Finite Local Quantum Dynamics: Relativistic Obstructions and Operational Reconstruction

We study an inverse problem for quantum cellular automata (QCA) and strictly locality-preserving discrete quantum dynamics: what can an observer infer about a finite local update from physical data, and which inferences remain impossible even with exact observations? We separate general interacting locality-preserving automorphisms from homogeneous finite-dimensional linear excitation sectors. In the latter, finite range implies a matrix Laurent-polynomial update and algebraic quasienergy branches. We derive a reciprocal-lattice monodromy obstruction: if repeated Brillouin windings generate infinitely many distinct analytic germs of a target quasienergy phase, the target cannot be an exact band of any finite-Laurent update; moreover, M distinct continuations require internal block dimension q ≥ M. Massive relativistic dispersion violates this criterion, and generic transverse slicing extends the obstruction to isotropic massive and massless dispersion in d ≥ 2. Independently, positive minimum site separation, exact finite-speed causality, and arbitrarily divisible microscopic time imply ultralocal dynamics. We then formulate the inverse direction. In an independently specified operational locality basis, finite full-matrix Fourier support plus pointwise unitarity is constructively equivalent to a homogeneous finite-range update, whereas spectra alone are insufficient. Exact reconstruction generally fixes only a blocking/refinement equivalence class. Conditional on such reconstruction, reachability is a compact-group control problem. Counterexamples accompany each nontrivial converse, and a complexity-controlled evidence hierarchy separates weak spectral consistency from process-level reconstruction.