Foundation Primitive Recognition Calculus Delta Amplitude

Finite amplitude vectors

In quantum mechanics, a wavefunction assigns a complex number to each possible state, and the square of that number's magnitude gives the probability of finding the system in that state. The framework called Recognition Science begins with a stripped-down version: a finite list of real numbers, one for each of N+1 alternatives. This is a delta amplitude, where "delta" signals the discrete, finite nature of the list, as opposed to the continuous functions of ordinary quantum theory.

The core definitions are simple. A real amplitude vector is just a function from the finite index set to the real numbers. Its squared norm is the sum of the squares of its entries. The Born weight of a single entry is its square. A vector is normalized when its squared norm equals one. The framework's library, a machine-checked collection of formal theorems, proves three facts about these objects: every Born weight is nonnegative, the Born weights of a normalized vector sum to one, and any norm-preserving transformation keeps a normalized vector normalized.

These three statements are the native finite core of unitary evolution. In ordinary quantum mechanics, unitary operators preserve the total probability. Here, the same idea appears without any Hilbert space machinery: a finite linear map that preserves the squared norm automatically preserves normalization. The library proves this directly for both real and complex amplitudes, with the complex version using |z|² = re² + im² as its Born weight.

The headline theorem, delta_amplitude_headline, packages all three properties into a single statement for any finite N. Its complex counterpart does the same for complex amplitudes. The framework's point is that these probabilistic rules do not require the infinite-dimensional apparatus of standard quantum theory; they emerge already at the level of finite lists. Hilbert space, in this account, is a later display completion, not the starting point.

What this establishes in plain language is that the basic rules of quantum probability are not an accident of continuous mathematics. They are forced by the simplest possible discrete structure: a finite list of numbers whose squares must behave like probabilities. The framework proves this once, for all finite N, and the same proof works for real and complex entries. This is a small but load-bearing result, because it shows the probabilistic core of quantum mechanics can stand on finite ground before any infinite limits are taken.

/-- **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⟩

/-- **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⟩

What this page does not claim

This module does not derive the full Born rule for continuous wavefunctions. It does not prove that Hilbert space is unnecessary for quantum mechanics. It does not connect these finite amplitudes to any specific physical measurement.

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)

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: