Design Low Flows (7Q10) at Six Long-Record US Gauges, 1900–2025: Log-Pearson III Quantiles With an Exact Derivation of the Wilson–Hilferty Frequency Factor, a Zero-Flow Correction, and a Check of Stationarity

The 7Q10, the lowest 7-day mean flow expected once in ten years, sets effluent permit limits and the dilution available to wastewater discharges in the United States. I compute it at six US Geological Survey gauges with 86 to 126 complete climatic years (April–March) of daily discharge up to March 2026. I derive the log-Pearson III frequency factor from the gamma distribution, show that the Wilson–Hilferty cube-root approximation follows from it in closed form, and prove the correction for years with zero flow. The fitted 7Q10 values are 80.9 ft³/s for the Fish River (Maine), 922 for the Potomac at Point of Rocks, 16.9 for the Rapidan, 16.1 for the St. Marys, 411 for the French Broad at Asheville and 0.26 for the Guadalupe at Comfort, Texas, where 7 of 86 years had a zero 7-day minimum. The number of years falling below each fitted 7Q10 is consistent with the nominal 10% (binomial p from 0.178 to 0.733). Splitting each record in half changes the 7Q10 by −21% (Rapidan) to +22% (Fish River); only the Potomac shows a significant monotonic trend in annual minima (Kendall τ = 0.139, p=0.021), and it is upward. A ten-year design flow is not rare over a project life: the chance of at least one year below it is 65% in 10 years and 96% in 30 years.