Uniqueness and polynomial gauge of the algebraic family

Determine whether the staggered algebraic matrix family is unique modulo its color-dependent auxiliary gauge and whether a polynomial gauge exists that removes the displayed denominators as gauge artifacts.

Background

The staggered matrices are defined over the quadratic function field associated with an irreducible algebraic curve and are presented in a particular rational gauge.

The appendix explicitly leaves unresolved both the existence of a polynomial gauge in the coordinate ring and uniqueness of the discovered algebraic component under the allowed color-dependent auxiliary gauge transformations.

References

Whether such a gauge exists, and whether the discovered algebraic component is unique modulo eq:m3-color-gauge, remain open.

— Local and quasilocal conservation laws of three-state IRF cellular automata and their quantum deformations  (2608.13080 - Prosen, 13 Aug 2026) in Appendix A, Section “Algebraic curve and reset data for the m=3 family”