Encoding and localization

The message and source are qubits. The encoding is the fixed unitary identification

where

The symbols $|+⟩$, $|-⟩$ here denote $Z_S$ eigenstates, not $X$ eigenstates.

The initial state is

There is no message dependence in the initial field and no additional information-bearing radiation factor.

Choose a real compact reduced profile $f$, and set $F=\mathfrak{F}[f]$. Its support is strictly exterior for every fixed compact spatial support. The same qubit controls the full shell. This is an effective extended source in a specified slicing, not a covariant microscopic matter field.