Counterexamples to a Cubic Nondegeneracy Conjecture

Abstract

Morier-Genoud and Ovsienko conjectured that a cubic Boolean generating function satisfying an ordered separation condition on its second polar must be equivalent to one of two symmetric normal forms. Two infinite families of counterexamples are constructed and proved analytically; both are direct sums of small seeds, and neither is equivalent to the proposed forms even after linear functions are added. The first adjoins a nondegenerate quadratic block to a ternary cubic seed and is distinguished from the proposed forms by the radical of its third polar. This radical determines the associative nucleus of every twisted quasialgebra with the given generating function, so the corresponding algebras are not isomorphic even when the grading is forgotten. The second family consists of direct sums of a five-variable seed and has zero third-polar radical. A criterion for quadratic-block presentations establishes the separation property of the seed; commutation-graph degrees, radical dimensions, and contraction ranks establish non-equivalence in all dimensions of the family.