One specified smooth packet

Choose $0<{\omega }_{-}<{\omega }_{+}$, and define

where $Z$ is its squared $L^2$ integral before normalization. This is a real normalized smooth compact-band packet.

For a chosen $\vartheta$, the construction consists of:

1. solve the same s-wave radial scattering equation used for $\tau ,{\rho }_{\mathrm{I}}$;

2. form $k_{\ast }$ and the real spectral Cauchy data (eq:targetCauchydata);

3. calculate the spatial derivative-tail certificate (eq:ordinarytailbound);

4. form $f_{R,T}$, its bulk representative, the coupling $\rho =-F_{R,T}$, and the explicitly time-sliced probe test;

5. calculate the constants $C_X,L_R,\Lambda ,b_g,b_J,N_R$;

6. choose $ϵ$ by Theorem thm:tolerance and prepare ${\sigma }_{R,ϵ}$.

These are analytical specifications. Numerical execution of this list is not asserted.

The inherited mean is

It determines the target probability in (eq:Hawkingbaseline); no numerical greybody table is inserted.

For $\vartheta \ne 0$, that target probability lies strictly between $1/2$ and $1$. Choosing an error smaller than half its distance from $1$ gives a nontrivial probability certificate rather than the vacuous upper bound one.