Reduced exponential current and defined spectral detector adoption

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). This is the mathematical context for the tested-current, Fourier and temporally smeared field constructions used below.

Use the length-coordinate, unit, and reference conventions of Thermodynamics-GR fields, currents, reference normalization and units. Adopt two distinct model branches.

Reduced exponential-current branch. Choose a constant $\kappa >0$ in inverse metres and a smooth Killing field $\xi$ on a declared domain in the TGR framework. The real parameter $u$, measured in metres, is the selected flow parameter for $\xi$; the reference event and normalization of this flow are fixed as part of the model. Set the exponential coordinate $U_{\kappa }(u)=-{\kappa }^{-1}{e}^{-\kappa u}$. Adopt the positive-chiral-current vacuum and tested current $j_{\kappa }$ constructed in Positive chiral current and exponential test-coordinate model. Its mathematical two-point covariance is the tested boundary $C_{\kappa }$ there; use its angular-frequency spectrum $S_{\kappa }$ and detailed-balance convention exactly as stated in Single-pole thermal contour and exact detailed balance. The exponential ray map and the choice of $\kappa$ are model inputs. If $\xi$ and $\kappa$ are identified with a generator and geometric surface gravity of a particular horizon, that identification is an additional physical assumption; no collapse geometry, four-dimensional horizon restriction, scattering factor, or backreaction is derived here.

Massless acceleration branch. Adopt the massless scalar and Minkowski vacuum of Specified bosonic scalar and CAR Dirac model adoption. Use the actually defined temporal smearing along the smooth proper-time hyperbola and its tested covariance from Constructed timelike smear and massless accelerated spectrum. The curve-smeared field is not a sharp-point field. Its accelerated spectrum and finite-switch coefficient are used only with the conditional hypotheses recorded in that lemma and in Exact defined Markov thermometer and switched coefficient bound.

For either branch, select an energy gap $E>0$ joules and define a two-state, continuous-time rate model with ground state $g$, excited state $e$, and model-time parameter $t\ge 0$ measured in seconds. In the reduced branch, no identification of $t$ with proper time along a spacetime curve is assumed. In the acceleration branch, take $t=\tau \ge 0$ along the proper-time hyperbola supplied by Constructed timelike smear and massless accelerated spectrum. In the reduced branch let ${\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}>0$ have units metres per second and set $r_{\uparrow }={\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}{S}_{\kappa }(E\mathrm{/}(\mathrm{\hslash }c))$ and $r_{\downarrow }={\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}{S}_{\kappa }(-E\mathrm{/}(\mathrm{\hslash }c))$. In the acceleration branch let ${\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}>0$ have units square metres per square second and set $r_{\uparrow }={\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}{S}_a(E\mathrm{/}\mathrm{\hslash })$ and $r_{\downarrow }={\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}{S}_a(-E\mathrm{/}\mathrm{\hslash })$. Here $c>0$ is metres per second, $\mathrm{\hslash }>0$ is joule-seconds, and the frequency arguments have units inverse metres and inverse seconds, respectively. These choices make both rates inverse seconds. The adopted population $p(t)$ of state $e$ is the solution of $p^{\mathrm{\prime }}(t)=r_{\uparrow }(1-p(t))-r_{\downarrow }p(t)$ with supplied $p(0)\in [0,1]$. This differential equation is the entire effective two-state model; the rates are adopted inputs, not exact QFT transition probabilities.

Only when a selected construction also supplies a density state and a normalized finite measurement instrument may an outcome be read using Physical interpretation of a supplied constructed quantum model. No such state, instrument, detector apparatus, or observed outcome is supplied here. The postulate asserts no general thermalization, exact all-order transition probability, KMS state, or experimental temperature reading.