Foundation Primitive Recognition Calculus Valid Comparison Examples

Valid comparison bridges

A ledger, a discrete record of events, needs a way to tell whether two entries show the same thing. The framework's answer is a valid comparison: a bridge that maps a protocol's native data to a display value, and then says two entries are equal exactly when their displays match. The module under this page proves that three such bridges exist and behave correctly.

The first bridge handles real numbers. A Delta-real protocol carries a value; the bridge displays that value directly, and the theorem real_display_valid_iff proves that two protocols compare as valid exactly when their underlying real values are equal. This is the simplest possible case: the display is the native value itself.

The second bridge handles finite probabilities. A native event in a finite probability space displays to its rational counting probability. The theorem probability_display_valid_iff proves that two events compare as valid exactly when their probabilities are equal. This is the bridge that lets a ledger treat "likely" and "certain" as comparable quantities.

The third bridge handles quantum amplitudes. A finite Hilbert state displays to its Born-rule weight, the sum of squared amplitudes over all outcomes. The theorem hilbert_display_valid_iff proves that two states compare as valid exactly when those summed weights are equal. This is the bridge that connects the framework's discrete recognition events to the continuous probabilities of quantum measurement.

In Recognition Science, these three bridges are the concrete proof that the framework's abstract comparison doctrine is not empty. They show that real numbers, probabilities, and quantum states can all be brought under the same comparison rule, each with a machine-checked guarantee that the display preserves equality. The headline theorem packages all three equivalences into one statement, confirmed in the framework's machine-checked library of formal theorems.

theorem real_display_valid_iff (x y : DeltaReal.Protocol) :
    ValidComparison.IsValidComparison realDisplayBridge x y ↔ x.value = y.value :=
  ValidComparison.validComparison_iff_native realDisplayBridge x y

theorem probability_display_valid_iff (N : ℕ) (E F : DeltaProbability.Event N) :
    ValidComparison.IsValidComparison (probabilityDisplayBridge N) E F
      ↔ DeltaProbability.prob E = DeltaProbability.prob F :=
  ValidComparison.validComparison_iff_native (probabilityDisplayBridge N) E F

theorem hilbert_display_valid_iff (N : ℕ)
    (ψ φ : FRSComplexAmplitude.FRSIAmp N) :
    ValidComparison.IsValidComparison (hilbertNormBridge N) ψ φ
      ↔ Finset.univ.sum (fun i : Fin (N + 1) => FRSComplexAmplitude.bornWeight ψ i)
        = Finset.univ.sum (fun i : Fin (N + 1) => FRSComplexAmplitude.bornWeight φ i) :=
  ValidComparison.validComparison_iff_native (hilbertNormBridge N) ψ φ

What this page does not claim

The module does not prove that every protocol has a valid comparison bridge. The Hilbert bridge does not establish the Born rule as a derived law; it assumes the Born weight as the display definition.

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

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