Foundation Primitive Recognition Calculus Hilbert Display Completion

Hilbert display completion

In quantum mechanics, a Hilbert space is a vector space with an inner product that lets you compute probabilities. The construction called Hilbert display completion takes the finite version of that idea and shows it is not an extra assumption in Recognition Science. It is a display: a way of writing the framework's native amplitudes, which are complex numbers attached to finitely many distinction alternatives, as ordinary Hilbert vectors.

The construction is simple. For a fixed number N of alternatives, a native amplitude assigns a complex number to each of N+1 slots. The display maps that amplitude to a vector in the finite Hilbert space of functions from those slots to the complex numbers. The squared norm of the displayed vector is defined as the sum of the squared magnitudes of its entries, and the Born weight at a slot is the squared magnitude of that entry.

The construction proves three equalities. First, the squared norm of the displayed vector equals the sum of the native Born weights. Second, the Born weight computed from the display equals the native Born weight at every slot. Third, a native amplitude is normalized exactly when the displayed vector has squared norm 1. These are established as theorems in the machine-checked library of formal theorems, so they hold by construction, not by convention.

Because those equalities hold, the display is a valid bridge for comparison. Two native amplitudes can be compared by their squared norms, and that comparison is exactly the same whether you use the native Born weights or the displayed Hilbert norm. The headline theorem bundles all of this into one statement: the finite Hilbert space is a faithful display of native amplitudes, preserving Born weights, norm, and normalization.

In plain language, this means the framework does not need to borrow quantum mechanics as an external ingredient. The finite Hilbert structure that quantum theory uses for probabilities is already present in the framework's own amplitudes. The construction establishes that the two descriptions are interchangeable for finite systems, so any calculation done in the Hilbert display is a calculation about the native amplitudes themselves.

/-- 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

What this page does not claim

The module does not derive the Born rule from recognition principles; it assumes squared magnitude as the observable protocol. The module does not establish that all of quantum mechanics follows from the framework, only that finite Hilbert structure is a faithful display.

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)

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