IndisputableMonolith.Foundation.MagnitudeOfMismatch
This module defines single-valuedness for comparison operators on unordered pairs and derives its consequences for symmetry and mismatch magnitude. Researchers following the Recognition Science forcing chain from primitive distinctions to logic realizations would cite it to confirm order-independent costs. The module consists of a short chain of lemmas linking the Sym2 factoring property to symmetry and a forced magnitude result.
claimA comparison operator $C : K → K → Cost$ is single-valued on the unordered pair when it factors through the symmetric square: there exists $f : Sym2 K → Cost$ such that $C(x,y) = f({x,y})$ for all $x,y$.
background
The module sits in the Foundation layer and imports PrimitiveDistinction, which supplies the base types K (configurations) and Cost (mismatch measures). Single-valuedness ensures that the cost assignment respects the unordered character of pairs, aligning with the J-cost and defect-distance notions used throughout the Recognition framework. The supplied doc-comment states that operationally this means the cost depends only on the set {x,y} and is therefore independent of presentation order.
proof idea
This is a definition module, no proofs. It introduces the predicate SingleValuedOnUnorderedPair together with four short implications (singleValued_implies_symmetric, symmetric_implies_factorsThrough, singleValued_iff_symmetric, asymmetric_not_singleValued) and closes with the derived statement magnitude_of_mismatch_forced.
why it matters in Recognition Science
The module supplies the magnitude_of_mismatch_forced result that is imported by the root IndisputableMonolith module (exposing the master T0-T8 forcing chain) and by RecognizerInducesLogic (unifying recognition geometry with the law of logic). It therefore closes the symmetry requirement on cost functions before the chain proceeds to J-uniqueness and the eight-tick octave.
scope and limits
- Does not derive the full T0-T8 forcing chain.
- Does not compute numerical constants such as phi or alpha.
- Does not treat higher-dimensional or continuous configuration spaces.
- Does not address Berry creation thresholds or Z_cf values.