Sharpness of the direct TAP approximation error

Establish whether the direct comparison error between the normalized spherical free energy and the finite-aspect-ratio TAP optimum, namely |F_p^S-F_TAP|=O_P(p^{-1}), is sharp.

Background

The paper proves an O_P(p{-1}) upper bound for the difference between the normalized spherical free energy F_pS and the TAP optimum F_TAP. It also proves that the individual fluctuations of F_pS and F_TAP around their deterministic equivalent occur at the larger p{-1/2} scale, but these separate fluctuation lower bounds do not imply a matching lower bound for their direct difference. The unresolved problem is therefore to determine whether the established p{-1} direct TAP error rate is optimal.

References

It remains open whether the direct $O_P(p{-1})$ TAP error is sharp.

— TAP Accuracy Below the Fluctuation Scale and Universal Posterior Geometry in Spherical Linear Models  (2609.20577 - Liu et al., 17 Sep 2026) in Section 'Future directions'