The Unequal-Arm Balance: A Bound on Short Weight from Arm Asymmetry, and the Weighing Procedure That Cancels It

An equal-arm balance delivers short weight if its arms are not equal, even when its weights are honest. Writing the two arms as L(1+e) and L(1−e), this paper shows that a single weighing errs by a fraction (1+e)/(1−e) − 1, approximately 2e: a half-percent asymmetry (e = 0.50%, a 3.0 mm arm difference on a 300 mm balance) already takes 10.1 g per kilogram, and stealing 20 g per kilogram requires an arm difference of only 5.94 mm on that balance. The paper then derives the double-weighing procedure — weigh once, swap goods and weights between the pans, weigh again, and take the geometric mean — which cancels the asymmetry exactly in the ideal model. The arithmetic mean, easier to compute in a market, leaves a second-order residual of only 0.20 g per kilogram at e = 1.00%. With 5 g weight steps and a ±2 g pivot deadband, the practical residual bound for the geometric mean is 2.26 g per kilogram root-sum-square (0.23%) and 3.20 g per kilogram worst case (0.32%). Any buyer who can insist on a swapped second weighing reduces an unknown arm error from percent level to a few grams per kilogram. AI disclosure: Dave, an AI research system, assisted in the preparation of this work.