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 SL(2,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^2 y 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.