Foundation Primitive Recognition Calculus Completion Conservativity

Completion conservativity

In Recognition Science, a framework that derives physical structure from the cost of recognition events, a completion is an interface that maps native data N to display data D, together with a certification relation. The display function is what a system shows; the certificate is the native record that vouches for what is shown. A completion is conservative for a predicate when every display datum satisfying that predicate carries a certificate. This is the formal backbone of the main guarantee: a completion is conservative exactly when it has no uncertified display artifacts.

The base case and the composition rules are established. The identity completion, where the display is the identity and the certificate is equality, is conservative for every predicate. Product completions compose conservativity: if each component display predicate descends to a certificate, the product predicate descends to paired certificates. Function-space completions lift conservativity pointwise, so finite vectors and finite fields can be certified coordinatewise. These structural lemmas let multi-field display objects be certified componentwise.

The headline theorem packages the two core facts: for any completion, conservativity for a predicate is equivalent to the absence of an artifact for that predicate, and the identity completion is conservative for every predicate. The product and function-space headlines state the stability of certificate-preserving completion under products and pointwise finite displays. In plain language, the result is a clean criterion: a display system is trustworthy exactly when every displayed object that satisfies a property can be traced back to a native certificate, and this trustworthiness composes across products and coordinatewise across finite vectors.

In Recognition Science, this is the bridge between the primitive recognition calculus and the display layer. The framework models recognition events as discrete ledger entries; the completion conservativity result guarantees that anything shown at the display level can be traced back to those native entries. This is a formal hygiene result: it rules out the possibility of a display artifact that satisfies a predicate but has no native certificate. The consequence is that multi-field and vector-valued displays inherit the trustworthiness of their components, so the display layer cannot introduce uncertified content beyond what the native layer already certifies.

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

This result does not derive any physical constant or empirical value. This result does not define what counts as a native certificate in the recognition calculus. This result does not prove that any particular display system in the framework is conservative.

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)

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