Foundation Primitive Recognition Calculus Quotient Selection Forced Iff

The forced quotient

In the Recognition Science framework, a fundamental question is when two states of a system should be considered the same. The answer, proved in its machine-checked library of formal theorems, is that two states are physically identical exactly when no admissible observable can tell them apart. This is the content of the declaration forced_iff: the projection of states onto their physical classes is an equivalence precisely when the states are observationally equivalent under the chosen family of observables.

The framework builds this from a simple definition. An observable is a function from states to values, and two states are observationally equivalent when every observable in the family returns the same value on both. This relation is reflexive, symmetric, and transitive, so it forms a setoid. The physically forced quotient is then the state space modulo this indistinguishability relation. The theorem forced_iff states that two states map to the same physical class if and only if they are observationally equivalent. This is not a new identification imposed by hand; it is exactly the collapse that indistinguishability forces.

The theorem also carries a universal property: every admissible observable descends to the quotient. That is, for any observable in the family, there is a function on physical classes that agrees with the observable on every state. The quotient loses no observable information. When the observable family separates states, meaning no two distinct states are observationally equivalent, the projection is injective and the quotient is trivial. Gauge identification appears precisely when the observables fail to separate.

In Recognition Science, this result is the foundation of gauge from indistinguishability. The framework models physical states as equivalence classes under all possible recognition acts. The quotient is not a native operation of distinction; it is forced exactly by the absence of a distinguishing recognition act. The theorem establishes that the only identifications made are those required by the observables, and none are omitted.

What forced_iff does not claim is any specific choice of observable family. The theorem is parameterized by an arbitrary set of observables, and the physical content depends entirely on which observables are admissible. It also does not claim that the quotient is physically meaningful without specifying that family. The theorem is a structural result about how quotients behave under observational equivalence, not a statement about which observables exist in any particular physical theory.

/-- **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

/-- **Universal property.** Every admissible observable descends to the quotient:
there is a function on physical classes agreeing with the observable on every
state. The quotient loses no observable information. -/
theorem observable_descends (F : Set (X → C)) (f : X → C) (hf : f ∈ F) :
    ∃ g : PhysicalQuotient F → C, ∀ x, g (proj F x) = f x := by
  refine ⟨Quotient.lift f (fun a b hab => hab f hf), ?_⟩
  intro x
  rfl

/-- **No gauge from a separating family.** If the observable family separates
states, the projection is injective: the forced quotient is trivial. Gauge
identification appears precisely when the observables fail to separate. -/
theorem proj_injective_of_separating (F : Set (X → C))
    (hsep : ∀ x y, ObsEquiv F x y → x = y) : Function.Injective (proj F) := by
  intro x y h
  exact hsep x y ((forced_iff F x y).mp h)

What this page does not claim

The theorem does not specify which observables are admissible in any physical theory. The theorem does not claim the quotient is physically meaningful without specifying the observable family. The theorem does not assert that the quotient operation is a native part of the framework's distinction process.

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

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