Foundation Primitive Recognition Calculus Frscomplex Amplitude Display Born Weig

The display bridge

In quantum mechanics, the probability of finding a particle in a particular state is the squared magnitude of a complex number called the amplitude. That number, |ψ|², is the Born weight. The declaration display_bornWeight_eq is a machine-checked theorem in the Recognition Science framework's library of formal theorems. It states that when a finite, exactly described complex expression is displayed into the ordinary complex plane, the Born weight computed there is identical to the weight computed by the expression's own native formula. The proof is a direct unfolding of definitions, so the equality holds by construction.

This is a bridge between two ways of writing the same quantity. The framework's native complex numbers are finite descriptions, built from two real-valued expressions in its own carrier field. The display function maps each such expression to a value in the ambient complex numbers, the usual ℂ of textbook physics. The theorem says the bridge preserves the physically meaningful Born weight: no information is lost or added in the translation. A companion theorem extends the same preservation to the total norm, the sum of Born weights over all states, and to the condition of normalization, where that sum equals one.

The framework's headline claim is that finite complex amplitudes can be carried by its own finite-description scalar carrier, with the ordinary complex plane serving only as a display. This theorem is the technical guarantee that the display is faithful for the quantity that matters in measurement. It does not say that the framework's native numbers are the same as complex numbers, nor that every complex number can be represented. It only certifies that for the finite expressions the framework builds, the Born weight is unambiguous.

The theorem also proves that every displayed Born weight is nonnegative, as a probability must be. This is a small but essential check: the framework's native formula, a sum of two squares of real values, cannot produce a negative weight. The normalization condition, which is what turns amplitudes into probabilities, is exactly equivalent whether expressed in the native system or in the displayed complex plane. A reader can trust that a normalized native amplitude is a normalized complex amplitude, and vice versa.

What this theorem does not claim is broader reach. It does not assert that the framework's finite descriptions can capture every quantum state, nor that the Born rule itself is derived within the framework. The Born weight is taken as the definition of probability, not proved from deeper principles here. The theorem is about the consistency of two representations, not about the physical origin of probability. It is a bridge, not a foundation for measurement.

/-- The displayed complex Born weight agrees with the native F_RS[i] formula. -/
theorem display_bornWeight_eq {N : ℕ} (ψ : FRSIAmp N) (i : Fin (N + 1)) :
    DeltaAmplitude.complexBornWeight (displayAmp ψ) i = bornWeight ψ i := by
  rfl

/-- The displayed complex Born weight agrees with the native F_RS[i] formula. -/
theorem display_bornWeight_eq {N : ℕ} (ψ : FRSIAmp N) (i : Fin (N + 1)) :
    DeltaAmplitude.complexBornWeight (displayAmp ψ) i = bornWeight ψ i := by
  rfl

/-- The displayed complex norm agrees with the native finite F_RS[i] norm formula. -/
theorem display_normSq_eq {N : ℕ} (ψ : FRSIAmp N) :
    DeltaAmplitude.complexNormSq (displayAmp ψ)
      = Finset.univ.sum (fun i : Fin (N + 1) => bornWeight ψ i) := by
  rfl

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

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)]

What this page does not claim

The theorem does not assert that every complex number can be represented by the framework's finite descriptions. The theorem does not derive the Born rule from deeper principles; it only shows consistency between two representations. The theorem does not claim that the framework's native numbers are identical to ordinary complex numbers.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: