Singlet-like correlations: equal peak work, unequal robustness

Initial system-environment correlations are a thermodynamic resource, enabling work extraction through their erasure. We compare three representative singlet-like, rotationally covariant correlation laws-a local classical benchmark, the quantum cosine law and an idealized stronger-than-quantum step law-under measurement misalignment. In the binary-outcome, uniform-marginal setting, all three can attain the same peak extractable work, $k_{B}Tln2$ . Their operational value differs, however, in robustness away from perfect alignment. For the chosen classical benchmark, the mutual information degrades as $\Theta (\delta \theta ln(1/\delta \theta \left)\right)$ , whereas for the quantum cosine law it degrades as $\Theta (\delta {\theta }^{2}ln(1/\delta \theta ))$ . The stronger-than-quantum step law is perfectly flat except at a critical angle. Accordingly, the paper establishes a robustness hierarchy within this restricted comparison class, rather than a no-go theorem for all classical correlations.