Foundation Primitive Recognition Calculus Valid Comparison Examples Valid Compar

Three concrete bridges

Recognition Science (a framework that models reality as a discrete record of recognition events) proves that its abstract ledger can be displayed in three familiar mathematical forms. The declaration valid_comparison_examples_headline, a theorem in the framework's machine-checked library of formal theorems, establishes that for real-valued protocols, finite probability events, and finite Hilbert-space amplitudes, a comparison is valid exactly when the displayed values agree. In plain terms: if two native events display to the same real number, the same rational probability, or the same Hilbert norm, then the framework treats them as indistinguishable at the display level.

For real displays, the theorem states that two Delta-real protocols are validly compared precisely when their values are equal. For finite probability events, the same holds for their rational probabilities. For finite Hilbert amplitudes, the condition is equality of the sum of Born weights over the finite index set, which is the standard quantum-mechanical probability of the state. Each of these is a bridge: a structure that maps native events to a display value while preserving the comparison relation.

These bridges are not derived from the core cost function J(x) = (x + 1/x)/2 - 1; they are separate definitions that connect the primitive recognition calculus to classical mathematics. The theorem simply packages three earlier results into one conjunction. It does not claim that real numbers, probabilities, or Hilbert spaces are the only possible displays, nor that these bridges are unique, nor that the displayed values themselves are physically meaningful beyond the comparison relation.

What the theorem changes is practical: it gives the framework a concrete vocabulary for talking about recognition events in terms of ordinary numbers. A reader can now see how the abstract ledger connects to real analysis, probability theory, and quantum mechanics, and can verify that the connection is formally checked, not assumed.

/-- **Valid-comparison examples headline.** The doctrine has concrete bridges for
real display, finite probability display, and finite Hilbert display. -/
theorem valid_comparison_examples_headline :
    (∀ x y : DeltaReal.Protocol,
        ValidComparison.IsValidComparison realDisplayBridge x y ↔ x.value = y.value)
      ∧ (∀ (N : ℕ) (E F : DeltaProbability.Event N),
          ValidComparison.IsValidComparison (probabilityDisplayBridge N) E F
            ↔ DeltaProbability.prob E = DeltaProbability.prob F)
      ∧ (∀ (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)) :=
  ⟨real_display_valid_iff, probability_display_valid_iff, hilbert_display_valid_iff⟩

/-- **Valid-comparison examples headline.** The doctrine has concrete bridges for
real display, finite probability display, and finite Hilbert display. -/
theorem valid_comparison_examples_headline :
    (∀ x y : DeltaReal.Protocol,
        ValidComparison.IsValidComparison realDisplayBridge x y ↔ x.value = y.value)
      ∧ (∀ (N : ℕ) (E F : DeltaProbability.Event N),
          ValidComparison.IsValidComparison (probabilityDisplayBridge N) E F
            ↔ DeltaProbability.prob E = DeltaProbability.prob F)
      ∧ (∀ (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)) :=
  ⟨real_display_valid_iff, probability_display_valid_iff, hilbert_display_valid_iff⟩

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 theorem does not prove that these are the only possible display bridges. The theorem does not assign physical meaning to the displayed values beyond the comparison relation. The theorem does not derive the bridges from the core cost function.

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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Derived articles

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