Foundation Primitive Recognition Calculus Valid Comparison Examples Real Display

The real display bridge

In the Recognition Science framework, a ledger is a discrete record of events, and the framework's core question is when two records can be considered to display the same thing. The declaration real_display_valid_iff answers this for the simplest case, real numbers. It establishes that two real-number protocols are a valid comparison exactly when their displayed values are equal. In plain terms: if the universe keeps two ledgers of real numbers, those ledgers agree precisely when the numbers they show are the same number.

The theorem is built on a bridge, which in this framework is a rule for connecting a native protocol to a displayed form. The real display bridge connects a Delta-real protocol to its real value, and the observable is that same value. The theorem states, for any two protocols x and y, that the comparison is valid if and only if x.value = y.value. This is a formal, machine-checked statement in the framework's library of formal theorems.

What the theorem does not claim is just as important. It does not say that real numbers are the only things that can be compared, nor that every comparison reduces to equality of displayed values. The framework also provides bridges for finite probability events and for finite Hilbert spaces, each with its own validity condition. The real display bridge is one concrete instance, not a universal law. It also does not claim anything about what the real numbers mean or how they are constructed; it only establishes the equivalence between valid comparison and equality of displayed values for this particular bridge.

The practical consequence is a clean test: when two real-valued protocols are candidates for the same observation, the framework says the comparison is valid exactly when the values match. This gives a precise, checkable criterion for agreement in the simplest case, and it provides a template for the more complex bridges that follow.

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

/-- Real display bridge: a Delta-real protocol displays to its real value, and
the observable is that same value. -/
noncomputable def realDisplayBridge :
    ValidComparison.Bridge DeltaReal.Protocol ℝ ℝ where
  display := DeltaReal.Protocol.value
  observeNative := DeltaReal.Protocol.value
  observeDisplay := id
  commutes := by intro x; rfl

/-- **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⟩

What this page does not claim

The theorem does not claim that all valid comparisons reduce to equality of displayed values. It does not claim anything about the meaning or construction of real numbers themselves.

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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