Foundation Primitive Recognition Calculus Orbit Euclidean Normalize Ratio Den Mu

The denominator identity

A ratio like 6/8 and the fraction 3/4 name the same quantity. The usual way to see this is to divide both numbers by their greatest common divisor, 2. The Recognition Science framework builds its own arithmetic for counting recognition events, and this theorem is the machine-checked version of that reduction step for the denominator.

The statement concerns recognition, a discrete record of events, and orbits, sequences of such events. A ratio of two orbit counts is stored as a pair of numbers. The function normalizeRatio divides both by their greatest common divisor, producing the reduced form. The theorem normalizeRatio_den_mul_gcd_toNat says that when you multiply the reduced denominator by the greatest common divisor, you recover the original denominator. In symbols: if q is a ratio orbit, then (normalizeRatio q).den.toNat * (gcd q.num.abs q.den).toNat = q.den.toNat.

This is the denominator half of a two-part identity. The companion theorem covers the numerator, with signs handled separately. Together they show that normalization does not change the value of the ratio, only its representation. The framework's library proves this by reducing the claim to the standard property of natural numbers: dividing and multiplying by the same factor cancels out.

In Recognition Science, this identity matters because it underpins the claim that every ratio orbit has a unique reduced form. The theorem ratio_normalization_target states that for every orbit there exists a reduced orbit with the same value and coprime numerator and denominator. The denominator identity is one of the two lemmas that make that existence proof go through.

The theorem does not claim that normalization is unique, nor that it preserves any ordering or metric on orbits. It does not say anything about the physical meaning of the ratio, only about the arithmetic of its representation. The framework's own documentation marks the bridge from this arithmetic to physical claims as open.

theorem normalizeRatio_den_mul_gcd_toNat (q : RatioOrbit) :
    (normalizeRatio q).den.toNat *
      (gcd q.num.abs q.den).toNat = q.den.toNat := by
  unfold normalizeRatio
  exact quotient_mul_divisor_toNat_of_divides
    (n := q.den) (d := gcd q.num.abs q.den)
    (gcd_ne_zero_of_right_ne_zero q.num.abs q.den q.den_ne_zero)
    (gcd_divides_right q.num.abs q.den)

theorem ratio_normalization_target : RatioNormalizationTarget := by
  intro q
  exact ⟨normalizeRatio q, normalizeRatio_crossEq q, normalizeRatio_coprime q⟩

What this page does not claim

The theorem does not claim that normalization is unique. The theorem does not claim anything about the physical meaning of the ratio. The theorem does not establish that every ratio has a reduced form with coprime parts; that is a separate theorem.

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/OrbitEuclidean.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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