Foundation Primitive Recognition Calculus Frscomplex Amplitude

Complex amplitudes from finite descriptions

In quantum mechanics, a complex amplitude is a number with a real and an imaginary part, and its squared magnitude gives a probability. The complex numbers form a continuous field, so a typical amplitude can require an infinite amount of information to specify exactly. Recognition Science asks whether a finite, discrete description can do the same job.

The framework's answer is a construction called F_RS[i], where F_RS is a minimal field of finite expressions and the i is the usual imaginary unit added as a formal symbol. An expression in this field is simply a pair: one F_RS expression for the real part and one for the imaginary part. The framework's machine-checked library of formal theorems defines this structure and shows that it behaves exactly like a complex number when displayed into the ordinary complex field.

The key theorems are all preservation results. When you evaluate an F_RS[i] expression into the complex numbers, its real and imaginary parts land back in the real F_RS carrier, so nothing escapes the finite description. For a finite list of such amplitudes, the Born weight, which is the squared magnitude used to get probabilities, is always nonnegative, and the total norm computed from the finite components equals the norm of the displayed complex vector. A state is normalized in the finite description if and only if its complex display is normalized. The framework proves all of this in one headline theorem, frsi_amplitude_headline.

In plain language, the construction establishes that complex quantum amplitudes do not require the full continuum of the complex numbers. A finite-description carrier, built from a minimal field plus a formal imaginary unit, can carry amplitudes, Born weights, and normalization without loss. The ambient complex field appears only as a display layer, not as the underlying data.

This matters because it narrows the gap between the continuous mathematics of quantum theory and the discrete ledger of recognition events that the framework uses as its starting point. If amplitudes can be finite, then the probabilistic core of quantum mechanics can live inside a discrete recognition calculus rather than being imported from outside it.

/-- A finite-description complex scalar in `F_RS[i]`: real and imaginary parts
are both `F_RS` expressions. -/
structure FRSIExpr where
  re : FRSCarrier.FRSExpr
  im : FRSCarrier.FRSExpr
  deriving DecidableEq, Repr

/-- Every displayed real part remains in the real RS carrier field. -/
theorem eval_re_mem (z : FRSIExpr) : (eval z).re ∈ MinimalField.rsField := by
  simpa [eval] using FRSCarrier.eval_mem z.re

/-- Every displayed imaginary part remains in the real RS carrier field. -/
theorem eval_im_mem (z : FRSIExpr) : (eval z).im ∈ MinimalField.rsField := by
  simpa [eval] using FRSCarrier.eval_mem z.im

theorem bornWeight_nonneg {N : ℕ} (ψ : FRSIAmp N) (i : Fin (N + 1)) :
    0 ≤ bornWeight ψ i := by
  unfold bornWeight
  nlinarith [sq_nonneg (FRSCarrier.eval (ψ i).re), sq_nonneg (FRSCarrier.eval (ψ i).im)]

theorem normalized_iff_display {N : ℕ} (ψ : FRSIAmp N) :
    Normalized ψ ↔ DeltaAmplitude.ComplexNormalized (displayAmp ψ) := by
  unfold Normalized DeltaAmplitude.ComplexNormalized
  rw [display_normSq_eq]

What this page does not claim

The construction does not prove that all quantum mechanics can be reduced to finite descriptions. The construction does not define the physical interpretation of the F_RS carrier beyond its algebraic role. The construction does not provide a numerical prediction for any measured quantity.

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)

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