Foundation Primitive Recognition Calculus Quotient Examples Quotient Examples He

The quotient examples

The recognition framework, a discrete record of which events a system can tell apart, defines a physical state as an equivalence class of raw states under the observables that are admitted. The declaration quotient_examples_headline packages three concrete demonstrations of this principle. First, if no observable is allowed, every phase state is indistinguishable, so the quotient collapses to a single class. Second, if all integer-valued observables are admitted, they separate every pair of distinct states, so the quotient is trivial and injective. Third, the projective-state display theorem states that two states belong to the same physical class exactly when the admitted projective observables cannot distinguish them.

These examples are toy illustrations, not a derivation of the physical world. They show the mechanics of quotienting by observables, but they do not claim that any particular set of observables is physically real, nor that the integers are the correct target for physical measurements. The theorem is a formal statement about the framework's own construction, not an empirical claim about nature.

The practical lesson is that the quotient structure is sensitive to the choice of observable family: empty families identify everything, separating families identify nothing. This is the pivot on which the framework's later results turn, but these examples themselves do not force any particular physical content.

/-- 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)

/-- 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

What this page does not claim

The examples do not assert that any particular observable family is physically real. The theorem does not derive the golden ratio, the fine-structure constant, or any measured physical constant. The examples do not claim that the integer-valued observable family is the unique separating family.

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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