Foundation Primitive Recognition Calculus Quotient Examples Projective State Dis

Projective state display

In the Recognition Science framework, the declaration projective_state_display establishes a precise criterion for when two states should be considered the same physical state. The framework models physical states as points in a space, and observables as functions that assign values to those points. The declaration states that two states belong to the same physical class if and only if every admitted observable assigns them the same value. This is the quotient construction: it identifies states that no measurement can distinguish, collapsing them into a single physical class.

The framework's ledger, a discrete record of recognition events, provides the context for this identification. The declaration is a theorem in the machine-checked library of formal theorems, meaning it is proved from the framework's axioms. It specializes a more general quotient theorem to the case of projective observables, which are functions from states to observation values. The theorem is stated for any type of states and any type of observations, making it fully general within the framework.

Two concrete examples illustrate the extremes. If the set of admitted observables is empty, then no observable can distinguish any two states, and the quotient collapses all states into a single class. Conversely, if all integer-valued observables are admitted, they separate integer states completely, and the quotient is trivial: each state remains its own class. These examples bracket the general principle that the physical quotient is determined entirely by the family of observables one chooses to admit.

What the declaration does not claim is equally important. It does not assert that any particular family of observables is the correct one for physics; that remains a modeling choice. It does not claim that the quotient construction is physically meaningful outside the framework's axioms. And it does not say anything about what observables exist in nature, only about the logical consequence of admitting a given family. The theorem is a structural fact about the framework's definitions, not an empirical claim about the world.

/-- A generic projective-state quotient: two states have the same physical class
exactly when the admitted projective observables cannot distinguish them. -/
theorem projective_state_display {State Obs : Type*} (F : Set (State → Obs)) (x y : State) :
    proj F x = proj F y ↔ ObsEquiv F x y :=
  forced_iff F x y

/-- A generic projective-state quotient: two states have the same physical class
exactly when the admitted projective observables cannot distinguish them. -/
theorem projective_state_display {State Obs : Type*} (F : Set (State → Obs)) (x y : State) :
    proj F x = proj F y ↔ ObsEquiv F x y :=
  forced_iff F x y

/-- If the admitted observable family is empty, every phase state is
indistinguishable and therefore identified by the physical quotient. -/
theorem empty_observable_phase_quotient (x y : PhaseState) :
    proj (X := PhaseState) (C := ℤ) (∅ : Set (PhaseState → ℤ)) x
      = proj (X := PhaseState) (C := ℤ) (∅ : Set (PhaseState → ℤ)) y := by
  apply identified_of_obsEquiv
  intro f hf
  cases hf

/-- If all integer-valued observables are admitted, they separate integer states,
so the quotient is trivial. -/
theorem separating_gauge_family_injective :
    Function.Injective (proj (X := ℤ) (C := ℤ) (Set.univ : Set (ℤ → ℤ))) := by
  apply proj_injective_of_separating
  intro x y h
  exact h (fun z => z) (by simp)

What this page does not claim

The declaration does not identify which observable family is physically correct. The theorem does not assert that any particular physical system realizes the empty or full observable families. The quotient construction is a logical consequence of the framework's definitions, not an empirical statement about nature.

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

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