Foundation Primitive Recognition Calculus Rational Field Positive Normalize

The positivity guarantee

A ratio is positive when its numerator is not balanced to zero by the framework's signed orbit structure and its denominator is a nonzero orbit position. The declaration positive_normalize proves that this positivity survives normalization: if a ratio is positive, then its normalized form is positive too. Normalization is the process of reducing a ratio to a canonical representative, the kind of simplification one performs to compare two fractions or to compute with them reliably.

The proof is short and direct. The framework defines positivity on its ratio orbits, then proves that a ratio is positive exactly when its rational value is greater than zero. The normalization step preserves this rational value, so the positivity condition carries through unchanged. A companion theorem, positive_not_zero, records that a positive ratio cannot have a rational value of zero. These results are part of a larger construction in which the framework builds its rational numbers, the PRC rationals, as a quotient of ratio orbits and then proves the standard field laws for them.

In Recognition Science, the framework models recognition events as discrete entries in a ledger, and its cost function J(x) = (x + 1/x)/2 - 1 is proved to be the unique function satisfying five plain conditions. The rational numbers built here are the arithmetic substrate on which that cost function and its consequences rest. The positivity guarantee is a technical lemma, but it is the kind of lemma that makes the whole edifice trustworthy: it ensures that a basic property is invariant under the simplification steps the framework uses throughout.

The declaration does not claim that positivity is part of the definition of a ratio. A ratio can be negative, zero, or positive; positivity is a property some ratios have. It also does not claim that the framework's rational numbers are the same as the ordinary rational numbers of classical mathematics, though their field laws match. The theorem is about the framework's own objects, built from its own primitives, and it is proved in its machine-checked library of formal theorems.

theorem positive_normalize {q : RatioOrbit}
    (h : positive q) : positive (DistinctionNat.normalizeRatio q) := by
  rw [positive_iff_toRat_pos, DistinctionNat.normalizeRatio_toRat]
  exact (positive_iff_toRat_pos q).mp h

/-- PRC-native positivity for a ratio orbit: positive signed numerator over a
nonzero orbit denominator. The denominator is an orbit position, so nonzero
means positive in the verifier display but is not part of the object definition. -/
def positive (q : RatioOrbit) : Prop :=
  SignedOrbit.nonneg q.num ∧ ¬ SignedOrbit.balanced q.num SignedOrbit.zero

theorem positive_not_zero {q : RatioOrbit}
    (h : positive q) : q.toRat ≠ 0 := by
  exact ne_of_gt ((positive_iff_toRat_pos q).mp h)

What this page does not claim

Positivity is not part of the definition of a ratio; ratios can be negative, zero, or positive. The framework's rational numbers are not claimed to be identical to the ordinary rational numbers, only to satisfy the same field laws. The declaration does not establish that every ratio has a positive normalized form, only that positivity is preserved by normalization.

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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