Foundation Universal Forcing Strict Positive Ratio Positive Ratio Strict Equiv E

The arithmetic equivalence

The natural numbers are the counting numbers 0, 1, 2, 3, and so on. In the Recognition Science framework, a ledger, a discrete record of recognition events, forces an arithmetic structure onto those events. The declaration positiveRatio_strict_equiv_existing states that two separate ways of building that arithmetic from a positive-ratio comparison operator produce the same natural numbers. One route starts from a strict continuous realization; the other starts from the existing positive-ratio wrapper. Both end at the same carrier set, the same objects that behave like the counting numbers.

The equivalence is a theorem in the framework's machine-checked library of formal theorems. It holds for any comparison operator that satisfies the five laws of logic: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. Under those conditions, the two constructions are provably isomorphic. The declaration is a def, a definition that packages a proof of this isomorphism. It does not add new axioms or new content; it shows that the strict path and the existing path agree.

What the declaration does not claim is broader. It does not say that the natural numbers are the only arithmetic the framework can produce, nor that the positive-ratio comparison operator is the only one satisfying the laws. It does not claim that the isomorphism extends to the full structure of arithmetic, only to the carrier set of the Peano model. It does not assert that the strict realization is unique, only that it agrees with the existing one. The declaration is a bridge, not a foundation stone.

/-- 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))

/-- 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 the natural numbers are the only arithmetic the framework can produce. It does not claim the positive-ratio comparison operator is the only one satisfying the laws of logic. It does not assert that the strict realization is unique.

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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