Encyclopedia Foundation Foundation Primitive Recognition Calculus Hilbert Display Completion Finite Hilb

ARTICLE 5 claims 4 theorems 1 model

Foundation Primitive Recognition Calculus Hilbert Display Completion Finite Hilb

A finite Hilbert space is a way of displaying the framework's own amplitudes, and the display provably preserves all the weights that matter.

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The finite Hilbert display

A Hilbert space is a vector space with an inner product, the standard mathematical setting for quantum states. The finite Hilbert display is a specific construction: it takes a complex amplitude on a finite set of N plus one alternatives, and presents it as a vector in a finite-dimensional Hilbert space. The definition is direct, each alternative gets one complex coordinate.

The declaration finite_hilbert_display_headline bundles four proved facts about this display. First, the squared norm of the displayed vector equals the sum of the native Born weights, the probabilities the framework assigns to each alternative. Second, the Born weight of each displayed coordinate equals the native Born weight for that alternative. Third, an amplitude is normalized in the native sense exactly when its displayed vector has squared norm one. Fourth, comparing two amplitudes by their native Born-weight sums is valid through the display, because the bridge that connects native amplitudes to displayed vectors commutes with the observable protocol.

In Recognition Science, the framework models recognition events as a discrete ledger, and these amplitudes are the framework's own objects. The headline theorem proves that the finite Hilbert space is not an independent structure but a faithful display of those native amplitudes. The bridge preserves the quantities that carry physical meaning: the Born weights, the squared norm, and normalization.

The theorem does not claim that every Hilbert-space construction applies to the framework's amplitudes. It proves a specific display and a specific bridge for finite alternatives. It does not introduce a new physical postulate, and it does not derive the Born rule from scratch. The Born weight is already defined in the framework; the theorem shows the display respects it.

MODEL FiniteHilbertDisplay · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean
FiniteHilbertDisplay · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean:28 
/-- The finite Hilbert display carrier: an ambient complex vector on the finite
distinction alternatives. -/
abbrev FiniteHilbertDisplay (N : ℕ) := Fin (N + 1) → ℂ
THEOREM finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean
finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean:72 
/-- **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⟩
THEOREM finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean
finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean:72 
/-- **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⟩
THEOREM finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean
finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean:72 
/-- **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⟩
THEOREM finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean
finite_hilbert_display_headline · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HilbertDisplayCompletion.lean:72 
/-- **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 theorem does not derive the Born rule from more basic principles. The theorem does not apply to infinite-dimensional Hilbert spaces. The theorem does not introduce a new physical postulate beyond the framework's existing amplitudes.

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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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