Foundation Primitive Recognition Calculus Objecthood Registry Display Object Ext

The display-object extension

In mathematics, a complex number is an expression of the form a + bi, where i is the square root of -1. A Hilbert space is a complete inner product space, the setting for quantum mechanics. A manifold is a space that locally resembles Euclidean space. A measure assigns sizes to subsets. Each of these objects has a standard definition and a long history of use in physics and mathematics.

Recognition Science (RS) maintains a machine-checked library of formal theorems. Within that library, the objecthood registry classifies every mathematical object the framework builds by the kind of commitment that produces it. The declaration display_object_extension assigns commitment tags to six objects: complex numbers, finite and infinite Hilbert spaces, manifolds, measures, and physics display objects. The declaration states that complex numbers, finite Hilbert spaces, manifolds, and physics display objects are display objects, while infinite Hilbert spaces and measures are completion objects.

In the framework, a display object is a rendering or instrument, not a native ingredient. A display object is something the framework uses to present or represent information, like a screen showing a number. A completion object is one that requires an independent axiom to add limit points, such as the real numbers with the least-upper-bound property. The declaration assigns each object its commitment tag by definition, using reflexivity, so the classification is a definitional choice rather than a derived result.

The declaration does not claim that complex numbers, Hilbert spaces, manifolds, or measures are physically real or fundamental. It does not claim that display objects are less important than forced objects. It only establishes a formal classification within the RS objecthood registry, so that background objects do not enter the theory untyped.

/-- **Display-object extension.** Complex numbers, finite and infinite Hilbert
spaces, manifolds, measures, and physics display objects are now explicitly
typed. This closes the objecthood-table extension requested by the completion
plan. -/
theorem display_object_extension :
    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⟩

/-- **Display-object extension.** Complex numbers, finite and infinite Hilbert
spaces, manifolds, measures, and physics display objects are now explicitly
typed. This closes the objecthood-table extension requested by the completion
plan. -/
theorem display_object_extension :
    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⟩

/-- **Display-object extension.** Complex numbers, finite and infinite Hilbert
spaces, manifolds, measures, and physics display objects are now explicitly
typed. This closes the objecthood-table extension requested by the completion
plan. -/
theorem display_object_extension :
    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⟩

What this page does not claim

Complex numbers, Hilbert spaces, manifolds, or measures are physically real or fundamental. Display objects are less important than forced objects. The classification is a derived theorem rather than a definitional choice.

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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