Universality of G-subshifts with specification

The paper proves universality for amenable-group subshifts with specification and a free element via a complete, technically careful construction: positive entropy from specification (Fact 1.15), quasitiling factors, a background subsystem X̄ that forbids a basic marker, unambiguous markers, and a multi-step, bounded-horizon filling scheme ensuring measurability, equivariance, and invertibility (Theorem 1.17). These ingredients are all present and coherently assembled in the manuscript . The model’s sketch matches the high-level blueprint, but it critically relies on an unproven Jewett–Krieger-type step for amenable groups and omits the X̄/marker machinery that prevents spurious marker occurrences and guarantees recognizability and measurability. Consequently, the model does not justify key steps (recognizable markers, entropy accounting on irregular cores, and an a.e. isomorphism), while the paper does.