What does this equation mean?

Two facts transfer exactly, with no correction term anywhere, because they follow from locality alone rather than from any emergent Lorentz symmetry. Front-speed causality is simply the Lieb-Robinson bound restated: $c_{\mathrm{LR}}$ is a hard limit, and the statement that nothing propagates outside the cone is a proof about commutators, not an analogy borrowed from relativity. Effective causal order follows immediately — two operations separated by more than $c_{\mathrm{LR}}$ t commute up to an exponentially small tail, so a well-defined partial order of what can affect what exists inside the crystal for the same structural reason it exists in the vacuum. Massless-cone kinematics is the third exact entry, carrying one scope condition: wherever a band is genuinely linear, its low-energy excitations obey the massless wave equation with $c_{\mathrm{LR}}$ , or the local Fermi velocity, standing in for c , and every kinematic consequence of that equation, Klein tunneling included, replays inside the window.