Characterize the intersection of the product-ramification locus with the remaining Karlsson classes

Determine whether the product-ramification locus \(\mathcal R_{6,\mathrm{prod}}\) intersects any Karlsson equivalence classes other than Tao’s class and the exceptional Karlsson class represented by \(H_\times\).

Background

The paper defines $\mathcal R_{6,\mathrm{prod}}$ as the set of order-six Hadamard classes possessing a product-regular frame whose residual product discriminant vanishes. Such points correspond to coalescence of the two roots of the product quadratic and therefore describe ramification of the quadratic–cubic reconstruction.

The authors prove that this locus is disjoint from Tao’s class and from the single Karlsson class $[H_\times]$ that lacks a product-regular frame. They do not determine whether the locus intersects the other classes in Karlsson’s three-parameter sector, leaving that intersection explicitly unresolved.