Theorem: Finite excess scalar energy

The excess outgoing-infinity and future-horizon energies are

They are finite, nonnegative, and $\delta E_{\mathrm{tot}}=\delta E_{\infty }+\delta E_{\mathrm{hor}}$.

For a packet supported in $B=[{\omega }_{-},{\omega }_{+}]$,

The exact energy of the projected classical signal ${\alpha }_{s,h}h$ is

This is not the energy of the whole band and does not imply that a nonmonochromatic packet has Hamiltonian ${\omega }_0{N}_h$.

Proof. Weyl displacement adds the rank-one mean-amplitude kernel to the centered occupation kernel. Pairing this positive change with frequency yields (eq:energy). Pointwise unitarity of the scattering matrix preserves the weighted squared norm. Lemma lem:compact, specifically (eq:energybound), proves integrability.

Cauchy–Schwarz gives

so the band lower bound follows. The energy of the specified projected signal is its ordinary frequency-weighted norm. Since frequency multiplication need not commute with the packet projection, energy cross terms with the complementary signal are not silently discarded.