arXiv:2610.32364v1 Announce Type: new Abstract: We analyze the long-time stability and convergence of a third-order backward differentiation formula (BDF3) for parabolic equations with exponentially fading memory. The BDF3 scalar resolvent has oscillatory coefficients, so the monotonicity argument used for lower-order formulas in [Math. Comp. 62 (1994) 477-496] does not apply. We instead control memory feedback by the absolute mass of its discrete Green sequence and establish the rigorous stability-constant bounds $23/20<\Lambda_3<579/500$. A causal third-order memory quadrature with nonnegative assembled weights then yields exponentially weighted stability without a parabolic step-size restriction. With third-order starting values, we prove a uniform-in-time third-order temporal error estimate and extend it to conforming finite elements. Numerical tests confirm the predicted convergence orders and the long-time behavior.