Foundation Universal Forcing Strict Positive Ratio Strict Positive Ratio Realiza

The strict positive-ratio realization

A positive ratio is a pair of positive numbers, one divided by the other, such as 3/2 or 7/4. In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of comparing two entries in that record. The declaration strictPositiveRatioRealization takes any comparison operator that satisfies the five laws of logic, and builds from it a strict continuous positive-ratio realization, meaning a model where the comparison structure is carried by ratios of positive numbers.

The key result is that this strict positive-ratio arithmetic is canonically the same as the natural numbers. The library constructs an equivalence between the arithmetic forced by the strict positive-ratio realization and the standard arithmetic of counting numbers. This is not a numerical coincidence; it is a structural isomorphism, a bijection that preserves the arithmetic operations. The declaration positiveRatio_arith_equiv_logicNat provides this equivalence, and a companion declaration positiveRatio_strict_equiv_existing shows that the strict version agrees with the previously built positive-ratio wrapper.

In Recognition Science, the framework models comparison as a forced cost, and the five laws of logic (reciprocal symmetry, zero cost at unity, composition, calibration, continuity) force a specific cost function. The strict positive-ratio realization is one concrete model of that forcing: it shows that a strict, continuous, ratio-based comparison operator satisfying those laws yields the same arithmetic as the natural numbers. This is a theorem in the machine-checked library of formal theorems, not a physical measurement or a heuristic guess.

What this declaration does not claim: it does not claim that the physical world uses positive ratios for comparison. It does not claim that the natural numbers are the only possible arithmetic; it shows an equivalence, not a uniqueness. And it does not claim that the five laws of logic are true of actual human reasoning; it shows what follows if they hold. The declaration is a piece of pure mathematics, a structural bridge between a ratio-based comparison model and the familiar counting numbers.

/-- Strict positive-ratio realization from the existing Law-of-Logic package. -/
noncomputable def strictPositiveRatioRealization
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    StrictLogicRealization where
  Carrier := {x : ℝ // 0 < x}
  Cost := ℝ
  zeroCost := inferInstance
  compare := fun x y => C x.1 y.1
  compose := fun x y => ⟨x.1 * y.1, mul_pos x.2 y.2⟩
  one := ⟨1, one_pos⟩
  generator :=
    let γ : ℝ := Classical.choose h.non_trivial
    ⟨γ, (Classical.choose_spec h.non_trivial).1⟩
  identity_law := by
    intro x
    exact h.identity x.1 x.2
  non_contradiction_law := by
    intro x y
    exact h.non_contradiction x.1 y.1 x.2 y.2
  excluded_middle_law := ExcludedMiddle C
  composition_law := RouteIndependence C
  invariance_law := ScaleInvariant C
  nontrivial_law := by
    exact (Classical.choose_spec h.non_trivial).2

/-- Strict positive-ratio forced arithmetic is canonically `LogicNat`. -/
noncomputable def positiveRatio_arith_equiv_logicNat
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    (StrictLogicRealization.arith (strictPositiveRatioRealization C h)).peano.carrier
      ≃ ArithmeticFromLogic.LogicNat :=
  (StrictLogicRealization.toLightweight (strictPositiveRatioRealization C h)).orbitEquivLogicNat

/-- The strict-derived lightweight realization has the same forced arithmetic
as the existing positive-ratio lightweight wrapper. -/
noncomputable def positiveRatio_strict_equiv_existing
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    (StrictLogicRealization.arith (strictPositiveRatioRealization C h)).peano.carrier
      ≃ (UniversalForcing.arithmeticOf (LogicRealization.ofPositiveRatioComparison C h)).peano.carrier :=
  ArithmeticOf.equivOfInitial
    (StrictLogicRealization.arith (strictPositiveRatioRealization C h))
    (UniversalForcing.arithmeticOf (LogicRealization.ofPositiveRatioComparison C h))

What this page does not claim

The declaration does not claim that physical comparison actually uses positive ratios. It does not claim that the natural numbers are the only possible arithmetic; it shows an equivalence, not a uniqueness. It does not claim that the five laws of logic are true of actual human reasoning; it shows what follows if they hold.

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/UniversalForcing/Strict/PositiveRatio.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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