E-process confidence sequences for unbounded means typically use a variance bound and a first-order score, leaving skewness and kurtosis unused. We construct a quadratic-envelope e-process from a conditionally centered vector score with a predictable second-moment bound. Under moment information specified independently of the tested stream, it stays positive on unbounded support and gives exact anytime-valid coverage under predictable strategies and fixed mixtures, without score clipping. For a location family with known central moments through order four, the centered observation and its centered square yield an exact confidence sequence whose local width shrinks, relative to a first-order construction, by a factor determined by skewness and kurtosis. At level 5%, the largest null ever-crossing rate was 4.63%; on two zero-inflated-lognormal families median stopping times fell by 36.8-45.8%. When the moments are calibrated on an independent reference sample, a centering-robust factor keeps exact coverage at a closed-form width cost: the median gain was 21.6-30.3% for Gamma(2) and lognormal errors with 10^6 reference observations, and a loss of up to 9.9% with 10^4. On public Fermi gamma-ray detector data with a declared stability margin, it crossed at the current location in 1.8% of 225 windows and stopped 30-35% earlier than the calibrated first-order method.

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