Energy-labelled emission

For an energy-eigenmode regulator, let ${\mathcal{H}}_B(M,\mathbf{q})$ be an internal source sector with energy $M$ and declared conserved labels $\mathbf{q}$. A one-step emission interface has the form

where the radiation occupation configurations $\alpha$ have energies $E_{\alpha }\ge 0$. Vacuum and no-emission configurations are included.

Writing

physicality requires

A specified occupation calibration on a code subspace is an operator constraint of the form

or a weaker expectation constraint on explicitly selected source states. Which of these is imposed must be declared.

Neither (eq:operatorcalibration) nor a diagonal probability distribution fixes the off-diagonal terms $K_{\alpha }\rho {K_{\beta }}^{†}$. Sector dimensions must also permit the isometry: a conservation label is not a proof that sufficiently many internal states exist.

A finite emission interface is used here as an effective completion on a prepared subspace. It is not identified with a microscopic black-hole Hilbert space or assigned dimension $e^{S_{\mathrm{BH}}}$ by assumption. Successive steps must act on the actual residual output; resetting that residual between steps would define a different dynamics.