Independence from the full local-charge tower

Determine whether the common non-staggered quasi-local charge family is linearly independent of the Hilbert-space closure of the complete staggered local-charge tower of the species-preserving rule.

Background

The species-preserving rule has both the common non-staggered quasi-local family and an infinite staggered tower of strictly local charges.

The paper computes finite truncations of the Euler spectrum and shows that the common quasi-local charge contributes nontrivially after projection against the first local charges. However, it does not determine whether this independence persists after the entire local tower is included.

References

They are not asserted to be the final Euler velocities: higher members of the local tower can modify them, and linear independence of $A_\lambda$ from the Hilbert-space closure of the full tower remains open.

— Local and quasilocal conservation laws of three-state IRF cellular automata and their quantum deformations  (2608.13080 - Prosen, 13 Aug 2026) in Section 3.1, subsection “Consequence for the preserving rule”

Determining the spectral atypicality of the exact states, the rigorous hydrodynamic completion of the species-preserving-rule tower, and the full algebraic classification of local and quasi-local charges are left 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 Conclusion