Two declared cutoff-dipole models with equal transition-energy density: the rate coefficient does not discriminate them, the formal photon-energy laws differ, with a detected-law reading at nonzero coupling

Adopt the declared finite-level cutoff dipole model The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector and its spectral-transition reading Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection, the declared ideal energy-window detection model The declared ideal counting, energy-window and analyser detection models of the radiation sector clause (c) and the preparation and apparatus discipline P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Declare three models, all with the same matter data $H_a$, $D_j^{\mathrm{c}\mathrm{h}}$ and the same cutoff $\mathrm{\Lambda }$ and coupling ${\eta }_0$, and with a declared upward level pair $(a,b)$, $\mathrm{\Delta }\in (0,\mathrm{\Lambda })$:

1. model $M_1$: cutoff mode data whose single-vertex photon density is ${\nu }_b^{(1)}=J_1\lambda$ with a declared nonnegative Lebesgue-measurable integrable representative $J_1:\mathbb{R}\to [0,\mathrm{\infty })$ of its $L^1(\lambda )$ class, $J_1=0$ $\lambda$-almost everywhere off $(0,\mathrm{\Lambda }]$ and $J_1$ continuous at $\mathrm{\Delta }$ with $J_1(\mathrm{\Delta })>0$;
2. model $M_2$: same matter and cutoff data, single-vertex photon density ${\nu }_b^{(2)}=J_2\lambda$, with a declared representative $J_2:\mathbb{R}\to [0,\mathrm{\infty })$ having the same properties and $J_2(\mathrm{\Delta })=J_1(\mathrm{\Delta })$;
3. model $L$ (the declared ideal sharp-line model): a declared spectral model in which the energy of an emitted and detected quantum is the value $\mathrm{\Delta }$ with probability one, so that every energy window $B\subseteq \mathbb{R}$ has probability ${\mathbf{1}}_B(\mathrm{\Delta })$.

Assume in addition that $J_1$ and $J_2$ are not proportional: there is no $\gamma \ge 0$ with $J_1=\gamma J_2$ $\lambda$-almost everywhere. The single-vertex laws below are the formal normalized spectral laws of the cited definition for all real ${\eta }_0$. Their reading as normalized one-photon preparations uses ${\eta }_0\mathrm{\ne }0$; at zero coupling the physical single-vertex vector is zero and no such preparation is formed. No zero-coupling conditioning is asserted. Then:

(a) (equal rate coefficients) the model rate coefficients of clause (c) of The golden-rule limit of the single-vertex emission coefficient and its reading as a model rate coefficient coincide, ${\mathrm{\Gamma }}_1=\frac{2\pi {\eta }_0^2}{\mathrm{\hslash }}{J}_1(\mathrm{\Delta })=\frac{2\pi {\eta }_0^2}{\mathrm{\hslash }}{J}_2(\mathrm{\Delta })={\mathrm{\Gamma }}_2;$ in particular a comparison of the leading rate coefficients alone does not discriminate $M_1$ from $M_2$;

(b) (different formal laws and conditional detected-energy reading) for every $t>0$ the normalised single-vertex photon-energy laws $P_t^{(1)}$ and $P_t^{(2)}$ of clause (e) of Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection differ: there is a Borel set $B\subseteq \mathbb{R}$ with $P_t^{(1)}(B)\mathrm{\ne }{P}_t^{(2)}(B)$; when ${\eta }_0\mathrm{\ne }0$, these are distinct predicted ideal energy-window probabilities in the declared normalized single-vertex one-photon preparations. This is a distinction between specified model laws, with no finite-sample discrimination guarantee; at ${\eta }_0=0$ only the formal laws are compared;

(c) (the sharp line is a limit, not a finite-time property) if $J_1>0$ $\lambda$-almost everywhere on a Borel set $B$ of positive Lebesgue measure with $\mathrm{\Delta }\mathrm{\notin }B$, then $P_t^{(1)}(B)>0$ for every $t>0$ whereas $L$ assigns probability $0$ to $B$; on the other hand $P_t^{(1)}(B)\to 0$ as $t\to \mathrm{\infty }$ for every Borel $B$ bounded away from $\mathrm{\Delta }$. Thus the declared sharp-line model is the idealisation that the single-vertex law approaches in the limit, not a property of the finite-time law;

(d) (what the comparison is) clauses (a)-(c) are statements about the three declared models. They construct no laboratory comparison and report no data: no laboratory apparatus or preparation protocol, calibration, detector response, efficiency, background, trial count, frequency or uncertainty budget is supplied, and none is invented. Comparing any of these predictions with reported data would test the combined model and apparatus assumptions under P7, would require a declared sampling and measurement model, and would carry finite-sample fluctuations and systematic uncertainties; it would not prove a model.

• What the intended discriminator is. The comparison identifies a concrete difference on which two specified models disagree — the energy-resolved law versus the leading rate coefficient — and not a comparison of "quantum theory" with "all classical physics". Neither outcome would establish or refute the framework; it would test the specified models together with the apparatus assumptions.
• Model data are not measured densities. The densities $J_1,J_2$ are declared model data; identifying either with a laboratory mode content or environment requires its own tested identification and uncertainty budget under P7.