The zero-lag endpoint of the exponentially weighted two-rate family, XEPMA, has zero first kernel moment at every smoothness order but is quadratic-exact (zero second moment as well) only at order s = 1. This note develops the two named endpoints that correct the second moment away from that order. A single moment knob, the term c (s) Δ² EMAˢ+2, adjusts the kernel's second moment while leaving its gain and lag untouched, so it composes with the endpoint at any order without re-deriving either. Setting c (s) to cancel the second moment gives QuadraticXEPMA, quadratic-exact at every real s >= 1; setting the same knob to the exact step-overshoot minimiser gives DampedXEPMA, whose step response sits below the endpoint's own overshoot at every order and period. We give the closed forms, the continuous-limit optima, and an honest account of the cost: quadratic exactness roughly doubles the endpoint's noise gain, so the full correction is a completeness result rather than a deployment default, while the damped endpoint is Pareto-useful on overshoot. Every displayed quantity is verified to machine precision by scripts released with the reference implementation.
Marcus John Hendry Don (Tue,) studied this question.