positiveRatio_hasIdentityStep
A comparison operator on positive reals obeying the Aristotelian constraints plus scale invariance produces a logic realization with non-zero comparison cost at the identity. Researchers embedding logic into arithmetic cite this to confirm the continuous positive-ratio model is non-trivial. The proof is a direct one-line application of the general non-triviality result for any logic realization.
claimLet $C$ be a comparison operator on positive reals that satisfies the laws of logic. The logic realization constructed by embedding $C$ into the setting-independent interface has a non-zero cost when comparing some positive element to the zero element.
background
ComparisonOperator is the type of functions from pairs of positive reals to reals that assign a comparison cost. SatisfiesLawsOfLogic is the structure requiring the four Aristotelian constraints (identity, non-contradiction, excluded middle) together with scale invariance and route independence. ofPositiveRatioComparison maps any such $C$ to a LogicRealization whose carrier is the positive reals and whose cost type is the reals. hasIdentityStep is the predicate asserting existence of a carrier element $x$ such that the comparison of $x$ to zero is nonzero; this predicate supplies the data from which arithmetic is extracted. The module supplies a setting-independent interface into which continuous positive-ratio, discrete, and categorical logic settings can be mapped for the universal forcing program.
proof idea
The proof is a one-line wrapper that applies hasIdentityStep_of_nontrivial to the realization produced by ofPositiveRatioComparison.
why it matters in Recognition Science
This theorem verifies that the continuous positive-ratio model satisfies the identity-step predicate required by the common LogicRealization interface. It thereby supports extraction of arithmetic from logic in the forcing chain and confirms non-triviality for the positive-ratio case. No downstream uses are recorded in the current graph, but the result closes a verification step for one of the concrete realizations admitted by the setting-independent interface.
scope and limits
- Does not prove injectivity of the orbit interpretation.
- Does not treat discrete proposition or categorical settings.
- Does not derive the full arithmetic structure from the realization.
- Does not connect to the phi-ladder or mass formulas.
formal statement (Lean)
152theorem positiveRatio_hasIdentityStep
153 (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
154 (ofPositiveRatioComparison C h).hasIdentityStep :=
proof body
Term-mode proof.
155 hasIdentityStep_of_nontrivial _
156
157/-- The continuous positive-ratio orbit interpretation is injective. -/