τ-site topology that turns it into an earned topos. The phrase 'τ-categorical' appears throughout the framework as the qualifier separating τ-internal claims (provable from K0–K6) from τ-external claims (requiring bridges into Mathlib or classical mathematics).
τ-Definition
The τ-categorical structure is the categorical content of the τ-kernel: τ as a category in its own right, equipped with the τ-site topology that turns it into an earned topos. The phrase 'τ-categorical' appears throughout the framework as the qualifier separating τ-internal claims (provable from K0–K6) from τ-external claims (requiring bridges into Mathlib or classical mathematics).
Categorical invariant. τ together with its canonical site topology, viewed as an (∞, 1)-topos via the earned-topos construction.
Primary registry anchor:
I.D56
τ-Derivation Chain
Mathematical content
A claim is τ-categorical iff it is provable in the τ-kernel signature K0 + K1–K6 by purely categorical reasoning (without invoking Mathlib bridges, ZFC, or classical analysis).
Role. scope-discriminator
Consequence. τ-categorical claims are the framework's fully-internal results. Bridges into orthodox mathematics (Mathlib, ZFC, classical analysis) are explicit declarations and live in Book III's bridge axioms (A01–A03), not in the τ-categorical core.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Topos.EarnedTopos
Lean kind: structure
Lean symbol: Tau.BookI.Topos.earnedTopos
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.