= Overlap multiplicity bound for piecewise difference operators
{title2=$\|R_\alpha\|\leq\sqrt{2m}/\alpha$}
Suppose a piecewise <finite difference> uses two translated evaluations divided by $\alpha$. If the pulled-back evaluation intervals cover almost every input point at most $m$ times, then $|a-b|^2\leq2(|a|^2+|b|^2)$ gives $\|R_\alpha f\|_2^2\leq2m\|f\|_2^2/\alpha^2$. This converts an interval-overlap count into an <operator norm> estimate without requiring pointwise values of an <L2 space> equivalence class outside the almost-everywhere evaluation rule.
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