How acausal equations emerge from causal dynamics

We construct a causal and covariantly stable kinetic model whose spectrum at real wavenumbers $k$ reproduces any rest-frame stable dissipative dispersion relation $\omega (k)$ via suitable initialization of the microscopic degrees of freedom. Macroscopic observables can therefore obey arbitrary linear evolution equations (including forms that would be acausal if taken as fundamental), while the underlying dynamics remains causal, and all apparent propagation is encoded in the initial data. This provides an explicit counterexample to the idea that microscopic causality alone constrains the analytic form of dispersion relations at real $k$. In particular, bounds on transport coefficients based solely on the analytic structure of $\omega (k)$, such as the hydrohedron bounds, require additional assumptions about the region in the complex $k$-plane where $\omega (k)$ corresponds to physical modes.