Foundation Primitive Recognition Calculus Hilbert Display Completion Display

The finite Hilbert display

A finite Hilbert space is a vector space over the complex numbers with a finite basis. In Recognition Science, the framework models a finite set of distinction alternatives as the basis of such a space. The declaration display maps a native F_RS[i] amplitude, a complex number attached to each distinction, to a finite Hilbert vector. The map is an abbreviation, not a new construction: it reuses the framework's existing amplitude display.

The framework defines a squared norm on the finite Hilbert display as the sum of the squared magnitudes of its components. It also defines a Born weight, the squared magnitude of a single component, which the framework uses as the probability-like observable. Two theorems establish that these display quantities equal the native F_RS[i] quantities. The squared norm of the display equals the sum of the native Born weights, and the display Born weight at each index equals the native Born weight at that index. A third theorem states that a native amplitude is normalized exactly when its display squared norm equals one.

These equalities compose into a bridge, a formal structure that compares native amplitudes by their display observables. The bridge's commutation condition, that display norm equals native Born-weight sum, is proved by the first theorem. The headline theorem packages the three equalities and the bridge validity condition into a single statement. It says that for any two native amplitudes, comparison by display norm is valid exactly when their native Born-weight sums are equal.

In Recognition Science, the framework's library, a machine-checked collection of formal theorems, proves this headline. The declaration establishes that the finite Hilbert display is a faithful representation of native finite amplitudes for the purposes of norm, Born weight, normalization, and comparison by norm. It does not claim that the display is the only representation, nor that it extends to infinite distinction sets. It also does not claim that the Born weight is a probability in the sense of a normalized measure over all alternatives; the framework only defines it as a weight and proves equality with the native quantity.

/-- The Hilbert-display norm equals the native F_RS[i] finite norm. -/
theorem display_normSq_eq {N : ℕ} (ψ : FRSComplexAmplitude.FRSIAmp N) :
    normSq (display ψ)
      = Finset.univ.sum (fun i : Fin (N + 1) => FRSComplexAmplitude.bornWeight ψ i) :=
  FRSComplexAmplitude.display_normSq_eq ψ

/-- The Hilbert-display Born weight equals the native F_RS[i] Born weight. -/
theorem display_bornWeight_eq {N : ℕ} (ψ : FRSComplexAmplitude.FRSIAmp N) (i : Fin (N + 1)) :
    bornWeight (display ψ) i = FRSComplexAmplitude.bornWeight ψ i :=
  FRSComplexAmplitude.display_bornWeight_eq ψ i

/-- Native F_RS[i] normalization is exactly display-Hilbert normalization. -/
theorem normalized_iff_display {N : ℕ} (ψ : FRSComplexAmplitude.FRSIAmp N) :
    FRSComplexAmplitude.Normalized ψ ↔ normSq (display ψ) = 1 := by
  rw [display_normSq_eq]
  rfl

/-- **Finite Hilbert display headline.** Finite Hilbert space is a display of
native F_RS[i] finite amplitudes. The bridge preserves Born weights, squared
norm, and normalization, and comparison by norm is valid through the
native/display/observable bridge. -/
theorem finite_hilbert_display_headline (N : ℕ) :
    (∀ ψ : FRSComplexAmplitude.FRSIAmp N,
        normSq (display ψ)
          = Finset.univ.sum (fun i : Fin (N + 1) => FRSComplexAmplitude.bornWeight ψ i))
      ∧ (∀ ψ : FRSComplexAmplitude.FRSIAmp N, ∀ i : Fin (N + 1),
          bornWeight (display ψ) i = FRSComplexAmplitude.bornWeight ψ i)
      ∧ (∀ ψ : FRSComplexAmplitude.FRSIAmp N,
          FRSComplexAmplitude.Normalized ψ ↔ normSq (display ψ) = 1)
      ∧ (∀ ψ φ : FRSComplexAmplitude.FRSIAmp N,
          ValidComparison.IsValidComparison (normBridge 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)) :=
  ⟨display_normSq_eq, display_bornWeight_eq, normalized_iff_display,
    fun ψ φ => ValidComparison.validComparison_iff_native (normBridge N) ψ φ⟩

What this page does not claim

The display is not the only faithful representation of native amplitudes. The display does not extend to infinite distinction sets. The Born weight is not claimed to be a probability measure over all alternatives.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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