Parseval’s Theorem · Transport Conservation

Claim: Parseval’s theorem expresses transport conservation.
Result: Motion content is invariant under change of basis (time ↔ mode).

1. The Classical Statement

For a function decomposed into Fourier modes, Parseval’s theorem states:

$$ \int |f(t)|^2 \, dt = \sum_{n=-\infty}^{\infty} |c_n|^2 $$

The total squared amplitude in time equals the total squared amplitude across modes.

This equality does not arise from algebraic symmetry.

It expresses conservation of motion under representation change.

Fourier analysis does not create or destroy motion.

It redistributes the same transported motion across phase modes.

Parseval guarantees that no motion is lost or gained in this redistribution.

The Basel sum appears when computing mean-square amplitude of periodic transport.

Parseval explains why the sum must converge to a finite value.

$$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \text{finite phase density} $$

The value $\pi^2/6$ is fixed by transport geometry.

$|f(t)|^2$ measures motion density in time.

$|c_n|^2$ measures motion density in phase space.

Parseval states that total motion density is invariant.

Motion adds linearly.

Energy and variance add quadratically.

Parseval enforces quadratic conservation across transport bases.

Method A: Transport Conservation
Motion content does not depend on representation.

Method B: Orthogonality of Modes
Independent phase directions sum without cross-terms.

Both interpretations agree.

Parseval’s theorem is not about Fourier series.

It is about conservation of motion.

Changing basis does not change what is transported.