Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Lt Q

The order on ratio orbits

In mathematics, an order relation lets you compare two objects and say which comes first. The declaration ltQ defines such a relation for a specific kind of object: a ratio orbit, which is a sequence of ratios generated by repeated growth steps in a discrete ledger, a record of events where each step multiplies the previous value by some factor. The definition is plain: one orbit is less than another when it is at most the other, and the two are not equal. Formally, ltQ p q means leQ p q and not crossEq p q, where leQ is the existing "at most" relation and crossEq is the equality relation between orbits.

The key result is a trichotomy theorem: for any two ratio orbits p and q, exactly one of three things holds: p is less than q, p equals q, or q is less than p. This mirrors the familiar property of real numbers, where any two numbers are either less, equal, or greater. The theorem is proved in the machine-checked library of formal theorems, and it relies on two supporting facts: the "less than" relation is irreflexive, meaning no orbit is less than itself, and equality between orbits is decidable, meaning a computer can always determine whether two orbits are equal. These together give a total order on ratio orbits, a structure where every pair is comparable.

The trichotomy matters because it turns a collection of growth processes into a well-behaved sequence. With a total order, one can sort orbits, find minima and maxima, and reason about limits in a principled way. This is a stepping stone in the framework's larger project of deriving physical structure from recognition events, but the declaration itself is purely combinatorial: it establishes an ordering property, not any physical law.

What ltQ does not claim is equally important. It does not assert that this order corresponds to numerical magnitude of the ratios themselves; the relation is defined on orbits, not on the individual ratio values. It does not claim that the order is dense, complete, or well-founded, only that it is total. And it does not derive any physical consequence, such as the golden ratio or the number of dimensions; those results live elsewhere in the framework and are not part of this declaration.

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

theorem ltQ_trichotomy (p q : RatioOrbit) :
    ltQ p q ∨ RatioOrbit.crossEq p q ∨ ltQ q p := by
  cases' leQ_total p q with hpq hqp
  · by_cases heq : RatioOrbit.crossEq p q
    · exact Or.inr (Or.inl heq)
    · exact Or.inl ⟨hpq, heq⟩
  · by_cases heq : RatioOrbit.crossEq q p
    · exact Or.inr (Or.inl (RatioOrbit.crossEq_symm heq))
    · exact Or.inr (Or.inr ⟨hqp, heq⟩)

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

instance crossEq_decidable (a b : RatioOrbit) : Decidable (RatioOrbit.crossEq a b) := by
  unfold RatioOrbit.crossEq
  infer_instance

What this page does not claim

This declaration does not claim that the order on ratio orbits corresponds to the numerical magnitude of the ratios themselves. This declaration does not claim that the order is dense, complete, or well-founded. This declaration does not derive any physical consequence such as the golden ratio or the number of spatial dimensions.

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)

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