Foundation Primitive Recognition Calculus Orbit Euclidean Normalize Ratio Num Mu

The normalization identity

In ordinary arithmetic, a ratio like 6/4 can be written in many equivalent ways: 3/2, 12/8, or 30/20. To pick one canonical representative, one divides both the numerator and denominator by their greatest common divisor (GCD), the largest number that divides both evenly. For 6 and 4, the GCD is 2, giving the reduced form 3/2. This reduction is so familiar that its correctness is rarely questioned.

The Recognition Science framework builds its foundational mathematics on discrete counting objects called DistinctionNat, which track how many times a recognition event has occurred. These are not the natural numbers of conventional arithmetic; they are a separate type with their own operations, and any claim about them must be proved from their definitions. The framework's library, a machine-checked collection of formal theorems, contains a function normalizeRatio that takes any ratio built from these objects and returns a reduced one. The theorem normalizeRatio_num_mul_gcd_toInt states a core identity: when the reduced numerator is multiplied by the GCD of the original numerator's absolute value and the original denominator, the result equals the original numerator. In symbols, for any ratio q, (normalizeRatio q).num.toInt * (gcd q.num.abs q.den).toNat = q.num.toInt.

This identity is the numerator half of a two-part proof that normalizeRatio preserves the ratio's value. The companion theorem normalizeRatio_den_mul_gcd_toNat establishes the same for the denominator: the reduced denominator times the GCD equals the original denominator. Together, they show that the reduced ratio represents the same fraction as the original, just with common factors divided out. The library also proves that the reduced numerator and denominator have no common factor, meaning the reduction is complete. These results are tagged THEOREM, meaning they are fully proved in the machine-checked library from the framework's own axioms, with no unverified steps.

What this theorem does not claim is equally important. It does not claim that the framework's normalization matches any particular conventional arithmetic result, though the framework's definitions are designed to mirror them. It does not claim that the GCD computation is efficient, only that it terminates and gives the correct answer. And it does not claim anything about the physical meaning of the reduced ratio; that is a separate question about how these objects are interpreted in the broader framework. The theorem is a piece of internal mathematical infrastructure, a guarantee that a bookkeeping operation does what it says.

theorem normalizeRatio_num_mul_gcd_toInt (q : RatioOrbit) :
    (normalizeRatio q).num.toInt *
      ((gcd q.num.abs q.den).toNat : ℤ) = q.num.toInt := by
  unfold normalizeRatio
  exact signedQuotient_mul_divisor_toInt_of_divides
    q.num (gcd q.num.abs q.den)
    (gcd_ne_zero_of_right_ne_zero q.num.abs q.den q.den_ne_zero)
    (gcd_divides_left q.num.abs q.den)

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 normalizeRatio_coprime (q : RatioOrbit) :
    coprime (normalizeRatio q).num.abs (normalizeRatio q).den := by
  rw [coprime_iff_nat_coprime]
  unfold normalizeRatio
  rw [signedQuotient_abs_toNat, quotient_toNat]
  have hgpos : 0 < (gcd q.num.abs q.den).toNat := by
    rw [gcd_toNat]
    apply Nat.gcd_pos_of_pos_right
    exact Nat.pos_of_ne_zero (by
      intro hzero
      apply q.den_ne_zero
      apply toNat_inj
      rw [hzero, toNat_zero])
  have hgposNat : 0 < Nat.gcd q.num.abs.toNat q.den.toNat := by
    rw [← gcd_toNat]
    exact hgpos
  rw [gcd_toNat]
  exact Nat.coprime_div_gcd_div_gcd
    (m := q.num.abs.toNat) (n := q.den.toNat) hgposNat

What this page does not claim

The theorem does not claim that the framework's normalization matches any particular conventional arithmetic result. The theorem does not claim that the GCD computation is efficient, only that it terminates and gives the correct answer. The theorem does not claim anything about the physical meaning of the reduced ratio.

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)

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.

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