Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy

Ordering the orbits

In mathematics, a trichotomy law is a guarantee that any two objects of a certain kind stand in exactly one of three relations: less than, equal to, or greater than. The real numbers have this property, and so do the natural numbers. The recognition calculus, a framework where reality keeps a discrete ledger of events, needs the same guarantee for its growth orbits, the paths that repeated scaling steps trace out.

The result RatioOrbitLtTrichotomy establishes this trichotomy for the ratio orbits of the primitive recognition calculus. A ratio orbit is the set of values reachable from a starting ratio by repeatedly applying a fixed growth step. The result defines a strict less-than relation ltQ on these orbits: one orbit is less than another when it is less than or equal and not equal. It then proves the trichotomy theorem: for any two orbits p and q, exactly one of p < q, p = q, or q < p holds.

The proof is short and direct. It uses the totality of the underlying less-than-or-equal relation on orbits, which says that for any two orbits, one is less than or equal to the other. It then splits on whether the two orbits are equal. If they are equal, the middle case holds. If they are not equal, the direction of the less-than-or-equal relation determines which of the two strict inequalities holds. The theorem also proves that the strict relation is irreflexive: no orbit is strictly less than itself.

In Recognition Science, this trichotomy matters because it gives the framework a total order on its growth orbits. A total order means every pair of orbits can be compared, with no ambiguity and no gaps. This is a structural precondition for later results that sort or rank orbits, and it is a proved theorem in the machine-checked library of formal theorems, not an assumption. The framework's library is a collection of formal proofs verified by a computer kernel.

What this result establishes in plain language is simple: the growth orbits of the primitive recognition calculus behave like numbers in this one respect. You can always ask which of two orbits comes first, and the answer is always exactly one of the three possibilities. This is a small but load-bearing piece of the framework's foundation, because later arguments that depend on comparing orbits can rely on this trichotomy without rechecking it.

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)

What this page does not claim

This result does not establish that the ratio orbits are ordered by magnitude in the usual real-number sense. This result does not prove that every pair of growth steps produces comparable orbits, only that the ratio orbits themselves are totally ordered. This result does not connect the trichotomy to the framework's derivation of three 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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