What limits local entropy change? A static spectral criterion for quantum chains

A closed finite quantum system can return arbitrarily close to its initial state, but how far can a region's entropy depart from its initial value during the evolution? For finite quadratic fermionic systems, we reduce the maximum over all initial states and all times to a static spectral calculation: universal two-sided bounds identify it with B = ‖P − P̄‖1, where P selects the region and P̄ is its spectral dephasing. The criterion includes mixed, non-Gaussian and initially entangled states. In an explicit random Kitaev chain of 2N sites, the disorder-averaged maximum stays bounded uniformly in length away from statistical self-duality. At self-duality it equals (ln 2/6) ln N + o(ln N), including after optimally freezing any fixed number of spectral modes. The same leading term holds throughout each fixed finite-size window |χ|√N ≤ M. A single preparation, vacuum on one half and maximally mixed on the other, already produces logarithmic growth in its long-time averaged local entropy. Cross-boundary spectral channels thus determine both the existence of an entropy budget and its exact critical growth, even when the many-body preparation is unknown.