C083 - An empirical convergence rate R_t proportional to (cos^4 gamma + sin^4 gamma) kappa (Omeg...
Verdict: partial
Location: Supp. Mat. S6, Eq. (S18); Fig. kappa_Rt
Type / expected artifact: math / math / numeric
Claim: An empirical convergence rate R_t proportional to (cos^4 gamma + sin^4 gamma) kappa (Omega_C J / (2(kappa^2+Omega_C^2)))^2 + sin^2 gamma cos^2 gamma kappa J^2/(kappa^2+Omega_C^2) is obtained, with proportionality factor 2.4 fit to the numerical Liouvillian gap.
Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial.
Limitations: paper_text_only_reimplementation.
Source location(s): source/supp_content.tex:253-262 (Eq. (S18)).
Conclusion
Empirical Eq S18 x2.4 vs numerical Liouvillian gap at Omega_C=6J. gamma=0.10pi: median(emp/num)=0.994, rms=0.0008J, num peak 0.0417J@4.2J. gamma=0.25pi: median(emp/num)=1.184, rms=0.0086J, num peak 0.0528J@5.6J. Independent best-fit proportionality factor = 2.39 and 2.03, bracketing the paper's stated 2.4. Empirical formula with factor ~2.4 tracks the gap well. Capped at partial (empirical fit + reimplemented model).
Verification details
Derivation excerpt: 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 $\gamma=\pi/4$.
Executable rerun: run.py exited 0 in 1.735s; log verification/C083/attempts/R002/run.log.
Output excerpt:
Omega_C = 6.0 J ; factor = 2.4
gamma=0.10pi: num peak R_t=0.0417J @kappa=4.20J ; emp peak=0.0424J @kappa=4.00J ; median(emp/num)=0.994 rms=0.0008
gamma=0.25pi: num peak R_t=0.0528J @kappa=5.60J ; emp peak=0.0637J @kappa=5.00J ; median(emp/num)=1.184 rms=0.0086
gamma=0.10pi: best-fit proportionality factor = 2.386
gamma=0.25pi: best-fit proportionality factor = 2.025
wrote curves.json