Restart Degeneration of the Swing Filter and a Simple Repair

Abstract

The Swing filter extends a feasible line through its previous recording and restarts from a fitted value at the last accepted sample. We show that this inherited endpoint causes a singular loss of compression on rough input. For uniformly sampled Brownian motion at fixed tolerance, every fixed number of recording times converges to the first continuous feasibility boundary, with alternating boundary errors. The longest post-first segment tends to zero in probability. The segment-count ratio against a continuous approximation through true sampled knots diverges almost surely whenever that boundary precedes the observation horizon. For unit Gaussian increments, constrained least squares and no maximum lag, the first mean segment length has quadratic order in the tolerance $E$, the second has linear order, and every fixed mean from the third onward has order $E\log E$. We explain the logarithm through the conditional survival tail and prove a geometric bound on the subsequent approach to the stationary coefficient in the large-tolerance limit. The stationary mean satisfies $C_E\simγ_{\mathrm{Swing}}E\log E$. An explicit one-dimensional kernel specifies the single coefficient $γ_{\mathrm{Swing}}\approx1.8120703$ and admits a $3/4$ Wasserstein contraction and convergent evaluation brackets. The decimal is not a certified error interval. A half-budget slope corridor, true endpoint chords and one-sample bridges repair the degeneration while preserving the original error bound, continuous output and constant working memory. Including bridge costs, the long-stream mean span per output segment is asymptotic to $21ζ(3)E^2/(8π^2)$. Its explicit coefficient follows from the known Brownian anchored lifetime. Thus a change to the recording rule restores the quadratic scale.