Foundation Primitive Recognition Calculus Completion Conservativity Completion

The completion interface

In the Recognition Science framework, a completion is a formal bridge between two kinds of data: native data, the raw internal form, and display data, the form a person or another system can read. The framework defines a completion as a pair of operations: one maps each native datum to its display, and the other certifies that a given display datum genuinely came from a native one. A certificate is a witness, a piece of evidence that a displayed value is not an invention but a faithful rendering of something real.

The central question is whether this bridge loses information. The framework proves a sharp answer: a completion is conservative, meaning every display datum that satisfies a given predicate carries a certificate, exactly when it has no artifacts. An artifact is a display datum that satisfies the predicate but has no certificate, a displayed value that looks right yet has no native source. The theorem conservative_iff_no_artifact states this equivalence directly: conservativity and artifact-freedom are the same property.

The framework then shows the property is stable under natural constructions. The identity completion, which displays every datum as itself and certifies each datum by itself, is conservative for every predicate. If two completions are each conservative for their own predicates, their product, which pairs displays and pairs certificates, is conservative for the paired predicate. The same lifting works for finite function spaces, where each coordinate is completed through the same interface, so a vector or field of certified values is itself certified coordinatewise. These are proved theorems in the framework's machine-checked library of formal theorems, not definitions chosen for convenience.

What the declaration does not claim is equally precise. It does not assert that every completion is conservative, only that conservativity is equivalent to having no artifacts. It does not say that certificates are unique, or that a display datum cannot have multiple native sources. It does not claim that the identity completion is the only conservative completion, nor that the product and function constructions cover every possible way to combine completions. The theorems establish a boundary, not a monopoly: they tell you exactly when a bridge loses nothing, and they leave open the many ways a bridge might still be useful even when it does.

theorem conservative_iff_no_artifact {N D Cert : Type*} (C : Completion N D Cert) (P : D → Prop) :
    ConservativeFor C P ↔ ¬ ArtifactFor C P := by
  unfold ConservativeFor CertificateCovered ArtifactFor
  constructor
  · intro h hc
    rcases hc with ⟨d, hP, hno⟩
    exact hno (h d hP)
  · intro h d hP
    by_contra hno
    exact h ⟨d, hP, hno⟩

theorem identity_conservative (N : Type*) (P : N → Prop) :
    ConservativeFor (identityCompletion N) P := by
  intro d _
  exact ⟨d, rfl⟩

/-- Conservative completions compose across products: if each component display
predicate descends to a certificate, the product predicate descends to paired
certificates. -/
theorem product_conservative
    {N₁ D₁ Cert₁ N₂ D₂ Cert₂ : Type*}
    (C₁ : Completion N₁ D₁ Cert₁) (C₂ : Completion N₂ D₂ Cert₂)
    (P₁ : D₁ → Prop) (P₂ : D₂ → Prop)
    (h₁ : ConservativeFor C₁ P₁) (h₂ : ConservativeFor C₂ P₂) :
    ConservativeFor (productCompletion C₁ C₂) (ProductPredicate P₁ P₂) := by
  intro d hd
  rcases hd with ⟨hP₁, hP₂⟩
  rcases h₁ d.1 hP₁ with ⟨c₁, hc₁⟩
  rcases h₂ d.2 hP₂ with ⟨c₂, hc₂⟩
  exact ⟨(c₁, c₂), ⟨hc₁, hc₂⟩⟩

/-- Conservativity lifts pointwise to finite function displays: if each coordinate
predicate has a certificate, the whole function has a coordinatewise certificate. -/
theorem function_conservative
    {I N D Cert : Type*} (C : Completion N D Cert) (P : D → Prop)
    (hC : ConservativeFor C P) :
    ConservativeFor (functionCompletion I N D Cert C) (AllPredicate P) := by
  intro d hd
  choose c hc using fun i : I => hC (d i) (hd i)
  exact ⟨c, hc⟩

What this page does not claim

Not every completion is conservative; the theorems characterize when one is. Certificates are not claimed to be unique for a given display datum. The identity, product, and function completions are not claimed to be the only conservative ones.

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/CompletionConservativity.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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