IndisputableMonolith.Foundation.LogicAsFunctionalEquation.Canonicality
The Canonicality module supplies the magnitude-of-mismatch interpretation for comparison operators. It encodes five properties: trivial self-match value, symmetry, total determinability, scale invariance, and nontriviality. Researchers formalizing the transition from reality to logic in Recognition Science cite this when building truth-evaluable structures; the module separates the finite-algebra condition for RCL.
claimLet $M$ denote the magnitude-of-mismatch interpretation of a comparison operator on positive ratios, with fields satisfying $M(x,x)=0$, $M(x,y)=M(y,x)$, total continuous determinability for every pair, scale-free invariance $M(\lambda x,\lambda y)=M(x,y)$, and nontriviality $M(x,y)\neq 0$ whenever $x\neq y$.
background
This module belongs to the LogicAsFunctionalEquation namespace and imports DirectProof, which isolates the finite pairwise polynomial closure hypothesis as the precise regularity condition needed to force the Recognition Composition Law family. The magnitude-of-mismatch interpretation is introduced with the five fields listed in the module doc-comment: trivial value at match, symmetry of the unordered pair, total/continuous determinability, scale-free comparison, and nontriviality. The local setting is the translation of logical primitives into functional equations via operative positive-ratio comparisons, prior to the RealityStructure construction.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module supplies the canonical interpretation that feeds the RealityStructure module, which formalises the Reality ⇒ Logic leg of the Logic Functional Equation paper. It ensures the comparison operator has truth-evaluable values by encoding the five mismatch properties, with the finite pairwise polynomial closure kept separate as the condition needed to force RCL.
scope and limits
- Does not incorporate the finite pairwise polynomial closure hypothesis.
- Does not prove the Recognition Composition Law.
- Does not construct the full reality structure.
- Does not address numerical evaluation or specific ratio examples.