Results Glossary Entry Canonical mathematics The Progression Operator ρ (I.D02) is the τ-internal advancement operator on kernel atoms — the canonical step that lifts content from one K1 strict-order position to the next. With 17 incoming edges, ρ is the framework's internal time/iter…
Results · Mathematics Glossary · Definition MathG-D12-progression-operator ρ Canonical

Progression Operator ρ

The Progression Operator ρ (I.D02) is the τ-internal advancement operator on kernel atoms — the canonical step that lifts content from one K1 strict-order position to the next. With 17 incoming edges, ρ is the framework's internal time/iteration mechanism: every K1-iterative construction in Books I-III is a finite ρ-composition.

Mathematics Glossary Primary: I.D02

τ-Definition

The Progression Operator ρ (I.D02) is the τ-internal advancement operator on kernel atoms — the canonical step that lifts content from one K1 strict-order position to the next. With 17 incoming edges, ρ is the framework's internal time/iteration mechanism: every K1-iterative construction in Books I-III is a finite ρ-composition.

Categorical invariant. An endo-functor ρ : τ → τ satisfying the K1 strict-order successor property — for every kernel atom A, ρ(A) is the immediate K1-successor.

Primary registry anchor: I.D02

Supporting items: I.K1

τ-Derivation Chain

  1. I.K0 — Universe Postulate
  2. I.K1 — K1 strict order on kernel atoms
  3. I.D02 — Progression Operator ρ — K1-successor as a functor

Lean modules referenced: TauLib.BookI.Orbit.TooFew

Mathematical content

Definition ρ
Definition

ρ : τ → τ is the endo-functor sending each kernel atom A to its immediate K1-successor ρ(A). For composite τ-objects, ρ extends via K3 composition: ρ(f ∘ g) = ρ(f) ∘ ρ(g). The operator is K1-monotone (a < b ⇒ ρ(a) < ρ(b)) and total.

Consequences:

  • Iteration ladder (I.D? — supporting) — built by repeated ρ-application.
  • Window-algebra integers W_n(k) (D03) — k counts ρ-iteration depth.
  • Holomorphy tower (S02) — built by ρ-iterated refinement.

Lean Coverage

Status: Formalized

Module: TauLib.BookI.Orbit.TooFew

Lean kind: def

Lean symbol: Tau.BookI.Orbit.progressionOperator

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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