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

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 energy and the TAP optimum differ by OP(p-1). Each is within OP(p-1/2) of its explicit deterministic equivalent, and this fluctuation scale is sharp. Uniformly over all global TAP maximizers, the normalized squared Euclidean distance to the spherical posterior mean is OP(p-1). We also prove that the posterior mass outside a data-dependent band determined by the ridge estimator has sharp exponential order. More precisely, uniformly over sufficiently small band widths varepsilon, the logarithm of this mass is at most -cpvarepsilon2+OP(1). For every fixed geometrically admissible width, a spherical-cap construction gives a matching exponential-order lower bound on this mass. For every deterministic sequence of widths varepsilonpgg p-1/2, the corresponding bands capture asymptotically all posterior mass.