IndisputableMonolith.Foundation.LogicAsFunctionalEquation.PositiveRatioForcing
This module defines ratioCost as the ratio-derived cost of two-argument comparisons and proves that scale-free comparison factors through positive ratios to force the RCL family. Researchers formalizing the Recognition Science derivation of physics from functional equations cite it as the bridge between finite logical comparison and the headline theorem. The structure imports FiniteLogicalComparison and establishes uniqueness plus equivalence properties for the ratio cost.
claimLet $\mathrm{ratioCost}(x,y)$ be the ratio-derived cost of a two-argument comparison. Scale-free comparison factors through positive ratios, ratio factors are unique, $\mathrm{ratioCost}$ equals the derived cost, and positive ratios are forced by scale-free comparison.
background
The module belongs to the LogicAsFunctionalEquation package. It imports FiniteLogicalComparison, whose documentation states that finite logical comparison on positive ratios forces the RCL family and that the finite-pairwise-polynomial condition is named as the finite algebraic content of logical comparison. Key objects are ratioCost together with the theorems scale_free_comparison_factors_through_ratio, ratio_factor_unique, ratioCost_eq_derivedCost, and positive_ratio_is_forced_by_scale_free_comparison. The setting is the sharpened theorem on finite logical comparison.
proof idea
The module contains the definition of ratioCost, the equivalence ratioCost_eq_derivedCost, and the supporting theorems on factoring, uniqueness, and forcing. It applies the upstream results from FiniteLogicalComparison to establish the positive-ratio properties without further heavy machinery.
why it matters in Recognition Science
This module supplies the positive-ratio forcing step that feeds MainTheorem. The downstream module collects the formal chain closest to the paper headline: scale-free comparison factors through positive ratios; no-hidden-state finite comparison gives counted-once composition; counted-once finite logical comparison forces the RCL family. It advances the Recognition Science program by linking finite comparison to the functional equation RCL.
scope and limits
- Does not extend results to non-positive arguments.
- Does not remove the finite-pairwise-polynomial condition.
- Does not treat infinite comparisons.
- Does not claim completeness beyond the RCL family.