Foundation Observable Floor Witness Observable Iff Bare For Eq

The equality special case

In mathematics, a setoid is a carrier set equipped with an equivalence relation, a rule for which elements count as the same. The Recognition Science framework builds its notion of a ledger, a discrete record of events, on a primitive floor: a pair of states that are not identified by some supplied relation. The declaration observable_iff_bare_for_eq, a theorem in the framework's machine-checked library of formal theorems, establishes the simplest case of that floor. When the supplied relation is plain equality, the observable floor exists exactly when there are two distinct elements. In symbols: ObservableFloorWitness K (fun x y => x = y) ↔ ∃ x y : K, x ≠ y. Two states are observably distinct in this special case precisely when they are unequal as terms.

The theorem is intentionally thin. It says nothing about what happens with a more interesting relation, such as a physical gauge equivalence that identifies many representatives as one state. The framework's library also proves the complementary fact: bare inequality does not imply observable distinguishability for an arbitrary relation. The counterexample is the indiscrete relation on the real numbers, where every pair is observationally equivalent, yet 0 and 1 are unequal. So the equality special case is a boundary, not the general story. The general story lives in the quotient-aware floor, where physical distinguishability means two representatives are not identified by the chosen equivalence relation, and a quotient carrier is non-singleton exactly when such a distinction exists.

What this declaration changes is the vocabulary of the framework's foundations. It separates two notions that an earlier audit found conflated: raw type-theoretic inequality and physical observability. The equality case recovers the legacy bare-distinguishability floor, so nothing breaks, but the framework now names the relation explicitly. The theorem is a bridge, not a destination. It certifies that the old floor is a special case of the new one, and it leaves the physical content where it belongs: in choosing the observational or gauge relation.

/-- Raw bare distinguishability is the equality-relation special case of an
observable floor. -/
theorem observable_iff_bare_for_eq (K : Type*) :
    ObservableFloorWitness K (fun x y => x = y) ↔ ∃ x y : K, x ≠ y :=
  Iff.rfl

/-- Bare inequality does not imply observable distinguishability for an
arbitrary observational relation. Take the indiscrete relation on `ℝ`, where
every pair is observationally equivalent. -/
theorem bare_distinction_does_not_imply_observable_distinction :
    ∃ (K : Type) (r : K → K → Prop),
      (∃ x y : K, x ≠ y) ∧ ¬ ObservableFloorWitness K r := by
  refine ⟨ℝ, (fun _ _ => True), ⟨?_, ?_⟩⟩
  · exact ⟨0, 1, by norm_num⟩
  · rintro ⟨x, y, hxy⟩
    exact hxy trivial

/-- A quotient carrier is non-singleton exactly when the original carrier has
an observable distinction relative to the quotienting setoid.

This is the quotient-aware repair to the raw-inequality objection: physical
distinguishability lives in `Quotient s`, or equivalently in pairs of
representatives not identified by `s.r`. -/
theorem quotient_nontrivial_iff_observableFloor
    {K : Type*} (s : Setoid K) :
    (∃ a b : Quotient s, a ≠ b) ↔ ObservableFloorWitness K s.r := by
  constructor
  · rintro ⟨a, b, hne⟩
    refine Quotient.inductionOn₂ a b ?_ hne
    intro x y hne'
    refine ⟨x, y, ?_⟩
    intro hxy
    exact hne' (Quotient.sound hxy)
  · rintro ⟨x, y, hxy⟩
    refine ⟨Quotient.mk s x, Quotient.mk s y, ?_⟩
    intro hq
    exact hxy (Quotient.exact hq)

What this page does not claim

The theorem does not claim that raw inequality ever suffices for observability under a non-trivial relation. It does not choose any particular observational relation for physical theories.

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/ObservableFloorWitness.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.

Derived articles

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