Foundation Universal Forcing Strict Positive Ratio

The strict positive-ratio realization

The positive-ratio realization is a way of building a ledger, a discrete record of events, from a comparison operator that satisfies the five laws of logic. The strict version adds a continuity condition: the comparison function must be strictly positive and continuous, ruling out pathological jumps. This construction creates that strict realization directly from the laws, without any extra assumptions.

The central result is that the arithmetic forced by this strict positive-ratio realization is canonically the natural numbers. In plain language: if you start with a continuous, strictly positive comparison operator obeying the five laws, the counting structure that emerges is exactly the usual numbers 0, 1, 2, 3, and so on. The formal library proves this by showing the strict realization's arithmetic is equivalent to the standard natural-number structure.

This matters because it shows the laws of logic alone, with continuity added, pin down arithmetic uniquely. The strict version also matches the existing positive-ratio wrapper: the arithmetic forced by the strict realization is the same as that forced by the non-strict version. So the continuity condition does not change the outcome; it just makes the construction more direct.

In Recognition Science, this is a step in the chain that forces structure from the cost of recognition. The strict positive-ratio realization is the cleanest path from the laws of logic to the natural numbers, and from there to the rest of the framework's derived constants and dimensions.

/-- 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

This construction does not derive the fine-structure constant or any specific physical constants. The equivalence to natural numbers does not prove the Riemann Hypothesis or any other open conjecture. The continuity condition is an added assumption, not itself forced by the five laws.

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)

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