positiveRatio_arith_equiv_logicNat
plain-language theorem explainer
The definition shows that arithmetic derived from a strict positive-ratio realization of the laws of logic is canonically equivalent to LogicNat. Foundation researchers in Recognition Science cite this when connecting functional equations to Peano arithmetic. It is realized as a direct one-line wrapper invoking the orbit equivalence on the lightweight realization.
Claim. Let $C : ℝ → ℝ → ℝ$ be a comparison operator and let $h$ witness that $C$ satisfies the laws of logic. Then the Peano carrier of the arithmetic structure extracted from the strict positive-ratio realization of $C$ is equivalent to the inductive type LogicNat of natural numbers forced by logic.
background
Strict positive-ratio realizations are built directly from a ComparisonOperator satisfying the laws of logic. ComparisonOperator is the type of cost functions on positive reals, while SatisfiesLawsOfLogic is the structure collecting the four Aristotelian constraints, scale invariance, and non-triviality. LogicNat is the inductive type with constructors identity (the zero-cost multiplicative identity) and step, generating the orbit {1, γ, γ², …} as the smallest subset of positive reals closed under multiplication by the generator.
proof idea
This is a one-line wrapper that applies the orbitEquivLogicNat lemma from StrictLogicRealization.toLightweight applied to the strictPositiveRatioRealization of C and h.
why it matters
The definition confirms that strict positive-ratio forced arithmetic coincides exactly with the logic-derived natural numbers, closing a foundational link between the functional equation and arithmetic. It supports the derivation of Peano structure from the Recognition Composition Law and the forcing chain steps T5–T8. No open questions are directly addressed.
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