Collision-Hull Compression for Homogeneous Keller Maps and a Forty-Variable Counterexample to Zhao's Vanishing Conjecture

Abstract

We formulate an elementary collision-generated compression principle for homogeneous Keller maps. In the language of polarization algebras, the construction takes the subalgebra generated by a collision; the additional point is that this subalgebra carries a noninjective Keller restriction and controls the dimension of the standard symmetric lift. For Thompson's 24-variable cubic-homogeneous map, exact polarization gives growth \(2,4,11,20,20\), and the resulting subalgebra is exactly MacFarlane's 20-dimensional invariant subspace. Thus the known \(24\)-to-\(20\) compression is recovered canonically from the collision itself. Although the resulting 40-variable lift had already been noted, we give an explicit 350-monomial homogeneous quartic for which Zhao's Vanishing Conjecture fails, together with an exact gradient collision over \(\Q(i)\). Over \(\Q\), \(\Q(i)\), and \(\C\), no proper invariant linear subspace containing this collision is available before the unchanged symmetric lift. No global minimality is claimed. The principal explicit calculations are checked by an accompanying exact calculation.

9 pages