Foundation Primitive Recognition Calculus Objecthood Registry Background Object

The objecthood audit

In mathematics and physics, a theory usually begins with a list of objects it takes for granted: the continuum of real numbers, points in space, sets, equality, infinity, functions, probability. These background objects rarely announce where they came from. The Recognition Science framework's ledger (a discrete record of recognition events) treats this as a problem: an object that enters a theory untyped can smuggle in assumptions the theory never earned.

The declaration background_object_audit in the framework's machine-checked library of formal theorems assigns each of seventeen background objects a commitment (the kind of act that produced it). The assignment is not decorative. Each object gets exactly one of seven tags: forced (uniquely determined by the law, no freedom), permitted (an admissible free choice), quotient (identification under an equivalence), completion (closure adding limit points, an independent axiom), display (a rendering or instrument, not a native ingredient), observable (defined by what can be measured), or convention (a gauge or labeling choice).

The audit's results read like a table. The continuum and infinity are tagged completion: they require an independent axiom asserting least-upper-bound completeness. Points, space, complex numbers, and finite Hilbert spaces are display: they are instruments for rendering, not native ingredients. Equality is a quotient, forced by indistinguishability. Sets and finite amplitudes are permitted: free but admissible choices. Functions and valid comparisons are observable, defined by what can be measured. Finite probability is forced, uniquely determined by the law. Conservative completion and infinite Hilbert space are completion; manifolds are display; measures are completion; physics display objects are observable.

The audit's force is that it prevents background objects from entering the theory untyped. Each object now carries a proved attribute stating what kind of commitment produced it. This is not a claim about how the objects behave mathematically; it is a claim about their status within the framework. The audit does not prove that the continuum exists, that points are displays, or that any particular physical theory is correct. It does not say which commitments are good or bad. It says only: here is what kind of act produced each object, and the assignment is machine-checked.

/-- **Background-object audit.** The conversation's broader target is now inside
the objecthood registry: continuum, point, space, set, equality, infinity,
function, probability, amplitude, valid comparison, and conservative completion
all receive explicit commitment tags. This prevents background objects from
entering the theory untyped. -/
theorem background_object_audit :
    commitmentOf RSObject.continuum = Commitment.completion
      ∧ commitmentOf RSObject.point = Commitment.display
      ∧ commitmentOf RSObject.space = Commitment.display
      ∧ commitmentOf RSObject.setObject = Commitment.permitted
      ∧ commitmentOf RSObject.equalityRegime = Commitment.quotient
      ∧ commitmentOf RSObject.infinityMode = Commitment.completion
      ∧ commitmentOf RSObject.functionTransport = Commitment.observable
      ∧ commitmentOf RSObject.finiteProbability = Commitment.forced
      ∧ commitmentOf RSObject.finiteAmplitude = Commitment.permitted
      ∧ commitmentOf RSObject.validComparison = Commitment.observable
      ∧ commitmentOf RSObject.conservativeCompletion = Commitment.completion
      ∧ commitmentOf RSObject.complexNumbers = Commitment.display
      ∧ commitmentOf RSObject.finiteHilbertSpace = Commitment.display
      ∧ commitmentOf RSObject.infiniteHilbertSpace = Commitment.completion
      ∧ commitmentOf RSObject.manifoldDisplay = Commitment.display
      ∧ commitmentOf RSObject.measureDisplay = Commitment.completion
      ∧ commitmentOf RSObject.physicsDisplayObject = Commitment.observable :=
  ⟨rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl,
    rfl⟩

/-- `completion`: ℝ has the least-upper-bound property. (That no countable
cost-closed carrier has it, so completeness is an independent axiom, is
`PRCCompletenessIndependence.completeness_is_exactly_the_continuum`.) -/
theorem classify_completion :
    ∀ S : Set ℝ, S.Nonempty → (∃ b, ∀ x ∈ S, x ≤ b) → ∃ s, IsLUB S s := by
  intro S hne hbdd
  obtain ⟨b, hb⟩ := hbdd
  exact Real.exists_isLUB hne ⟨b, fun x hx => hb x hx⟩

/-- `display`: the protocol-real value map is surjective onto ℝ and faithful
(observational equality = equal value). ℝ is the forgetful display of the `ℝδ`
protocol interface. -/
theorem classify_display :
    Function.Surjective DeltaReal.Protocol.value
      ∧ (∀ x y : DeltaReal.Protocol,
          DeltaReal.Protocol.ObsEq x y ↔ x.value = y.value) := by
  obtain ⟨hsurj, _, hfaithful, _⟩ := DeltaReal.Protocol.display_real_forgetful
  exact ⟨hsurj, hfaithful⟩

/-- `quotient`: the physical quotient identifies two states iff no observable
distinguishes them. The quotient is forced by indistinguishability. -/
theorem classify_quotient :
    ∀ {X C : Type} (F : Set (X → C)) (x y : X),
      QuotientSelection.proj F x = QuotientSelection.proj F y ↔ QuotientSelection.ObsEquiv F x y :=
  fun F x y => QuotientSelection.forced_iff F x y

/-- `forced`: the rationals are forced into every carrier; they are the unique
prime subfield. No carrier of the framework can omit them. -/
theorem classify_forced_rationals :
    ∀ (K : Subfield ℝ) (q : ℚ), (q : ℝ) ∈ K :=
  fun K q => SubfieldClass.ratCast_mem K q

What this page does not claim

The audit does not prove that the continuum exists or that any particular physical theory is correct. The audit does not rank commitments as good or bad, only assigns each object its kind. The audit does not claim that background objects are dispensable or that the framework can do without them.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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