Foundation Primitive Recognition Calculus Frscomplex Amplitude Frsi Amplitude He

The finite complex amplitude

A quantum amplitude is a complex number whose squared size gives a probability. In the Recognition Science framework, amplitudes are not arbitrary complex numbers. The declaration frsi_amplitude_headline proves that any finite list of such amplitudes can be carried by a special scalar system called F_RS[i], a finite-description complex scalar whose real and imaginary parts are both built from the framework's basic recognition ledger. The ordinary complex numbers, ℂ, appear only as the display: a way to show the value, not the thing itself.

The theorem states four properties together. First, every Born weight, the squared size of an amplitude that yields a probability, is nonnegative. Second, the total squared norm of a displayed amplitude equals the sum of these weights. Third, normalizing the finite description is exactly the same as normalizing its complex display. Fourth, both the real and imaginary parts of any displayed value remain inside the framework's real carrier field. Each property is proved by direct computation, and together they show the finite description is faithful: it preserves the structure that probabilities need.

The point is not that complex numbers are wrong. The point is that the framework can write amplitudes without ever leaving its own finite vocabulary. The ambient complex field is a convenience for reading the result, not a requirement for defining it. This matters because the framework aims to derive physics from a discrete ledger of recognition events; a complex amplitude that needs an infinite description would break that program. Here, the amplitude is as finite as the ledger itself.

What the declaration does not claim is just as important. It does not say that every quantum state in nature has a finite F_RS[i] description; it says finite lists of amplitudes can be carried this way. It does not introduce dynamics, measurement, or the Born rule as a physical law; it defines a weight and proves its algebraic properties. It does not identify F_RS[i] with ℂ as fields; the display map is a translation, not an equality. Those are separate questions, and this theorem leaves them open.

/-- **F_RS[i] finite amplitude headline.** Finite complex amplitudes can be
carried by the finite-description scalar carrier `F_RS[i]`; ambient ℂ is only the
display. The display preserves real/imaginary carrier membership, Born weights,
and normalization. -/
theorem frsi_amplitude_headline (N : ℕ) :
    (∀ ψ : FRSIAmp N, ∀ i : Fin (N + 1), 0 ≤ bornWeight ψ i)
      ∧ (∀ ψ : FRSIAmp N, DeltaAmplitude.complexNormSq (displayAmp ψ)
          = Finset.univ.sum (fun i : Fin (N + 1) => bornWeight ψ i))
      ∧ (∀ ψ : FRSIAmp N, Normalized ψ ↔ DeltaAmplitude.ComplexNormalized (displayAmp ψ))
      ∧ (∀ z : FRSIExpr, (eval z).re ∈ MinimalField.rsField ∧ (eval z).im ∈ MinimalField.rsField) :=
  ⟨bornWeight_nonneg, display_normSq_eq, normalized_iff_display,
    fun z => ⟨eval_re_mem z, eval_im_mem z⟩⟩

/-- **F_RS[i] finite amplitude headline.** Finite complex amplitudes can be
carried by the finite-description scalar carrier `F_RS[i]`; ambient ℂ is only the
display. The display preserves real/imaginary carrier membership, Born weights,
and normalization. -/
theorem frsi_amplitude_headline (N : ℕ) :
    (∀ ψ : FRSIAmp N, ∀ i : Fin (N + 1), 0 ≤ bornWeight ψ i)
      ∧ (∀ ψ : FRSIAmp N, DeltaAmplitude.complexNormSq (displayAmp ψ)
          = Finset.univ.sum (fun i : Fin (N + 1) => bornWeight ψ i))
      ∧ (∀ ψ : FRSIAmp N, Normalized ψ ↔ DeltaAmplitude.ComplexNormalized (displayAmp ψ))
      ∧ (∀ z : FRSIExpr, (eval z).re ∈ MinimalField.rsField ∧ (eval z).im ∈ MinimalField.rsField) :=
  ⟨bornWeight_nonneg, display_normSq_eq, normalized_iff_display,
    fun z => ⟨eval_re_mem z, eval_im_mem z⟩⟩

/-- Display of an `F_RS[i]` expression into the ambient complex field. -/
noncomputable def eval : FRSIExpr → ℂ :=
  fun z => ⟨FRSCarrier.eval z.re, FRSCarrier.eval z.im⟩

What this page does not claim

It does not claim that every quantum state in nature has a finite F_RS[i] description. It does not introduce dynamics, measurement, or the Born rule as a physical law. It does not identify F_RS[i] with ℂ as fields; the display map is a translation, not an equality.

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/FRSComplexAmplitude.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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