Theorem: Evolution, channels, and conserved free energy

The specified dynamics has a unique finite-dimensional unitary evolution. Its radiation prefixes are CPTP channels on all message operators:

A complementary channel is obtained by tracing the accessible prefix instead, retaining the whole source, the key purification, the dephasing environment, and unused cells.

Every printed interaction commutes with $H_0$. Consequently the expectation and full probability distribution of $H_0$ are conserved.

Proof. All generators are bounded matrices with smooth coefficients. Existence and uniqueness follow from the finite linear evolution equation; unitarity follows from self-adjointness. The pure-environment evolution composed with the encoding is an isometry. Partial trace gives CPTP channels and their complements.

The internal key and message controls act in degenerate sectors. For key emission, the source fuel decreases by ${\epsilon }_{\mathrm{cell}}$ while $C_1$ gains the same amount. For ciphertext emission the analogous equality holds for $F_2,C_2$; $Q$ is degenerate. Every term of occupation emission takes source energy $E_i$ to $E_i-n{{\omega }_i}^{\mathrm{cell}}$ and adds $n{{\omega }_i}^{\mathrm{cell}}$ in the cell. Thus every interaction commutes with $H_0$. The full evolution commutes with $H_0$, proving the conservation statement.