All-dimensional generic gauge completeness for six qubit-only families

Establish generic gauge completeness, equivalently prove $\ker J_A=\operatorname{im}\Phi_A$, for independent weighted graph-state tensor networks in dimensions $q>2$ on the triangular lattice, centered-square connectivity, and the 2-uniform tilings t2008, t2014, t2015, and t2019, under the period conditions specified for each family.

Background

The paper proves generic gauge completeness for thirty-three families at every $q\ge2$ and for six additional families only at $q=2$. The unresolved cases are higher-dimensional extensions of the qubit results.

The six families are explicitly identified in the generic-completeness tables and later appendices: triangular, t2008, t2014, t2015, t2019, and centered-square. Resolving them would complete the claimed generic comparison for all thirty-nine families at arbitrary allowed dimension.