Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Lt Q Ir

The strict order

In mathematics, a strict order is a relation that behaves like "less than" for numbers: it is transitive, and no object is strictly less than itself. That last property, called irreflexivity, is what the declaration ltQ_irrefl establishes. It states that for any ratio orbit p, the proposition ltQ p p is false. A ratio orbit is a discrete record of growth states in the Recognition Science framework, where each state encodes a ratio between successive quantities in a ledger of recognition events.

The proof is direct. The relation ltQ is defined as: ltQ p q holds exactly when leQ p q holds and the orbits p and q are not cross-equal. Cross-equality is a separate equivalence relation on orbits. To prove ltQ p p is false, assume it holds. Then by definition, leQ p p holds and crossEq p p is false. But crossEq is reflexive, so crossEq p p is true. Contradiction. The theorem thus follows from the reflexivity of crossEq and the structure of the definition.

This theorem is part of a trichotomy law: for any two ratio orbits p and q, exactly one of ltQ p q, crossEq p q, or ltQ q p holds. The irreflexivity result is the foundational piece that makes this trichotomy coherent, ensuring the strict order has no loops. Without it, the ordering would collapse and the framework's growth dynamics would lose their directional meaning.

In Recognition Science, this strict order underpins how the framework models the forced progression of growth ratios. The framework's library of machine-checked formal theorems proves this as a logical consequence of the definitions, not as an assumption. The theorem does not claim that ratio orbits are totally ordered by ltQ alone, nor that crossEq is the only equivalence relation on orbits, nor that the trichotomy holds for any other relation besides ltQ.

theorem ltQ_irrefl (p : RatioOrbit) : ¬ ltQ p p := by
  intro h
  exact h.2 (RatioOrbit.crossEq_refl p)

def ltQ (p q : RatioOrbit) : Prop := leQ p q ∧ ¬ RatioOrbit.crossEq p q

What this page does not claim

ltQ_irrefl does not claim that ratio orbits are totally ordered by ltQ alone. It does not claim that crossEq is the only equivalence relation on orbits. It does not claim that the trichotomy holds for any relation other than ltQ.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/Grow/RatioOrbitLtTrichotomy.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: