Foundation Primitive Recognition Calculus Quotient Selection Identified Of Obs E

Collapsing the indistinguishable

In mathematics, a quotient is a way of declaring certain things equal. The declaration identified_of_obsEquiv is a theorem inside the Recognition Science framework that says: if two states give the same value for every observable in a given family, then those two states are identified in the quotient. Plainly: if no measurement you are allowed to make can tell two situations apart, the framework treats them as the same physical state. This is the formal statement that indistinguishability forces identification.

The theorem is a direct consequence of a stronger result, forced_iff, which states the equivalence in both directions: two states map to the same physical class if and only if no admissible observable separates them. The framework's library of machine-checked formal theorems proves this as a theorem, not as a definitional choice. The identification is forced by the absence of a distinguishing recognition act, not imposed by hand.

The theorem does not claim that all observables are preserved. The companion theorem observable_descends shows that every admissible observable still descends to the quotient, meaning no observable information is lost by the collapse. But the theorem itself only establishes the one-way implication: observational equivalence implies identification. The converse, that identification implies observational equivalence, is part of the stronger forced_iff theorem, not of identified_of_obsEquiv alone.

The theorem also does not claim that the quotient is nontrivial. If the observable family separates states, meaning any two distinct states can be distinguished by some observable, then the projection is injective and the quotient collapses to the identity. Gauge identification appears precisely when the observables fail to separate. The theorem is agnostic about whether such separation holds; it only says what happens when it does not.

What this establishes for the framework is a precise sense in which gauge freedom is not an added structure but a consequence of limited observation. The physical state space is the state space modulo indistinguishability, and the theorem guarantees that this quotient is exactly the indistinguishability collapse, adding no identifications beyond what the observables force and omitting none. This is the foundation for treating gauge symmetries as derived from observational limits rather than as independent postulates.

/-- Indistinguishable states are identified in the quotient. -/
theorem identified_of_obsEquiv (F : Set (X → C)) {x y : X} (h : ObsEquiv F x y) :
    proj F x = proj F y :=
  (forced_iff F x y).mpr h

/-- **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 claim that the quotient is nontrivial or that gauge identification always occurs. The theorem does not claim that all observables are preserved, only that admissible ones descend. The theorem does not claim that observational equivalence is the same as physical identity in all contexts.

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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