An Infinite Family of Exact Counterexamples to Quadratic Stationarity of Simplex Grids

For each level H ≥ 3 of the complete three-objective Das–Dennis grid, we exhibit an interior point
and a two-sided feasible simplex-tangent direction along which the anchored-box magnitude increases
by the exact amount 2H−3
8H ε2 − 1
4 ε3 > 0. This disproves the nonpositive quadratic-variation assertion
in Emmerich’s Conjecture 12.7 at every such level, without any boundary-tangent ambiguity, and
proves that these complete grids are not local maximizers under single-point perturbations. We
also establish the exact negative first variation at simplex vertices, the exact quadratic loss along
grid edges, and a family of inward edge-variation identities. The counterexamples leave first-order
stationarity and Pareto compliance logically distinct.