{"id":"b93ee8d9-f3fb-4d43-b4f1-7fc671696872","arxiv_id":"2607.06952","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Random matrices with independent ε-balanced entries in a log(n)^{1+δ} band—and arbitrary entries outside—have cokernels approaching the Cohen–Lenstra distribution; up to αn per column and βn per row bad entries are also tolerated.","lead":"This paper proves that the cokernel of a random integer matrix still follows the Cohen–Lenstra distribution even when a positive fraction of entries are arbitrary or mildly dependent, and for band matrices with a very thin random band. It extends Wood's 2019 universality theorem and answers open questions about how few random entries are needed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; moment estimates are internally consistent and the stated assumptions cover the proofs.","rationale":"The reader's verdict was ACCEPT with moderate confidence, and my stress-test does not change that. The most fragile part of the argument is indeed the set of assumptions around column independence and the geometric spread condition (1.1), but these are explicitly stated hypotheses, not hidden or circular steps. The paper flags the column-independence reliance in Section 1.3, and the necessity of (1.1) is demonstrated by the paper's own Example II. I checked the main algebraic transitions: the moment factorization (2.1), the code/non-code decomposition in Section 4, the dimension-counting Lemma 5.3, and the path-graph Lemma 6.2 together with the telescoping product in Lemma 6.1. No contradiction or unjustified equality surfaced. The only textual error I found is the definition of o(g(n)) in Section 2.1, which is reversed from standard usage but is not invoked with that meaning anywhere. Because the central claims are scoped by the stated assumptions and the proof appears coherent, no verdict adjustment is warranted.","tokens_in":34483,"tokens_out":58992,"duration_ms":427543,"concrete_test":"Independently re-derive the pivotal inequality in Lemma 5.3: from Eq. (5.12) and the bounds dim V0 ≥ (1−α)n, dim V_{ℓ+1} ≤ αn(β/θ)^ℓ, verify that #Sur ≤ K e^{(ℓ+1)C log^3 n} |G|^n D1^{-(1−α)n} D2^{-αn(1−(β/θ)^ℓ)}. If the algebra fails, Proposition 4.2(a) is unsupported; if it holds, the main counting step is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not identify a load-bearing soundness gap. The core reductions are sound: Eq. (2.1) justifies the moment method under column independence; Theorem 2.1/2.2 follow from Proposition 4.2 by the standard code/non-code decomposition; and the counting lemmas in Sections 5–6 check out. In particular, Lemma 5.3's bound follows from Lemma 5.4 and the dimension estimates, Cases 1–3 in Proposition 4.2(a) cover the D1=D2 regime with the promised (1−ε)^r or 2^{−φn} savings, and Lemma 6.2 is supported by Lemmas 6.3–6.4, so Lemma 6.1's product bound (6.12) is credible. The genuine limitation—independence of distinct columns and the geometric condition (1.1)—is explicitly stated (Section 1.3) and is not a hidden assumption; the paper's Example II shows (1.1) is necessary. The only concrete slip is the nonstandard definition of o(g(n)) in Section 2.1, which is not used in that sense and has no effect on estimates.","agreement_with_reader":"partial"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-04T01:42:02.183400+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":2}