Foundation Primitive Recognition Calculus Grow Ratio Orbit Dense Mediant Lt Q Me

The mediant between ratios

A ratio orbit is a discrete record of two counters that grow together, each step adding to one side or the other. The mediant of two such ratios is the fraction you get by adding their numerators and adding their denominators: for 1/2 and 2/3, the mediant is 3/5, which always lies between them. The declaration ltQ_mediant proves a basic ordering fact: if one ratio is less than another, then the mediant sits strictly between the two. This is a theorem in the machine-checked library of formal theorems, meaning the proof has been verified step by step by a computer.

The statement is simple but the consequence is not. Repeatedly taking mediants builds a dense set: between any two distinct ratios, no matter how close, another ratio appears. This is the same property that makes the rational numbers fill the number line without gaps, even though each individual fraction is isolated. The theorem guarantees that a process which only ever forms mediants can approach any ratio from below or above, never needing to jump over a value. It is a structural fact about how ordered fractions behave, not a claim about any particular physical system.

In Recognition Science, this ordering result supports the growth of a recognition cycle: the framework models how a discrete ledger of events can pass through ratios in an orderly way. The theorem ltQ_mediant is one link in that chain, showing the mediant operation respects the ordering that the framework uses to track relative sizes. What the theorem does not claim is that the mediant process reaches every ratio in finitely many steps. Density means approximation, not completion; the orbit gets arbitrarily close but never lands on every value. It also does not claim that the mediant is the only way to build such a dense order, nor that any physical process must follow this path. The theorem is about the arithmetic of ratios, and the framework's use of it is a separate modeling choice.

theorem ltQ_mediant : ∀ p q, ltQ p q → ltQ p (mediant p q) ∧ ltQ (mediant p q) q := by
  intro p q h
  rw [ltQ_iff_toNat] at h
  refine ⟨?_, ?_⟩
  · rw [ltQ_iff_toNat]
    simp only [mediant, SignedOrbit.add_pos, SignedOrbit.add_neg,
               DistinctionNat.toNat_add, Nat.mul_add, Nat.add_mul]
    generalize p.num.pos.toNat * p.den.toNat = e1 at *
    generalize p.num.pos.toNat * q.den.toNat = e2 at *
    generalize p.num.neg.toNat * p.den.toNat = e3 at *
    generalize q.num.neg.toNat * p.den.toNat = e4 at *
    generalize q.num.pos.toNat * p.den.toNat = e5 at *
    generalize p.num.neg.toNat * q.den.toNat = e6 at *
    omega
  · rw [ltQ_iff_toNat]
    simp only [mediant, SignedOrbit.add_pos, SignedOrbit.add_neg,
               DistinctionNat.toNat_add, Nat.mul_add, Nat.add_mul]
    generalize p.num.pos.toNat * q.den.toNat = e2 at *
    generalize q.num.pos.toNat * q.den.toNat = e7 at *
    generalize q.num.neg.toNat * p.den.toNat = e4 at *
    generalize q.num.neg.toNat * q.den.toNat = e8 at *
    generalize q.num.pos.toNat * p.den.toNat = e5 at *
    generalize p.num.neg.toNat * q.den.toNat = e6 at *
    omega

What this page does not claim

The mediant process reaches every ratio in finitely many steps. The mediant is the only operation that builds a dense order on ratios. Any physical process must follow the mediant path.

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/RatioOrbitDenseMediant.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: