Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Cross E

The decidable equality

A ratio orbit is a sequence of ratios generated by a fixed growth rule, one step after another. The declaration crossEq_decidable establishes a precise fact about such orbits: for any two of them, a computer can always decide, in finite time, whether they are equal. This is a recognition ledger fact, where the ledger is a discrete record of events. The proof works by unfolding the definition of equality and letting the underlying logic supply the decision procedure. It is a small, clean result, but it is the load-bearing floor for everything else in the file.

What follows from this decidability is a trichotomy: for any two ratio orbits p and q, exactly one of three things holds. Either p is strictly less than q, or p equals q, or q is strictly less than p. The theorem ltQ_trichotomy proves this, and it relies on the decidability result to split cases cleanly. The strict less-than relation itself is defined as "less than or equal, but not equal", and the theorem ltQ_irrefl confirms that no orbit is strictly less than itself. These three results together give the ratio orbits a total order, a structure where every pair can be compared.

The classical cousin of this idea is the real number line. Every pair of real numbers can be compared, and the comparison is decidable in principle. The ratio orbits inherit that same feel, but with a discrete, combinatorial flavor. The trichotomy is what lets later arguments speak of the "next" orbit, or the "smallest" orbit above a given one, without ambiguity. It is the kind of fact that a working mathematician uses without thinking, but which a machine-checked library must prove explicitly.

In Recognition Science, this is one more rung in the ladder from the cost function to physical structure. The framework models growth as a sequence of ratios, and the ability to compare any two such sequences is what makes the model well-founded. Without decidability, the whole edifice would rest on a comparison that might never terminate. With it, the framework can proceed to the next step: showing that the growth rule itself forces a particular scaling, the golden ratio, as the unique self-similar choice.

What the declaration does not claim is more modest than it might appear. It does not say which orbit is larger, only that the question has an answer. It does not say anything about the values of the ratios, only about their order. And it does not, by itself, establish that the golden ratio is special; that is a separate theorem further along the chain. The decidability result is a tool, not a conclusion. It is the kind of fact that makes the rest of the framework possible, but it is not itself a statement about the physical world.

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

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)

What this page does not claim

The declaration does not say which orbit is larger, only that the question has an answer. The declaration does not establish that the golden ratio is special; that is a separate theorem. The declaration makes no claim about the physical world on its own.

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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Derived articles

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