Stochastic Bin Packing with Travel Costs: The Exact Policy Threshold and Why Convex-Order Monotonicity Fails

A moving company loading a truck chooses between \emph{stacking} (highutilisation, high handling time) and \emph{tossing} (lower utilisation, lowerhandling time); the better policy depends on the travel distance $D$ through acritical threshold $D^{*}$. We give the \emph{exact} asymptotic threshold fori.i.d.\ box volumes and clarify the model's overhead coefficient. We then examinethe natural conjecture that $D^{*}$ is monotone in box-size variability. We showthat $D^{*}=c/\Delta\eta$ is \emph{antitone} in the trip-rate gap $\Delta\eta$(so $D^{*}$ falls whenever tossing utilisation falls), which fixes the directionof the effect. However, we prove that the stronger convex-order statement usedin an earlier draft is \emph{false}: there are pairs $F\le_{\cv}G$ of equal-meandistributions for which $D^{*}$ moves \emph{both} ways --- we give exactfinite-sample counterexamples and an exact computation under an explicitfixed-stacking-utilisation surrogate. The obstruction is that convex ordercontrols the spread of a distribution but not its \emph{tileability}, and thewaste function is not convex. The contribution is a correct threshold formulatogether with a certified negative result delimiting the convex-order template.The threshold algebra and the overhead identity are kernel-checked in Lean~4(Appendix~\ref{app:lean}); the probabilistic and convex-order statements areestablished by exact rational computation, not formalised.