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TAP Accuracy Below the Fluctuation Scale and Universal Posterior Geometry in Spherical Linear Models

arXiv · AI, language, vision and robotics · article · Sep 17, 2026 · UTC

We study the Bayes-optimal spherical linear model as the ambient dimension and sample size grow proportionally, under a quantitative Marchenko--Pastur spectral-regularity condition on the design. This condition is satisfied by normalized i.i.d. designs with standardized entries of finite fourth moment, but does not require entrywise independence or impose conditions on the singular vectors. Under this condition, we prove a quantitative all-temperature TAP approximation and characterize the posterior geometry. For the natural finite-aspect-ratio TAP functional, the normalized spherical free ene

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