IndisputableMonolith.Foundation.LogicAsFunctionalEquation.DirectProof
This module defines the operative positive-ratio comparison as a continuous nontrivial operator on positive reals obeying identity, symmetry, and scale invariance. Researchers tracing the direct path from mismatch magnitude to the Recognition Composition Law cite it when establishing canonical operator conditions. The module imports the parent LogicAsFunctionalEquation setup and exports the core definition plus supporting lemmas used by downstream canonicality and counterexample arguments.
claimA continuous nontrivial $C: (0,∞)×(0,∞)→ℝ$ is an operative positive-ratio comparison when it satisfies $C(x,x)=0$, $C(x,y)=-C(y,x)$, and $C(x,y)=C(λx,λy)$ for every $λ>0$.
background
The module belongs to the Foundation.LogicAsFunctionalEquation hierarchy and imports the base functional-equation machinery. It introduces the operative positive-ratio comparison, which encodes the magnitude of mismatch between positive ratios under the Recognition Composition Law. The parent module supplies the J-uniqueness (T5) and forcing-chain context (T0–T8) that motivate reading comparison operators as defect measures. The definition requires continuity, the identity condition, single-valued symmetry, and scale invariance on positive arguments.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the operative positive-ratio comparison that feeds the canonicality argument, which states that once a comparison operator is read as a magnitude of mismatch the operator-level conditions become the canonical structural content of that reading. It also supports the quartic-log counterexample module. The definition fills the direct-proof step of the paper’s logic functional equation, connecting to the RCL and the eight-tick octave in the Recognition Science framework.
scope and limits
- Does not derive the J-cost function or phi-ladder from the operative comparison.
- Does not treat negative ratios or non-positive arguments.
- Does not prove uniqueness of the comparison operator.
- Does not address the quartic-log counterexample construction itself.