IndisputableMonolith.Foundation.MagnitudeOfMismatch
The MagnitudeOfMismatch module introduces single-valued comparison operators that factor through unordered pairs for costs arising from distinctions. Researchers building the forcing chain or recognizer-induced logic cite it to establish order-independent mismatch magnitudes. The module supplies the core definition plus short equivalence lemmas between single-valuedness and symmetry.
claimA comparison operator $C : K → K → Cost$ is single-valued on the unordered pair when it factors through the type of unordered pairs $Sym2 K$, i.e., there exists $f : Sym2 K → Cost$ such that $C(x,y) = f(s(x,y))$ for the unordered pair $s(x,y)$.
background
The module imports Mathlib and PrimitiveDistinction. It defines SingleValuedOnUnorderedPair together with the auxiliary notions singleValued_implies_symmetric, symmetric_implies_factorsThrough, singleValued_iff_symmetric, asymmetric_not_singleValued, equalityCost_singleValued, forces, and magnitude_of_mismatch_forced. These capture the requirement that mismatch cost between two elements of K is independent of presentation order and therefore descends to the symmetric square.
proof idea
This is a definition module. The listed lemmas are one-line wrappers that apply the definition of factoring through Sym2 together with the definition of symmetry on the codomain Cost.
why it matters in Recognition Science
The module supplies the magnitude_of_mismatch_forced result used by the root IndisputableMonolith module (master forcing-chain theorem T0–T8) and by RecognizerInducesLogic (unification of recognition geometry with the law of logic). It therefore supplies the order-independence step required before the J-cost and phi-ladder constructions can be applied to mismatch magnitudes.
scope and limits
- Does not specify the concrete form of the Cost type or the J-function.
- Does not prove existence of single-valued operators for arbitrary K.
- Does not connect the mismatch magnitude to the phi-ladder or the eight-tick octave.
- Does not derive numerical bounds on mismatch values.