Foundation Primitive Recognition Calculus Quotient Selection Proj Injective Of S

When observation settles everything

In mathematics, a quotient is a way of declaring certain things equivalent and then treating the equivalence classes as the new objects. The Recognition Science framework builds this from a concrete starting point: a set of states X and a family of observables F, where each observable is a function from states to some value set. Two states are observationally equivalent under F when every observable in F returns the same value on both of them. This is the framework's ledger, a discrete record of what the observables can distinguish.

The framework then forms the physical quotient: the state space modulo this observational equivalence. The projection proj sends each state to its equivalence class. The central theorem forced_iff states that two states map to the same class if and only if no admissible observable separates them. This is the exact collapse: the quotient adds no identifications beyond indistinguishability and omits none. A second theorem, observable_descends, shows every admissible observable still works on the quotient, so no observable information is lost in the collapse.

The declaration proj_injective_of_separating addresses the opposite case. If the observable family separates states, meaning any two distinct states have some observable that distinguishes them, then the projection is injective: distinct states map to distinct classes. In plain terms, when the observables can tell every state apart, the quotient is trivial. It collapses nothing. Gauge identification, where distinct states are treated as the same physical class, appears precisely when the observables fail to separate them.

This is a theorem in the framework's machine-checked library of formal theorems, proved from the definitions of observational equivalence and the quotient. It does not claim that any particular physical theory has a separating family of observables. It does not claim that gauge freedom is absent in any real system. It establishes a conditional: if separation holds, then the quotient is injective. Whether separation holds for a given family is a separate question, not settled by this theorem.

/-- **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)

/-- **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

What this page does not claim

No particular physical theory is asserted to have a separating family of observables. The theorem does not establish that gauge freedom is absent in any real system. Separation is a hypothesis, not a consequence; the theorem only states what follows if it holds.

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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