Foundation Primitive Recognition Calculus Completion Conservativity Conservative

Conservativity and artifacts

A completion is a bridge from raw data to a richer display. Think of a census: the raw data are household forms, and the display is a table of population counts. The bridge is conservative when every displayed fact that satisfies a given predicate can be traced back to a certificate, a piece of the original data that vouches for it. The central theorem, conservative_iff_no_artifact, states that a completion is conservative for a predicate precisely when it has no artifact: no displayed datum that satisfies the predicate while lacking any certificate.

The proof is a short logical equivalence. If every qualifying display carries a certificate, then none can be an artifact, since an artifact is exactly a qualifying display without one. Conversely, if no artifact exists, then any qualifying display must have a certificate, because its absence would make it an artifact. The same theorem also shows that the identity completion, which displays data as itself with the data as its own certificate, is conservative for every predicate. This is the base case that makes larger constructions work.

The framework then shows that conservativity composes. If two completions are each conservative for their own predicates, their product completion is conservative for the paired predicate, so a multi-field display can be certified componentwise. The same lifting works for finite vectors and finite fields: if each coordinate predicate has a certificate, the whole function display has a coordinatewise certificate. These closure properties are what let the framework certify complex displays from simple ones without losing the guarantee.

In Recognition Science, this theorem is the guardrail for the ledger, the discrete record of recognition events. The ledger must not show a fact that no event supports. Conservativity is the formal statement of that requirement: every displayed fact descends to a certificate from the native data. The theorem does not say which predicates are conservative, nor does it construct certificates for any particular display. It only establishes the equivalence between conservativity and the absence of artifacts, and the closure of conservativity under products and pointwise function displays.

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

The theorem does not construct certificates for any particular display predicate. The theorem does not say which predicates are conservative, only that conservativity equals having no artifacts. No claim that any specific physical display in the framework is artifact-free.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: