Foundation Primitive Recognition Calculus Quotient Selection Obs Equiv Refl

The reflexivity of indistinguishability

In mathematics, an equivalence relation is a way of saying that two things are the same in some specific respect. The relation is reflexive if everything is related to itself. The theorem obsEquiv_refl establishes exactly this for observational equivalence: for any state x and any family of observables F, the state x is observationally equivalent to itself. The proof is immediate from the definition, since every observable f in F satisfies f x = f x by the fundamental law of equality.

Observational equivalence is defined by the framework as the relation holding between two states x and y when every observable in the family F returns the same value on both. The theorem obsEquiv_refl is the first of three properties that together make this relation an equivalence relation, alongside symmetry (if x is equivalent to y then y is equivalent to x) and transitivity (if x is equivalent to y and y to z, then x to z). These three properties are packaged together in the definition obsSetoid, which turns observational equivalence into a mathematical structure called a setoid, a set equipped with an equivalence relation.

The point of this construction is to build a quotient: a new space where states that are observationally indistinguishable are identified. The theorem forced_iff states that two states map to the same element of this quotient if and only if they are observationally equivalent. The reflexivity result is a necessary foundation for this quotient to be well-defined, since it guarantees that the relation used to form it is a genuine equivalence relation.

In Recognition Science, the quotient is called the physically forced quotient, and it represents the collapse of the state space under indistinguishability. The reflexivity theorem is a small but essential piece of this structure. It does not claim that any two distinct states are equivalent, nor that observables are capable of distinguishing all states. It merely asserts the minimal logical requirement that a state is always the same as itself, a property so basic that it often goes unstated in informal mathematics but must be explicitly proved in a machine-checked library of formal theorems.

This result is part of a larger framework that derives physical structure from the forced cost of recognition. The reflexivity of observational equivalence is a prerequisite for the quotient construction, which in turn is a prerequisite for the framework's treatment of gauge symmetry. The theorem itself is elementary, but it is a load-bearing element in the formal architecture: without it, the setoid structure would not exist, and the quotient could not be formed.

theorem obsEquiv_refl (F : Set (X → C)) (x : X) : ObsEquiv F x x := fun _ _ => rfl

/-- Two states are observationally equivalent under the observable family `F`
when every observable in `F` returns the same value on them. -/
def ObsEquiv (F : Set (X → C)) (x y : X) : Prop := ∀ f ∈ F, f x = f y

/-- **The quotient is exactly the indistinguishability collapse.** Two states map
to the same physical class iff no admissible observable separates them. The
forced quotient adds no identifications beyond indistinguishability and omits
none. -/
theorem forced_iff (F : Set (X → C)) (x y : X) :
    proj F x = proj F y ↔ ObsEquiv F x y :=
  Quotient.eq

What this page does not claim

The theorem does not claim that any two distinct states are observationally equivalent. The theorem does not claim that observables are capable of distinguishing all states. The theorem does not establish that the quotient is nontrivial.

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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:

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expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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