Foundation Primitive Recognition Calculus Rational Field On Prcrat Normalized Re

The normalized representative

A rational number can be written in many ways: 1/2, 2/4, 3/6 all name the same value. When a calculation is defined on fractions, a well-built definition must give the same answer for every equivalent form, or the result depends on notation rather than mathematics. The declaration onPRCRat_normalized_representative is the framework's guarantee that its recognition cost, the forced price of a recognition event, satisfies exactly this requirement on its own rational number system.

The framework builds its rationals not from integers but from ratio orbits, classes of signed ratios that carry the same value. Each orbit has a canonical normalized representative, the simplest form of the ratio. The theorem states that applying the cost function to any ratio orbit gives the same result as applying it to that orbit's normalized representative. In symbols, if q is a ratio orbit and normalizeRatio q is its canonical form, then onPRCRat (mk q) = onPRCRat (mk (normalizeRatio q)). The proof is short: both sides map to the same rational value under the system's embedding into the ordinary rationals, and the embedding is injective.

The declaration does not define the cost function itself. It assumes the cost is already defined on ratio orbits and on the quotient of rationals, and it proves a consistency property of that definition. It does not claim that normalization is part of the object's definition; the docstring is explicit that nonzero denominators are positive in the verifier display but are not part of the object definition. The theorem does not compute any numerical cost, does not mention the golden ratio, and does not connect to the forcing chain that derives physical constants.

What the declaration changes is the framework's internal hygiene. A cost function that depended on the chosen representative would be ill-defined on rationals, and every later theorem that uses the cost on rationals would inherit that flaw. This theorem closes that gap at the foundation, so the framework's rational arithmetic and its cost function can be composed freely. It is a small but load-bearing piece of the machine-checked library of formal theorems that the framework uses to keep its derivations honest.

theorem onPRCRat_normalized_representative (q : RatioOrbit) :
    onPRCRat (PRCRat.mk q) =
      onPRCRat (PRCRat.mk (DistinctionNat.normalizeRatio q)) := by
  apply PRCRat.toRat_injective
  rw [onPRCRat_toRat, onPRCRat_toRat, PRCRat.toRat_mk, PRCRat.toRat_mk,
    DistinctionNat.normalizeRatio_toRat]

theorem onPRCRat_normalized_representative (q : RatioOrbit) :
    onPRCRat (PRCRat.mk q) =
      onPRCRat (PRCRat.mk (DistinctionNat.normalizeRatio q)) := by
  apply PRCRat.toRat_injective
  rw [onPRCRat_toRat, onPRCRat_toRat, PRCRat.toRat_mk, PRCRat.toRat_mk,
    DistinctionNat.normalizeRatio_toRat]

theorem onPRCRat_normalized_representative (q : RatioOrbit) :
    onPRCRat (PRCRat.mk q) =
      onPRCRat (PRCRat.mk (DistinctionNat.normalizeRatio q)) := by
  apply PRCRat.toRat_injective
  rw [onPRCRat_toRat, onPRCRat_toRat, PRCRat.toRat_mk, PRCRat.toRat_mk,
    DistinctionNat.normalizeRatio_toRat]

What this page does not claim

The declaration does not define the recognition cost function. The declaration does not compute any numerical cost value. The declaration does not connect to the golden ratio or the forcing chain.

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/RationalField.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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