Foundation Primitive Recognition Calculus Valid Comparison Valid Comparison Doct

The comparison doctrine

A comparison is an act of judgment: two displays are put side by side and pronounced alike or different. The doctrine of valid comparison says when that judgment is sound. It is sound exactly when the two displays, whatever they are, trace back to native objects whose own observable properties already agree. A picture of a thing, observed carefully, must agree with the thing itself; if two pictures agree with each other but disagree with what they depict, the comparison is an illusion.

Formally, the framework models this with a bridge: a display function that turns a native object into a display object, plus an observation function that reads a native object directly and another that reads a display object. The bridge commutes when reading the display gives the same answer as reading the native object directly. A comparison of two native objects through the bridge is valid when the two displayed observations are equal. The central theorem proves that this validity is exactly equivalent to the native observations being equal. The display adds no information and removes none; it is a faithful window.

The doctrine also holds under composition. If a first bridge turns native objects into displays, and a second bridge turns those displays into further displays, the composite bridge is valid whenever both bridges commute with the observable protocol. The theorem states that a comparison through the composite bridge is legitimate exactly when the native observations of the intermediate displays agree. Validity is stable: chaining faithful windows cannot manufacture a false agreement.

In Recognition Science, this doctrine is the guardrail for every measurement and every comparison that the framework makes. It does not say which observation protocol to choose, nor what a native object is. It says only that whatever protocol is chosen, a comparison in the display carrier is legitimate precisely when it descends to equality in the native protocol. The framework proves this as a theorem in its machine-checked library of formal theorems, with the equivalence and the composition stability both established.

The practical consequence is a rule of thumb for anyone building on the framework: never trust a comparison that cannot be traced back to the native objects it compares. If a display disagrees with its native source, the display is the thing to doubt. The doctrine turns that intuition into a formal equivalence, and it guarantees that the equivalence survives when displays are stacked.

theorem validComparison_iff_native {N D O : Type*} (B : Bridge N D O) (x y : N) :
    IsValidComparison B x y ↔ B.observeNative x = B.observeNative y := by
  unfold IsValidComparison
  rw [B.commutes x, B.commutes y]

theorem validComparison_compose {N D E O : Type*}
    (B₁ : Bridge N D O) (B₂ : Bridge D E O) (x y : N) :
    IsValidComparison (compose B₁ B₂) x y ↔ B₂.observeNative (B₁.display x) = B₂.observeNative (B₁.display y) :=
  validComparison_iff_native (compose B₁ B₂) x y

/-- **Valid comparison doctrine.** A comparison in a display carrier is legitimate
exactly when it descends to equality of the native observable protocol, and this
legitimacy is stable under composition of display bridges. -/
theorem valid_comparison_doctrine {N D E O : Type*}
    (B₁ : Bridge N D O) (B₂ : Bridge D E O) :
    (∀ x y : N, IsValidComparison B₁ x y ↔ B₁.observeNative x = B₁.observeNative y)
      ∧ (∀ x y : N, IsValidComparison (compose B₁ B₂) x y
          ↔ B₂.observeNative (B₁.display x) = B₂.observeNative (B₁.display y)) :=
  ⟨validComparison_iff_native B₁, validComparison_compose B₁ B₂⟩

What this page does not claim

The doctrine does not choose any particular observation protocol or define what a native object is. It does not claim that any specific display is faithful; it only states what validity means when a bridge commutes. It does not prove that a valid comparison exists for any given pair of native objects.

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

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