Replay-Gap Thresholds: Sharp Burst Guarantees and Coupled Retention

Sparse replay can retain training examples even when their update frequency vanishes, but the placement of updates matters. We prove a sharp worst-case replay-gap threshold for constant-step online logistic regression on any separable interfering pair. For a fixed burst of $B$ rare updates, let $S_B(m)$ be its total scalar logistic response from margin $m$, let $\kappa =b/(a{c}^2)$ in coordinates $(a,0),(-b,c)$, and set $R_{c,B}=\exp (S_B(0)/\kappa )$. Under an explicit sufficient step-size bound, every replay-time ratio sequence confined eventually to a fixed interval $1<R_{-}\le R_k\le R_{+}<R_{c,B}$ guarantees eventual all-time retention from every finite initialization. Geometric schedules above the threshold establish worst-case sharpness; equality remains open. The proof uses a common invariant region controlled by the derivative of the full burst response. We also extend the centered-map method to any finite number of orthogonal common examples coupled through one interfering rare example, obtaining an exact geometric sign criterion in the stated exposure regime and a computable local guarantee for arbitrary persistent ratio variation. In this family, two schedules whose rare-update counts differ by at most one at every prefix can have opposite retention outcomes, robustly under small bounded shared log-time jitter. The results concern logarithmically sparse replay and raw training margins, not population accuracy.