Physical regulator and thermal calibration

The following construction supplies a concrete, conserving completion rather than an unspecified perturbation ${\rho }_{\mathrm{thermal}}+\delta {\rho }_x$.

Consider two identical neutral scalar species in the same s-wave energy-eigenmode regulator, both with frequency $\omega >0$. They are additional declared field degrees of freedom in this phenomenological example, not a claim that such particles have been observed. There is no imposed conserved flavor charge. Every radiation configuration below has zero angular momentum and zero electric charge.

The stationary outgoing mean occupation of either mode is

For $0<q<1$, the two-mode thermal state is

Its total radiation energy in this regulator is $n\omega$ on the $n$-particle sector.

A normalized time-localized packet is not an exact energy eigenmode. Exact conservation in this section refers to the energy-eigenmode regulator. Transferring the construction to finite-time packets requires an independently bounded spectral-width and dynamics error; no sharp-frequency, sharp-time identification is made.

Fix a finite integer $L\ge 0$ with

and define

This is the source state of a finite-energy completion, not a data-selection rule. All attempts in that model are retained.

Because the discarded thermal sectors are orthogonal,

Unbounded observables require their own moment audit; trace distance alone does not control energy or number expectation. Here that audit is explicit: