Exact finite scattering for declared incoming preparations

For mean occupation at most $E$, using a common cutoff and common flag extension gives the additional bound

It is not inferred from matching mean counts.

Frequency-dependent amplitudes generally leave the selected packet span:

A piecewise-constant unitary $S_a$ on subbands preserves the packet span when each packet belongs to one subband. Its discrepancy must include phases and be bounded as above.

For covariance only, contraction-valued amplitudes with $\sup |\tau -\tilde{\tau }|\le {ϵ}_{\tau }$ imply

This does not control coherent scattering phases.