Conceptual derivation and integrable structure of the staggered family

Derive the staggered matrix-product generating family conceptually and determine whether it admits a Yang–Baxter or Lax-pair interpretation, rather than relying only on its algebraic construction and verification.

Background

The paper constructs the staggered family over an algebraic curve and proves that its logarithmic derivatives generate an infinite tower of local conserved charges for the species-preserving rule.

Although the tower establishes extensive local conservation laws, the underlying conceptual origin of the tensors and any conventional integrable structure remain unresolved.

References

The theorem gives an infinite sequence of independent local charges, but it does not prove their (hydrodynamic) completeness, nor does it reveal a possible Yang-Baxter or Lax pair structure. The latter questions remain an outstanding problem for future research.

— Local and quasilocal conservation laws of three-state IRF cellular automata and their quantum deformations  (2608.13080 - Prosen, 13 Aug 2026) in Section 3.3, subsection “Low orders and integrability”