Foundation Magnitude Of Mismatch Magnitude Of Mismatch Forced

The forced symmetry

A comparison between two things can be ordered: you might compare x to y, or y to x. The framework's ledger, a discrete record of events, treats every comparison as a single entry. The declaration magnitude_of_mismatch_forced proves that if such a comparison returns one value regardless of order, then that value must be symmetric: the cost of comparing x to y equals the cost of comparing y to x. This is not a choice about how to interpret a cost; it is a structural consequence of asking for a single answer.

The proof runs in both directions. If a comparison operator factors through the unordered pair, meaning it depends only on the set {x, y} and not on which element is named first, then it is symmetric. Conversely, any symmetric comparison can be written as a function on the unordered pair. The two conditions are equivalent, and the equivalence is what the declaration establishes. An asymmetric reading, where C x y and C y x differ, is not a single comparison at all; it is two distinct directional functions pretending to be one.

In Recognition Science, this symmetry is the operator form of the principle of non-contradiction: a thing cannot be both more and less mismatched than another in the same comparison. The declaration shows that this principle is not an extra assumption layered onto the cost structure. It is forced by the act of making a single-valued comparison. The equality-induced cost, the canonical cost that arises from equality itself, already has this shape automatically, with no further choice.

The declaration does not claim that all costs are symmetric, only that a single-valued comparison must be. It does not derive the specific form of the cost function J(x) = (x + 1/x)/2 - 1; that is a separate theorem. It also does not assert that any particular physical process is symmetric. The claim is narrower and precise: single-valuedness on an unordered pair and symmetry are the same property, and the framework's machine-checked library of formal theorems has verified this equivalence.

What this changes is the space of possible cost functions. If a theory wants a comparison that yields one number for a pair, it cannot also allow that number to depend on which item is named first. The directed revision reading, where correcting x to y costs something different from correcting y to x, requires two separate functions. The framework's declaration closes that door for single-valued comparisons, leaving symmetry as the only consistent option.

/-- **Magnitude-of-Mismatch theorem (combined).**

On any carrier and cost type, a comparison operator is single-valued on
the unordered pair if and only if it satisfies (L2) Non-Contradiction
in operator form (`C x y = C y x`). The two are equivalent statements;
neither is more primitive than the other. The asymmetric reading of
Non-Contradiction does not produce a single comparison operator. -/
theorem magnitude_of_mismatch_forced
    {K Cost : Type*} (C : K → K → Cost) :
    SingleValuedOnUnorderedPair C ↔ ∀ x y : K, C x y = C y x :=
  singleValued_iff_symmetric C

/-- 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)

/-- 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

This declaration does not derive the specific form of the cost function J(x). This declaration does not claim that all possible cost functions are symmetric, only that single-valued ones must be. This declaration does not assert that any particular physical process is symmetric.

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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