Proposition: Energy interpretation is not identified by the frozen process data

The unchanged XXII-C data do not determine the energy observable of a laboratory qubit.

More precisely, the same valid process and complete instrument tables admit distinct appended energy assignments with different expectation values, because energy assignments and energy-conservation restrictions are not part of the frozen declaration.

Proof. Use $W_0=I_{\mathcal{H}}/8$. A definite-order realization prepares an independent maximally mixed input qubit for each laboratory and discards its outgoing qubit. For local instrument Choi matrices,

which is the product of the corresponding maximally mixed input probabilities.

Append either of the laboratory energy assignments

Their input expectations are respectively $0$ and $\omega /2$. Neither assignment changes $W_0$, an admitted instrument map, or any frozen probability table. The frozen declaration contains no restriction requiring the instruments to conserve either assigned energy.

If a physical implementation is required to obey an energy balance, its apparatus, work supply, generators, and preparation constraints must be added and checked. They are absent from this data set.