Foundation Primitive Recognition Calculus Delta Amplitude Delta Complex Amplitud

Finite complex amplitudes

In quantum mechanics, a system's state is often written as a vector of complex numbers, one per possible outcome. The standard rule, named for Max Born, turns each complex number into a probability by taking the square of its absolute value, |z|² = re² + im². The framework's machine-checked library of formal theorems examines what this rule requires when the vector is finite, a list of N+1 complex numbers rather than a function on an infinite space.

The central result, a theorem named delta_complex_amplitude_headline, bundles three facts about such finite complex amplitude vectors. First, every Born weight is nonnegative, so no outcome ever gets a negative probability. Second, if the vector is normalized, meaning the sum of all Born weights equals one, then those weights form a valid probability distribution. Third, any finite linear transformation that preserves the squared norm also preserves normalization, so a normalized state stays normalized after evolution. This last point is the finite native core of unitary evolution, the kind of change that quantum theory treats as reversible and probability-preserving.

In Recognition Science, the declaration establishes that this finite layer is self-contained. The framework models amplitudes as primitive data on a discrete ledger of recognition events, a record of which outcomes have been considered. The theorem shows that the Born rule and norm preservation do not require an infinite Hilbert space to be stated or proved; they hold already for finite lists. The framework's library treats Hilbert space as the display completion, the limit one reaches by letting the list grow, not as the starting point.

The declaration does not claim that finite complex amplitudes are the whole of quantum mechanics. It does not derive the Born rule from deeper principles; it assumes the rule as a definitional choice. It does not prove that every norm-preserving map is linear, nor that any particular physical system is described by a finite list. The theorem concerns the internal consistency of the finite amplitude model, not its adequacy for all of physics.

What the result changes is the order of explanation. A reader can now see that the probabilistic core of quantum theory, nonnegative weights, total probability one, and norm-preserving evolution, is a finite combinatorial fact, not a consequence of infinite-dimensional analysis. The framework's claim is that this finite layer is the primitive one, and Hilbert space is the picture one gets by completing it.

/-- **Complex finite-amplitude headline.** Complex amplitudes already have a
native finite layer: Born weights are nonnegative, normalized finite complex
amplitudes sum to one, and norm-preserving finite complex transformations
preserve normalization. Hilbert space remains the display completion. -/
theorem delta_complex_amplitude_headline (N : ℕ) :
    (∀ ψ : ComplexAmp N, ∀ i : Fin (N + 1), 0 ≤ complexBornWeight ψ i)
      ∧ (∀ ψ : ComplexAmp N, ComplexNormalized ψ →
          Finset.univ.sum (fun i : Fin (N + 1) => complexBornWeight ψ i) = 1)
      ∧ (∀ U : ComplexAmp N → ComplexAmp N, ComplexNormPreserving U →
          ∀ ψ : ComplexAmp N, ComplexNormalized ψ → ComplexNormalized (U ψ)) :=
  ⟨complexBornWeight_nonneg, fun _ hψ => complex_born_weights_sum_one hψ,
    fun _ hU _ hψ => complex_normalized_of_normPreserving hU hψ⟩

theorem complexBornWeight_nonneg {N : ℕ} (ψ : ComplexAmp N) (i : Fin (N + 1)) :
    0 ≤ complexBornWeight ψ i := by
  unfold complexBornWeight
  nlinarith [sq_nonneg (ψ i).re, sq_nonneg (ψ i).im]

theorem complex_born_weights_sum_one {N : ℕ} {ψ : ComplexAmp N} (hψ : ComplexNormalized ψ) :
    Finset.univ.sum (fun i : Fin (N + 1) => complexBornWeight ψ i) = 1 := by
  simpa [ComplexNormalized, complexNormSq] using hψ

theorem complex_normalized_of_normPreserving {N : ℕ} {U : ComplexAmp N → ComplexAmp N}
    (hU : ComplexNormPreserving U) {ψ : ComplexAmp N} (hψ : ComplexNormalized ψ) :
    ComplexNormalized (U ψ) := by
  unfold ComplexNormalized
  rw [hU ψ, hψ]

What this page does not claim

The declaration does not derive the Born rule from deeper principles. The declaration does not prove that every norm-preserving map is linear. The declaration does not claim that any particular physical system is described by a finite list of amplitudes.

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/DeltaAmplitude.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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