The Rational-Angle Lock

Abstract. A short note dissolving an open item from a companion paper. The cusp law governing where the minimum of a von Mangoldt Toeplitz symbol sits is the Hardy-Littlewood singular series multiplying the Lerch-Wood expansion of the polylogarithm at z = 1. That fixes the exponent, the sign, the strength and the transition; rescaling every measured edge collapses them onto one prime-free universal cusp. The phenomenon is the major-arc approximation of the circle method drawn as a graph.