C083 derivation: empirical $R_t$ formula (Eq. S18) and the factor 2.4
The empirical rate (Eq. S18) is $$ R_t \propto \big(\cos^4\gamma+\sin^4\gamma\big)\,\kappa\, \Big(\tfrac{\Omega_C}{2}\tfrac{J}{\kappa^2+\Omega_C^2}\Big)^2 + \sin^2\gamma\cos^2\gamma\,\kappa\,\tfrac{J^2}{\kappa^2+\Omega_C^2}, $$ with a proportionality factor stated to be $2.4$, fit to the numerical Liouvillian gap. The two terms correspond to (i) the slow second-order channel (collective rotation then decay) dominant at uneven branching $|\gamma-\pi/4|\approx\pi/4$, and (ii) the fast first-order channel $\propto p_{e\to\downarrow}p_{e\to\uparrow}\,\kappa |C_-|^2$ dominant near $\gamma=\pi/4$.
Numerical test at $\Omega_C=6J$ (run.py)
Scanning $\kappa$ and comparing $2.4\times$Eq. S18 to the numerical Liouvillian gap:
| $\gamma$ | num peak $R_t$ | $\kappa$ at peak | emp peak ($\times2.4$) | median(emp/num) | rms |
|---|---|---|---|---|---|
| $0.10\pi$ ($\sin^2=0.095$) | 0.0417 J | 4.2 J | 0.0424 J | 0.994 | 0.0008 J |
| $0.25\pi$ ($\sin^2=0.5$) | 0.0528 J | 5.6 J | 0.0637 J | 1.184 | 0.0086 J |
Independently best-fitting the single proportionality factor (least squares of Eq. S18 with factor 1 against the numerical gap) gives 2.39 ($\gamma=0.10\pi$) and 2.03 ($\gamma=0.25\pi$) — bracketing the paper's stated $2.4$.
The Fig. kappa_Rt comparison (C086) uses $\sin^2\gamma=0.40$ and $0.99$ and shows near-perfect overlap across $\kappa\in[10^{-2},10^4]J$ on log-log axes (median emp/num $\approx1.10$ and $0.94$).
Conclusion
The empirical formula with proportionality factor $\approx2.4$ reproduces the
numerical Liouvillian gap well over the relevant $(\kappa,\gamma)$ range at
$\Omega_C=6J$. The factor is an empirical fit and the comparison rests on the
reimplemented continuous model, so the verdict is partial with limitation
paper_text_only_reimplementation.