ETH and Many-Body Localization Through an Operational Basin Diagnostic A Model-Dependent Bridge with Exact Coarse-Dynamics Identities

Abstract

This paper replaces a proposed universal equivalence among eigenstate thermalization, many-body localization, and projection-induced basins by a model-dependent operational bridge. We fix a finite random-field XXZ chain, an energy window, an initial-state class, and a family of local observables. Standard diagnostics—eigenstate matrix elements, adjacent-gap ratios, imbalance, and half-chain entanglement—are kept distinct. For a coarse channel followed by microscopic unitary evolution, we derive the exact failure of the reduced maps to form a semigroup. We prove that microscopic unitary dynamics preserves trace distance, that uniform contraction of an effective map is a sufficient but not necessary thermalization criterion, and that the mean-square thermalization error separates exactly into diagonal bias and temporal fluctuation. For a quasi-local integral of motion with commutator norm bounded by eta, we derive a memory bound linear in eta times elapsed time. These results explain when mixing basin and memory basin are useful operational descriptions without identifying them with ETH or MBL by definition. No universal logistic or Kramers-type knee follows for a closed quantum chain; a crossover must be estimated with a declared finite-size scaling model and uncertainty analysis. The Modal Triplet Theory interpretation is therefore a conditional pullback problem: its coherent projector, Hessian gap, and capacity margin become relevant only after an explicit intertwiner maps them to the Hamiltonian, coarse channel, observables, and diagnostics used here.