Foundation Magnitude Of Mismatch Equality Cost Single Valued

The forced symmetry of comparison

In mathematics, a function of two arguments is symmetric when swapping the arguments leaves the result unchanged: f(x, y) equals f(y, x). A comparison operator that measures the cost or magnitude of mismatch between two entities is a function of this kind. The classical question is whether such an operator can be asymmetric, giving a different value when the order is reversed, while still being a single, well-defined function.

The answer, formalized in the machine-checked library of formal theorems, is no. A comparison that is single-valued on the unordered pair, meaning there is one function f such that the cost C x y equals f applied to the set {x, y}, must be symmetric. The proof is direct: if C factors through the unordered pair, then C x y and C y x both equal the same value f({x, y}), so they are equal. Conversely, any symmetric comparison factors through the unordered pair. The two properties are equivalent, and this equivalence is the theorem singleValued_iff_symmetric.

This equivalence has a sharp consequence for the equality-induced cost, the cost that measures how far two entities are from being equal. The framework proves that this cost is symmetric, via the non-contradiction principle that C x y equals C y x. By the equivalence, it follows immediately that the equality-induced cost is single-valued on the unordered pair. This is the declaration equalityCost_singleValued: it establishes that the canonical cost derived from equality is automatically in the single-valued shape, with no further interpretive choice needed.

In Recognition Science, this result removes a degree of freedom. The asymmetric reading of comparison, where C x y and C y x are allowed to differ, does not produce a single binary function; it produces two distinct directional functions. The framework's library shows that any such asymmetry splits the operator, and the single-valued reading is the unique one consistent with treating comparison as a predicate on a distinguished pair. The theorem is not a claim about any specific cost function, but about the structure of comparison itself.

What this does not claim is broader. It does not say that all comparisons in all contexts must be symmetric; it says that if a comparison is single-valued on the unordered pair, symmetry is forced. It does not derive the specific form of the equality-induced cost, only its symmetry and single-valuedness. And it does not claim that asymmetry is impossible, only that an asymmetric comparison is not a single function on pairs but a pair of directional functions.

/-- **Equivalence: single-valuedness on the unordered pair is symmetry.** -/
theorem singleValued_iff_symmetric
    {K Cost : Type*} (C : K → K → Cost) :
    SingleValuedOnUnorderedPair C ↔ ∀ x y : K, C x y = C y x :=
  ⟨singleValued_implies_symmetric C,
   fun h => symmetric_implies_factorsThrough C h⟩

/-- The equality-induced cost is single-valued on the unordered pair. -/
theorem equalityCost_singleValued (K : Type*) (weight : ℝ) :
    SingleValuedOnUnorderedPair (equalityCost K weight) :=
  symmetric_implies_factorsThrough (equalityCost K weight)
    (non_contradiction_from_equality K weight)

/-- The negation: if `C` is asymmetric on at least one pair, it cannot
factor through the unordered pair. Single-valuedness fails the moment the
two orderings give different values.

This is the Lean form of Theorem 3 of the companion paper: asymmetry
splits a single binary function into two directional functions. -/
theorem asymmetric_not_singleValued
    {K Cost : Type*} (C : K → K → Cost)
    (h : ∃ x y : K, C x y ≠ C y x) :
    ¬ SingleValuedOnUnorderedPair C := by
  rintro hSV
  rcases h with ⟨x, y, hxy⟩
  exact hxy (singleValued_implies_symmetric C hSV x y)

What this page does not claim

This does not claim that all comparison operators must be symmetric in every context. This does not derive the specific numerical form of the equality-induced cost. This does not claim that asymmetry is impossible, only that it is not a single function on pairs.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/MagnitudeOfMismatch.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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