= Order preservation for scalar entropy solutions
Subtract the weak equation for $u-v$ from its <Kato inequality for scalar conservation laws> and divide by two. The result is an inequality for $(v-u)_+$ with a flux bounded in magnitude by $M(v-u)_+$. The same shrinking-interval argument proves $u_0\ge v_0\Rightarrow u\ge v$ almost everywhere. Choosing the constant solution $v=0$ proves preservation of nonnegativity even when $f(0)\ne0$. Two-sided absolute-value contraction alone should not be mistaken for this one-sided comparison proof.
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