Foundation Primitive Recognition Calculus Frscomplex Amplitude Born Weight Nonne

The nonnegative weight

The Born rule is the standard recipe that turns a quantum state into a probability: for a state described by a complex number, the probability of an outcome is the square of that number's magnitude. Since a square can never be negative, any honest implementation of the rule must produce a nonnegative weight. The declaration bornWeight_nonneg is the machine-checked proof that the Recognition Science framework's native version of this weight satisfies exactly that requirement.

The framework builds its amplitudes from a finite-description scalar carrier it calls F_RS[i], a structure whose real and imaginary parts are both expressions in the framework's own real carrier field. The bornWeight definition takes such an amplitude and an index, evaluates the real and imaginary parts, squares each, and adds them. The theorem bornWeight_nonneg states, for every amplitude and every index, that this sum is greater than or equal to zero. The proof is short: it invokes the algebraic fact that a square is nonnegative, applied to each of the two squared components.

What the theorem does not claim is just as important. It does not assert that the weights sum to one; that separate condition, called normalization, is a different definition in the same file, and the framework proves it is equivalent to normalizing the displayed complex amplitude. The theorem also does not claim that the weights describe actual measurement statistics, nor that the framework's amplitudes correspond to any particular physical system. It is a structural guarantee: if the framework's amplitudes are used, the Born-weight formula cannot produce a negative number.

In Recognition Science, this result is one small piece of a larger program: the framework aims to derive physical structure from a primitive recognition ledger, and here it shows that its finite-description amplitudes behave like ordinary quantum amplitudes in at least this basic respect. The practical consequence is that any probability interpretation built on these weights starts from a sound foundation, free of the sign errors that can plague hand-written implementations.

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)]

/-- Normalization over `F_RS[i]` is exactly normalization of the ambient complex
display. -/
def Normalized {N : ℕ} (ψ : FRSIAmp N) : Prop :=
  Finset.univ.sum (fun i : Fin (N + 1) => bornWeight ψ i) = 1

What this page does not claim

The theorem does not claim the weights represent actual measurement probabilities. The theorem does not claim the framework's amplitudes describe any specific physical system. The theorem does not claim the weights sum to one; normalization is a separate condition.

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)

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