Unimodular Random Graphs with Property (T) Have Cost One

The paper proves that unimodular random graphs (URGs) with Property (T) have cost one, defining URG cost via extended rewirings and establishing the main theorem (Theorem 1.2) together with its equivalence for pmp cbers (Theorem 5.1) . The proof builds Kazhdan-optimal partitions and uses a clusterwise Bernoulli extension to produce, for any ε>0, an ergodic realization with a set A of measure <ε having finitely many A-clusters, then applies Gaboriau’s induction formula to obtain cost ≤ 1+ε(D−1), hence cost 1 by letting ε→0 (Theorem 5.2 and the ensuing argument) . It also develops URG analogues of Connes–Weiss/Glasner–Weiss to control almost-invariant colorings and establish the spectral-gap framework (Theorems 4.1 and 4.3) . By contrast, the candidate solution hinges on a stronger, unsubstantiated step asserting the existence of connected generating graphings of arbitrarily small excess degree above 2 for measured equivalence relations/URGs under Property (T). The 2025 paper does not claim this 2+ε-degree rewiring; instead, it proves cost 1 via the small-cluster set/induction route. While the candidate’s Step 1 (identifying URG cost with the measured cost of a realization) aligns with standard definitions, Step 2’s “2+ε” rewiring is neither established nor used in the paper’s proof. Hence, the paper is correct and complete; the model’s proof is incorrect/incomplete at a crucial step.