Proposition: Number-preserving Fock extension

Embed the code basis into a finite set of oscillator occupation vectors. Extend $H,L_{\alpha }$ diagonally by zero on its orthogonal complement. The resulting bounded operators define a CPTP semigroup on the full finite-mode Fock space. It preserves the code, finite number moments, and total number expectation:

Consequently, residual states subjected to this extension retain the number-based concentration bounds of Section sec:static.

Proof. The extended operators are bounded, mutually commuting, and commute with $N$. The same random-unitary construction defines the normal channel on trace-class operators. Each unitary preserves number spectral subspaces and number moments. Average the exact number identity. The concentration theorem then applies to the resulting residual states.