# Verification report

When Binary-Form Norm Balls Have Finite Haar Volume, version 1.0.
Alper Ferudun, Mercury Software GmbH, 18 October 2026.

Accepted scope: all-radii Haar-volume finiteness is equivalent to properness;
for real binary forms it is equivalent to every real-root multiplicity being
strictly below half the degree. For the compatible Bombieri norm, a nonproper
vector has only empty or infinite-volume balls. The trivial-stabilizer cubic
x^2y has infinite-volume balls for every positive radius and every norm, and
the displayed coordinate cutoff has the stated exact volume.

The written proofs check unimodularity, thickening dimension, action order,
Haar density, compact minima, real-root scope, critical threshold and cutoff
endpoints. The critical quartic supplies a negative control.

Normal and optimized Python/SymPy runs agree on 1,270 exact supplementary
checks, including action identities, Jacobians, stabilizer ideals, norm
identities and the cutoff antiderivative; zero floating point checks.
These are not a formal proof or an exhaustive numerical search.

The original cubic example is credited to the frozen upstream report.
The general equidistribution question with corrected hypotheses is not
resolved. This is an AI-assisted, self-audited, unrefereed preprint, not an
independently reviewed or formally certified result. No absolute novelty
or priority claim is made.