Adapting the compression technique of MIP*=RE to the restricted class of tailored non-local games -- a subclass of synchronous games, generalizing linear-constraint-system games, whose perfect strategies are Z-aligned permutation strategies that commute along edges (ZPC) -- this work proves TailoredMIP*=RE: a polynomial-time algorithm sends any Turing machine M to a tailored game G_M that admits a perfect ZPC strategy when M halts and has synchronous quantum value at most 1/2 when M never halts. The central technical advance is answer reduction for this restricted class, a careful adaptation of probabilistically checkable proofs, combined with question reduction by introspection and parallel repetition. It follows that distinguishing a tailored game with a perfect strategy from one whose every strategy is far from perfect is undecidable; via the companion paper's reduction this yields non-sofic unimodular networks, a negative resolution of the Aldous-Lyons conjecture, and reproves the negation of Connes' embedding problem through a more streamlined argument.
This paper, and its companion [BCLV24], are devoted to a negative resolution of the Aldous--Lyons Conjecture [AL07, Ald07]. In this part we study tailored non-local games. This is a subclass of non…