Foundation Primitive Recognition Calculus Delta Amplitude Complex Norm Sq Nonneg

The nonnegative norm

In quantum mechanics, a complex amplitude is a number whose squared magnitude gives a probability weight. The Recognition Science framework works with finite lists of such amplitudes, and the declaration complexNormSq_nonneg proves a basic property: for any finite list of complex numbers, the sum of the squared magnitudes is never negative. This is the finite, discrete version of the familiar fact that a vector's squared length is always nonnegative.

The proof is direct. Each complex number z has a squared magnitude |z|² = re² + im², which is a sum of two squares and therefore nonnegative. The declaration sums these nonnegative terms over a finite index set, and a sum of nonnegative reals is nonnegative. The machine-checked library records this as a proved theorem, not an assumption.

In Recognition Science, this fact is part of a larger structure. The framework models a ledger, a discrete record of recognition events, and assigns amplitudes to possible alternatives. The nonnegativity of the squared norm is the first step toward interpreting these weights as probabilities: if the total weight is one, then each individual weight is between zero and one, and the sum of all weights is one. The declaration complexNormSq_nonneg establishes only the nonnegativity part; it does not by itself prove that weights sum to one or that they behave like probabilities.

The declaration also does not claim anything about infinite-dimensional spaces, continuous wavefunctions, or the full apparatus of Hilbert space. It is a finite, discrete statement. It does not assert that any particular amplitude vector is normalized, only that the squared norm is nonnegative. It does not relate this norm to any physical measurement or to the framework's other constants. It is a building block, not a complete theory.

theorem complexNormSq_nonneg {N : ℕ} (ψ : ComplexAmp N) : 0 ≤ complexNormSq ψ := by
  unfold complexNormSq
  exact Finset.sum_nonneg (fun i _ => complexBornWeight_nonneg ψ i)

What this page does not claim

The declaration does not prove that weights sum to one. The declaration does not apply to infinite-dimensional spaces. The declaration does not connect the norm to any physical measurement or framework constant.

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: