Foundation Primitive Recognition Calculus Quotient Selection Gauge From Indistin

The gauge from indistinguishability

In the Recognition Science framework, a state space comes with a family of observables, functions that assign a value to each state. Two states are observationally equivalent under that family when every observable returns the same value on both. The declaration gauge_from_indistinguishability packages three theorems about this setup, and together they define what the framework means by a gauge identification.

The first theorem, forced_iff, states that two states map to the same physical class if and only if no admissible observable separates them. The quotient adds no identifications beyond indistinguishability and omits none. The second, observable_descends, says every admissible observable still works on the quotient: there is a function on physical classes that agrees with the observable on every state, so no observable information is lost. The third, proj_injective_of_separating, covers the opposite case: if the observables separate states, meaning any two distinct states differ on some observable, then the projection is injective and the quotient is trivial. Gauge identification appears precisely when the observables fail to separate.

In Recognition Science, the quotient is not a native operation of distinction. It is forced exactly by the absence of a distinguishing recognition act. The framework models this as a machine-checked theorem in its library of formal proofs, with the three properties bundled into a single declaration. The practical consequence is that gauge freedom is not an assumption; it is a derived consequence of what the observables can and cannot see.

The declaration does not claim that any particular physical theory has this structure. It does not say which observables a real system admits, nor does it assert that the quotient matches any specific gauge group from conventional physics. It establishes a formal relationship: given a family of observables, the quotient is exactly the indistinguishability collapse, no more and no less.

/-- **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 declaration does not specify which observables a real physical system admits. It does not assert that the quotient matches any specific gauge group from conventional physics. It does not claim that gauge freedom is a physical phenomenon rather than a formal construction.

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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