Zero-State Entropy: Residual Complexity of Lifted Histories in Null-Invariant Dynamical Systems

This preprint introduces a lifted-history entropy framework for null-invariant dynamical systems. In the base setting, a distinguished null state is fixed by every admissible drive, so the state trajectory initialized at zero is dynamically silent. The paper shows that this state-level silence does not preclude residual complexity in explicitly chosen lifted layers, such as control tapes, cocycles, tangent-response operators, or finite-resolution observational structures. The framework defines topological and metric lifted zero-state entropy through separated observed histories, proves silence for state-factorizing observers, gives lower bounds from separated lifted histories, and illustrates the theory through counting lifts, finite-resolution collapse/persistence, shear-matrix cocycles, and a closure-motivated toy model.