IndisputableMonolith.Foundation.LogicAsFunctionalEquation.Canonicality
This module defines the magnitude-of-mismatch interpretation for comparison operators that underpins the Recognition Composition Law derivation. It encodes five properties: trivial self-match value, symmetry, continuous determinability for positive pairs, scale invariance, and nontriviality, while isolating finite pairwise polynomial closure as a separate hypothesis. Researchers citing the Logic Functional Equation paper use it to bridge direct proofs to truth-evaluable reality structures. The module consists of core definitions and translation
claimA comparison operator admits a magnitude-of-mismatch interpretation when there exists $M$ satisfying $M(x,x)=0$, $M(x,y)=M(y,x)$, $M$ continuous and total on positive ratios, $M$ invariant under positive scaling, and $M$ nontrivial.
background
The module sits between the DirectProof module, which isolates finite pairwise polynomial closure as the regularity condition forcing the RCL family, and the downstream RealityStructure module. It introduces the MagnitudeOfMismatch structure whose five fields match the doc-comment: trivial value at match, symmetry of the unordered pair, total/continuous determinability, scale-free comparison, and nontriviality. Upstream doc-comment states: 'This module gives the paper-facing operative comparison wrapper around the already-formalised translation theorem.'
proof idea
This is a definition module, no proofs. It supplies the MagnitudeOfMismatch structure together with translation lemmas (mismatch_to_operative, canonical_identity, canonical_non_contradiction, canonical_excluded_middle, canonical_scale_invariance, canonicality_of_encoding, rcl_from_canonical_mismatch_encoding) that convert the five properties into operative comparisons.
why it matters in Recognition Science
The module supplies the canonicality layer required by the RealityStructure module, which formalises the Reality ⇒ Logic leg: 'The starting point is a comparison operator whose values are truth-evaluable: self-comparison has a trivial value, reordering is single-valued, every positive pair has a determinate continuous comparison, and composite comparisons have a determinate finite pairwise combiner.' It separates the interpretation from the polynomial-closure hypothesis that forces RCL.
scope and limits
- Does not incorporate the finite pairwise polynomial closure hypothesis.
- Does not derive the Recognition Composition Law.
- Does not treat non-positive ratios or multi-operand comparisons.
- Does not supply numerical evaluations or concrete examples.