Strengthened surface-dynamics core conjecture

Prove that every pair of interactions in $sd\times sd$ satisfies the (SD+) property, including the requirement that the local observables form a core for the transported generator in addition to the (SD) surface-dynamics conditions.

Background

The strengthened (SD+) property supplements the surface-dynamics conditions with a generator-core condition. This condition enables a converse relation between invariance of an interaction and commutation of the corresponding dynamics. The paper establishes (SD+) only under substantially stronger decay assumptions.

References

Any pair $(\Phi, \Psi)\insd\timessd$ has the \preSDplus{} property.

— Approach to Equilibrium in Translation-Invariant Quantum Systems: Structural Results II  (2610.21475 - Jakšić et al., 15 Oct 2026) in Section 3, subsection “Dynamical characterization”

Unfortunately, we do not know how to verify the condition $\Psi_t\insd$ in Proposition~\ref{prop:hc} in non-trivial examples.

— Approach to Equilibrium in Translation-Invariant Quantum Systems: Structural Results II  (2610.21475 - Jakšić et al., 15 Oct 2026) in Section 3, subsection “Dynamical characterization”

Establishing that compatibility and preservation of the core from rough initial data remains outside the present continuum theorem.

— Stochastic Shockwave Dynamics in First-Order Traffic Flow  (2610.23022 - Zhang et al., 16 Oct 2026) in Appendix, Section “Common-core compatibility,” following Theorem \ref{thm:generator-lax}