= Pairwise improvement from nonparallel utility gradients
Two strictly positive nonproportional utility gradients $g,h$ admit a transfer direction $\delta$ with $g\cdot\delta>0$ and $h\cdot\delta<0$. Choose $\delta=g-th$ with $(g\cdot h)/\|h\|^2<t<\|g\|^2/(g\cdot h)$; strict <Cauchy-Schwarz inequality> makes this interval nonempty. Small opposite transfers in this direction strictly improve both differentiable utilities. Absence of such trades therefore forces positive proportionality of all gradients in an unconstrained allocation space.
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