Finite-Cadence Refresh Equilibria for Streaming Prior-Anchored Ridge: A Closed-Form Forgetting Window and Spectral Correction

Recursive estimators on drifting streams must forget stale data fast enough to track the target, yet slowly enough to average out noise. In prior-anchored ridge regression, this trade-off admits a closed-form spectral decomposition in which each eigen-direction interpolates between the data solution and the prior with pull weight s_i = ρ/ (μ_i + ρ). When the anchor is refreshed every k steps, we show that the moving-frame offset converges to a finite-cadence equilibrium that lags the pure-leaky path by an explicit Δ_k(s) while reducing estimation variance by a factor Ψ(s, r_k). Minimizing the cycle-averaged risk gives ω^* = (4ν^2 Σ/ c')^{1/3}, c' = σ^2 d_{eff}(ρ), and the plug-in strength ρ^* = σ^2 / (nδ^2). A random-matrix deterministic equivalent extends both laws to finite dimension, renormalizing the anchor to ρ_{eff} = ρ(e^{ζq} - 1) / (ζq); it matches simulation within 5% at fixed operating points and within 13% at the method's own, and rank-deficient populations reduce exactly to their active subspace. The resulting method, SPAR, calibrates window and strength online from residuals. On a 27-configuration grid the calibrated window opens on drift without divergence, and the grid defaults transfer to an hourly electricity stream and to BCI IV 2b. The core algebra, refresh chain, and regret bounds are formalized in Lean 4.