Nonmonotone Potential Means for a Convex Dirichlet Schrodinger Operator

For the Dirichlet Schrodinger operator -d^2/dx^2 + V on (0,1), define M_j(V) as the expectation of V in the j-th L2-normalized eigenfunction. We prove M_2(t x)-M_3(t x)>t^2/20000 for every 0 1/40000. In particular, V(x)=x+10^-5 x^2 is an explicit smooth strictly convex counterexample to Lavine's potential-mean monotonicity question. The proof combines the classical second-order Stark coefficient, explicitly credited to prior work, with a complex-analytic remainder bound and a quantitative stability estimate. This answers question (2) in Section 10 of OWR 2009/06, p424, also AIM Low Eigenvalues Section 3(2), p12. The other questions bundled in frozen UnsolvedMath record OWR-3389-023 remain separate; no whole-record closure is claimed. Includes a complete written proof and portable exact and symbolic checks. AI-assisted self-audited preprint; no independent peer review, formal verification or absolute-priority certification is claimed.