The golden-rule limit of the single-vertex emission coefficient and its reading as a model rate coefficient

Adopt the model, the declared upward level pair $(a,b)$ and the page data $c_j,M,{\nu }_b,A_b$ of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector and Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection, and the single-vertex emission coefficient ${\gamma }_b(t)$ of First-order emission and absorption amplitudes of the cutoff dipole model, and the exact model probability of the one-photon level-b projection up to the Dyson tail clause (a). Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice) and the density hypothesis: ${\nu }_b=J_b\lambda$ for a declared Lebesgue-measurable representative $J_b:\mathbb{R}\to [0,\mathrm{\infty })$ of an $L^1(\lambda )$ class, with $J_b=0$ $\lambda$-almost everywhere on $\mathbb{R}\setminus (0,\mathrm{\Lambda }]$, continuous at $\mathrm{\Delta }$. All point values below refer to this representative; continuity means that for every $ϵ>0$ there is $\delta >0$ such that $\mathrm{\mid }{J}_b(E)-J_b(\mathrm{\Delta })\mathrm{\mid }<ϵ$ whenever $\mathrm{\mid }E-\mathrm{\Delta }\mathrm{\mid }<\delta$. Two representatives of the same class that are continuous at $\mathrm{\Delta }$ have the same value there: otherwise continuity of their difference gives a nonzero difference throughout a neighborhood of positive measure, contradicting equality almost everywhere. Then:

(a) (resonant limit) if $0<\mathrm{\Delta }<\mathrm{\Lambda }$ and $J_b(\mathrm{\Delta })>0$, then ${\lim }_{t\to \mathrm{\infty }}\frac{{\gamma }_b(t)}{t}=\frac{2\pi {\eta }_0^2}{\mathrm{\hslash }}{J}_b(\mathrm{\Delta });$

(b) (above-cutoff limit) if $\mathrm{\Delta }>\mathrm{\Lambda }$, then ${\lim }_{t\to \mathrm{\infty }}{\gamma }_b(t)\mathrm{/}t=0$; the boundary case $\mathrm{\Delta }=\mathrm{\Lambda }$ is not asserted;

(c) (rate reading) the theorem declares the model rate coefficient ${\mathrm{\Gamma }}_b:=\frac{2\pi {\eta }_0^2}{\mathrm{\hslash }}{J}_b(\mathrm{\Delta })(0<\mathrm{\Delta }<\mathrm{\Lambda }),$ read in ${\mathrm{s}}^{-1}$ by the unit convention of the model (the interaction term is read in joules), as the model's reading of the limit in (a); the identification of the mode density $J_b$, the cutoff $\mathrm{\Lambda }$ or the coupling ${\eta }_0$ with laboratory data is a tested identification with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions);

(d) (scope) clauses (a)-(c) concern the limit of the explicit single-vertex coefficient per unit time. They assert no exact exponential decay of any population, no lifetime or linewidth, no weak-coupling, Markov or environmental property of the model's exact evolution, no all-time validity of a rate law and no measured rate; in particular ${\mathrm{\Gamma }}_b$ is a property of the declared model together with its declared mode density, not an observed decay constant.

• What the limit is not. The limit in clause (a) is a limit of ${\gamma }_b(t)\mathrm{/}t$ for the single-vertex coefficient; it is not derived from the exact evolution, and its interpretation as a rate law over a time window requires the separate physical assumptions of a weak-coupling window and a mode-density model, which this pair states but does not prove.
• Cutoff behaviour. Clause (b) records the model's cutoff: a transition whose energy exceeds the declared cutoff has single-vertex coefficient per unit time tending to zero in the limit, because the photon density vanishes above $\mathrm{\Lambda }$; the boundary case is left unasserted rather than assigned a value.