When Binary-Form Norm Balls Have Finite Haar Volume

For a nonzero real binary form of degree \(d\), we prove that all norm sublevel sets in \(\operatorname{SL}(2,\mathbb{R})\) have finite Haar volume if and only if every real projective root has multiplicity strictly less than \(d/2\). For the Bombieri norm, failure makes every nonempty sublevel set have infinite volume. The proof uses a positive-weight criterion for properness and a full-dimensional Haar-measure thickening. The cubic \(x^2y\) has trivial stabilizer but infinite-volume balls at every positive radius in every norm. We compute its exact cutoff volume and give a critical quartic with stabilizer \(\{I,-I\}\) and empty small balls. The cubic example is credited to the frozen UnsolvedMath research report; its Haar-volume calculation is completed here. This is a normalization obstruction in the setup preceding AIM Question 19, not a solution of the corrected equidistribution question. AI-assisted, self-audited and unrefereed; no absolute-priority or formal-certification claim.