Foundation Magnitude Of Mismatch Forces

The forced symmetry

When a comparison takes two things and returns one value, the order of the two things cannot matter. A function that reports the cost of comparing x with y and a different cost for comparing y with x is not one function on a pair; it is two separate functions, one for each direction. The Recognition Science framework formalizes this as a theorem in its machine-checked library: any comparison that factors through the unordered pair is symmetric, meaning C x y = C y x for every x and y.

The framework calls this the magnitude of mismatch. The idea is that a comparison measures how far two items are apart, and a single-valued measure of that distance must give the same answer regardless of which item is named first. The theorem is proved in both directions. If a comparison is single-valued on the unordered pair, it is symmetric. And if it is symmetric, it factors through the unordered pair. The two properties are equivalent, and the framework states this as a single theorem, singleValued_iff_symmetric, with the plain meaning that single-valuedness and symmetry are the same condition.

The framework connects this to a classical principle. In its account, the symmetry of comparison is the operator form of the Aristotelian law of non-contradiction: a thing cannot both be and not be the same in the same respect. The framework argues that any asymmetric reading of comparison does not produce a single binary function but splits into two directional functions. The equality-induced cost, the canonical cost derived from whether two things are equal, is automatically symmetric by this theorem, so it falls into the forced shape without any additional choice.

What the declaration does not claim matters as much as what it proves. The theorem is about symmetry of a comparison operator, given that the operator is single-valued on an unordered pair. It does not say that any particular comparison function is symmetric; it says that if a comparison is a single function on an unordered pair, then it must be symmetric. It does not derive the specific cost function J(x) or any numerical constant. It does not prove that the world must contain comparisons; it proves a conditional statement about any comparison that exists. The framework's own account argues the encoding is structurally necessary, but the theorem itself establishes the equivalence, not the metaphysical necessity.

/-- **Single-valued predication forces symmetry.**

If a comparison operator factors through the unordered pair, then the
order of its arguments does not matter. The asymmetric reading (where
`C x y` and `C y x` are different values) is not a single function on
pairs; it is two distinct directional functions. -/
theorem singleValued_implies_symmetric
    {K Cost : Type*} (C : K → K → Cost)
    (h : SingleValuedOnUnorderedPair C) :
    ∀ x y : K, C x y = C y x := by
  intro x y
  rcases h with ⟨f, hf⟩
  have hxy := hf x y
  have hyx := hf y x
  have hsym : (s(x, y) : Sym2 K) = s(y, x) := Sym2.eq_swap
  rw [hxy, hyx, hsym]

/-- **Conversely: symmetric comparisons factor through unordered pairs.**

Any symmetric function on `K × K` is the lift of a single function on
`Sym2 K`. So symmetry and single-valuedness on the unordered pair are
equivalent. -/
theorem symmetric_implies_factorsThrough
    {K Cost : Type*} (C : K → K → Cost)
    (hsymm : ∀ x y : K, C x y = C y x) :
    SingleValuedOnUnorderedPair C := by
  refine ⟨Sym2.lift ⟨fun a b => C a b, fun a b => hsymm a b⟩, ?_⟩
  intro x y
  simp [Sym2.lift_mk]

/-- **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)

What this page does not claim

The theorem does not prove that any particular comparison function is symmetric; it proves a conditional statement about comparisons that are single-valued on an unordered pair. The theorem does not derive the specific cost function J(x) or any numerical constant such as the golden ratio. The theorem does not establish that comparisons must exist in the world; it establishes an equivalence about any comparison that does exist.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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