Foundation Primitive Recognition Calculus Quotient Examples Separating Gauge Fam

The separating gauge family

In mathematics, a quotient identifies points that a chosen set of tests cannot tell apart. If the tests are few, many points collapse together; if the tests are rich enough, each point keeps its own identity. The theorem separating_gauge_family_injective in the machine-checked library of formal theorems shows the extreme case: when the set of tests is the full set of integer-valued functions on the integers, the quotient map is injective, meaning no two distinct integers are ever identified.

This is a toy example inside the framework's recognition calculus, where a state is a discrete record of events and an observable is a test that reads a number from a state. The theorem proves that if you admit every integer-valued observable, then for any two different states x and y, there is at least one observable f with f(x) ≠ f(y). The proof uses the identity function f(z) = z, which trivially separates any two distinct integers. The result is a direct consequence of a more general lemma: a family of observables that separates points yields an injective projection onto physical classes.

The companion theorem empty_observable_phase_quotient shows the opposite extreme: if no observables are admitted, every state is identified with every other state. Together these two results bracket the spectrum of possible quotient behavior. The headline theorem quotient_examples_headline packages both extremes and the general projective-state display theorem into one statement.

What the declaration does not claim is broader than what it proves. It does not claim that any particular physical theory uses the full set of integer-valued observables; it is a toy example, not a physical postulate. It does not claim that the quotient is injective for any smaller family of observables, or for observables valued in other sets. It does not claim anything about the cost function J or the forcing chain; this theorem lives in the quotient-examples file and concerns only the separation property of observable families.

/-- 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)

/-- 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)

/-- 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

What this page does not claim

This theorem does not claim that any physical theory uses the full set of integer-valued observables. It does not claim injectivity for smaller observable families or for observables valued in other sets. It does not claim anything about the cost function J or the forcing chain.

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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