Diagonal Discipline
The Diagonal Discipline (I.D03) is the τ-internal coherence rule: every τ-categorical morphism that factors through the diagonal Δ : A → A × A must commute with K3 composition along that factorization. With 17 incoming edges, it is the structural reason τ-internal Hartogs extensions (T12) are unique and the central theorem's categoricity check is well-posed.
τ-Definition
The Diagonal Discipline (I.D03) is the τ-internal coherence rule: every τ-categorical morphism that factors through the diagonal Δ : A → A × A must commute with K3 composition along that factorization. With 17 incoming edges, it is the structural reason τ-internal Hartogs extensions (T12) are unique and the central theorem's categoricity check is well-posed.
Categorical invariant. A coherence axiom asserting that diagonal-factoring morphisms in τ commute with K3 composition; this is what makes τ a 'disciplined' (∞, 1)-category.
Primary registry anchor:
I.D03
Supporting items:
I.K3
τ-Derivation Chain
Mathematical content
For every τ-morphism f : A → B that factors as f = Δ_B ∘ g for some g : A → B and Δ_B : B → B × B the diagonal: K3 composition commutes with the factorization (i.e. for any h composable, K3(h, f) = K3(h, Δ_B) ∘ g).
Consequences:
- Global Hartogs Extension (T12) — uniqueness rests on diagonal discipline.
- Central theorem (T04) — categoricity check is well-posed because diagonal-factored expressions don't ambiguate.
- Yoneda enrichment ladder (T05) — each step preserves discipline.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.MetaLogic.LinearDiscipline
Lean kind: axiom
Lean symbol: Tau.BookI.MetaLogic.diagonalDiscipline
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.