Spectral Dynamics Beyond the Leading Eigenvector

Naive mean-field variational inference on a dense Ising model gives a damped hyperbolic-tangent iteration that reuses one spiked Wigner matrix at every step. We hold that update fixed and ask what state it already requires, tracking not one leading direction but the residual spectrum in the noise eigenbasis. An exact transport identity moves that measure, and a matched-state theorem, certified at the gains we run, shows squared modal energy is insufficient, because the next signal update reads a signed spectral phase. Signed signal-residual atoms then recover exactly the first two input-field moments. For a matrix-independent initialization we prove a Gaussian signal law, a global modal regression whose gain is affine in the noise eigenvalue, and the first-update spectral law. We then cross the first matrix reuse: exact GOE conditioning and a pseudo-Lipschitz empirical law separate the new field into retained Gaussian correlation, response memory, and fresh Gaussian noise, giving the second signal update in closed form, and the response is resolved even on symmetric seeds whose overlap cannot see it. Beyond the first reuse, unsigned spectral transport is a diagnostic, not a closure: its per-step error settles on a floor that stays flat from n=200 to 1200, and an exact error budget attributes that floor mainly to correlation between the nonlinear remainder and the residual, with the signed channel contributing for concentrated seeds. Across five independent matrices at dimension 1200, restoring the signed moments cuts the median free-running signal-update error from 0.112 to 0.012 and, in trajectory-median error, beats a control that drops only the signed atoms on every matrix and seed class. A frozen spectral preconditioner then tests how far the diagnosis transfers to a feasible operator. It does not overtake AMP, and controls attribute its gain to soft preservation of the leading outlier.