Foundation Primitive Recognition Calculus Delta Amplitude Normalized Of Norm Pre

Normalization survives

In quantum mechanics, a state is usually represented by a vector of complex numbers called an amplitude vector, and the squared size of that vector is its norm. A vector is normalized when its norm equals 1, which is what makes the squares of its entries behave like probabilities: they are nonnegative and sum to 1. A linear transformation that preserves the norm of every vector is called unitary, and unitaries are exactly the transformations that can describe how a closed quantum system evolves over time.

The declaration normalized_of_normPreserving states a simple conservation fact: if a transformation preserves the norm of every amplitude vector, and a particular vector is normalized, then the transformed vector is also normalized. In symbols, if normSq(ψ) = 1 and normSq(U ψ) = normSq(ψ) for all ψ, then normSq(U ψ) = 1. The proof is immediate: substitute the norm-preserving hypothesis and then the normalization hypothesis. The declaration is a theorem in the framework's machine-checked library of formal theorems, meaning the one-line argument has been verified by a computer.

This fact matters because it makes normalization a stable property under the allowed dynamics. If a system starts in a state whose probabilities sum to 1, and it evolves by a norm-preserving transformation, then it stays in a state whose probabilities sum to 1. Without this, the probabilistic interpretation would break after the first step of evolution. The theorem is the finite-dimensional core of the statement that unitary evolution preserves total probability; it does not require the infinite-dimensional machinery of Hilbert spaces.

The declaration does not claim that every norm-preserving transformation is linear, nor that every linear norm-preserving transformation is unitary in the sense of preserving inner products. It only concerns the squared norm, not the full inner product structure. It also does not claim that any particular physical system evolves this way; the theorem is about the mathematics of finite amplitude vectors, not about which transformations nature actually uses.

In Recognition Science, this declaration is part of a larger framework that derives physical structure from a ledger of recognition events. The framework models amplitudes as finite vectors and proves that Born weights are nonnegative, that normalized amplitudes yield total probability one, and that norm-preserving transformations preserve normalization. The declaration is one of the three components of the delta-amplitude headline theorem, which packages all three facts together. The framework's claim is that this finite structure is the native core, with Hilbert space as a later display completion.

theorem normalized_of_normPreserving {N : ℕ} {U : Amp N → Amp N}
    (hU : NormPreserving U) {ψ : Amp N} (hψ : Normalized ψ) : Normalized (U ψ) := by
  unfold Normalized
  rw [hU ψ, hψ]

/-- **Delta-native amplitude headline.** Finite amplitude data has nonnegative
Born weights; normalized finite amplitudes yield total probability one; and
norm-preserving finite transformations preserve normalization. -/
theorem delta_amplitude_headline (N : ℕ) :
    (∀ ψ : Amp N, ∀ i : Fin (N + 1), 0 ≤ bornWeight ψ i)
      ∧ (∀ ψ : Amp N, Normalized ψ →
          Finset.univ.sum (fun i : Fin (N + 1) => bornWeight ψ i) = 1)
      ∧ (∀ U : Amp N → Amp N, NormPreserving U →
          ∀ ψ : Amp N, Normalized ψ → Normalized (U ψ)) :=
  ⟨bornWeight_nonneg, fun _ hψ => born_weights_sum_one hψ,
    fun _ hU _ hψ => normalized_of_normPreserving hU hψ⟩

What this page does not claim

The theorem does not assert that any particular physical system evolves by a norm-preserving transformation. The theorem does not establish that norm preservation implies linearity or inner-product preservation. The theorem does not derive the Born rule from more primitive principles; it assumes the squared norm as the definition of probability weight.

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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