Fourth-order asymptotics and monotonicity in slope-constrained moment design

Abstract

We study the least amplitude required to satisfy finitely many linear moment conditions under a Lipschitz constraint. Starting from a sign certificate for the unconstrained problem, we obtain a fourth-order expansion in the reciprocal slope budget. Its coefficient contains a nonnegative quadratic correction caused by the lower moment constraints. The result covers both finitely many switches on a compact interval and countably many separated switches on the half-line, under an explicit weighted summability condition. Examples show that the fourth-order coefficient can have either sign, and that twice continuous differentiability alone does not imply a fourth-order expansion. For a positive theta-kernel family along a short, validated cusp branch, we give uniform, explicit sixth-order error bounds. A weighted change of scale then proves strict ordering of the actual optimal values for every pair of distinct branch parameters, including arbitrarily close pairs. The numerical claims are accompanied by interval certificates and reproducible checks. This preprint is prepared for independent mathematical review; neither a formal verification nor an external referee report is claimed.