Proposition: Heating is not source identification

For a pure force-diffusion term $-D(t){\mathcal{K}}_x$, its contribution to momentum heating is

If two models have identical centre-of-mass generators throughout a heating calibration and the interferometer, all admitted centre-of-mass calibration instruments have identical laws. Total heating alone does not identify the environmental part of $D_{\mathrm{total}}$.

Proof. On a common moment domain,

Move the double commutator to the observable. This is the diffusion contribution; Hamiltonian or damping terms may also change $⟨p^2⟩$.

Identical generators give identical states before each heating instrument and hence identical recorded distributions, as in Theorem thm:width.