Warm-Started Oja Matches Pooled PCA to First Order

Suppose historical observations are no longer available, but their leading empirical principal component, sample count, and spectral calibration have been retained. Can one-pass continuation track the PCA estimates that would be obtained by pooling the old data with every incoming prefix? In a fixed-dimensional, stationary rank-one isotropic Gaussian spike with iid historical and streaming observations and comparable sample sizes, we answer yes to first order. We prove that the actual normalized Oja recursion with a sample-count-shifted harmonic effective step is, uniformly over every prefix and on the same realization, within $o(N^{-1})$ expected maximal squared projector distance of both projector-regularized PCA and pooled empirical PCA. All three share the same influence path and, uniformly in the prefix, attain risk $(d-1)({\sigma }^2+\Delta ){\sigma }^2/\{{\Delta }^2(\nu +k)\}+o(N^{-1})$. At the final prefix this is the Gaussian local asymptotic minimax constant. Clipped old-sample plug-in calibration preserves the conclusion, and the historical sample size is the unique first-order pseudocount. The projector-risk comparison extends to $d+\log N=o(N)$ at the natural $o(d/N)$ scale. The Oja proof is entirely discrete. Empirical Riemannian flows restarted on the same prefixes track the Oja path at calibrated times; a separate fixed-data time profile identifies pooled-efficient early stopping.