Bound-state and incomplete-decay warnings

For another regular coupling, an energy $E$ outside the coupled support may solve

When the vector is square integrable, an eigenvector is proportional to

with prepared spectral weight

These formulas follow directly from the eigenvalue equations. In-band eigenvectors require their own integrability conditions.

An atom does not imply $q(T)\ge {Z_E}^2$ at every time, because other spectral components can interfere. A valid uniform statement is: if an atom has weight $Z>1/2$, then

so all-field recovery error is at least $c_d(2Z-1{)}^2$ for every $T$.

If the remaining measure has an $L^1$ absolutely continuous density, its Fourier transform tends to zero. The long-time average of $q(T)$ is the sum of squares of the distinct atomic weights. For infinitely many atoms, finite-sum approximation in total mass justifies passage to the limit.

Theorem thm:C5 avoids these obstructions for its prepared code only. It does not assert decay for every regular coupling. For example, $g=0$ gives $q(T)=1$ and error $c_d$ for all $T$.