Foundation Primitive Recognition Calculus Frscomplex Amplitude Eval Im Mem

The imaginary part stays in the field

In quantum mechanics, a complex amplitude is a pair of real numbers, one for the real part and one for the imaginary part. The Recognition Science framework, which builds physics from a discrete record of events called a ledger, represents these amplitudes with expressions built from its own finite-description scalar field. The declaration eval_im_mem is a theorem in the framework's machine-checked library of formal theorems. It states that when such an expression is evaluated into the ordinary complex numbers, its imaginary part always lands back inside the framework's own real carrier field, the set of values the ledger can actually produce.

The theorem is a small but load-bearing piece of a larger construction. The framework defines a complex expression as a pair of real expressions, one for each component. Its evaluation function maps this pair to a complex number by evaluating the two parts separately. The theorem eval_im_mem proves that the imaginary component of that result is not just any real number; it is a member of the framework's minimal field, the smallest collection of values generated by the ledger's operations. The proof is immediate from the definition, a reflection that the framework's complex numbers are built to stay inside its own arithmetic.

The consequence is that the framework can carry finite complex amplitudes without leaving its own descriptive universe. The headline theorem of the same file bundles this with three companions: Born weights, the squared magnitudes that give probabilities, are nonnegative; the squared norm of a displayed amplitude equals the sum of its Born weights; and normalization in the framework's terms is exactly normalization in the ordinary complex display. Together they show that the familiar quantum-mechanical machinery of amplitudes, probabilities, and normalization can be reproduced using only the ledger's finite descriptions, with the ambient complex numbers serving as a display layer rather than a source of new content.

What the theorem does not claim is just as precise. It does not assert that every complex number can be represented, nor that the framework's field is closed under complex conjugation or other operations beyond those it defines. It does not say that probabilities derived from these amplitudes match experiment; that is an empirical question outside the theorem's scope. And it does not claim that the framework's complex numbers are the same as the ordinary complex field, only that their evaluations land in the carrier field. The theorem is a structural guarantee about where the imaginary part lives, not a statement about what physics follows.

/-- 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

/-- 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

/-- **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⟩⟩

What this page does not claim

The theorem does not claim every complex number can be represented by the framework's expressions. The theorem does not claim probabilities derived from these amplitudes match experimental results. The theorem does not claim the framework's complex numbers are identical to the ordinary complex field.

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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