{"id":"3d0890c0-ac04-4b0b-b5f5-e516f252e025","arxiv_id":"2505.14281","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves asymptotic and exact extremal densities for (s,q)-multigraphs in the large- and small-multiplicity regimes.","lead":"Extremal multigraphs allow several edges between two vertices, but only a bounded total number inside every small set. This paper determines the largest possible sum and product of edge multiplicities for wide parameter ranges, settling two open conjectures.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.7, the step-up lemma, is asserted with proof 'essentially identical' to [35, Theorem 3.11] and is never written out; Theorems 1.15 and 1.17 for every s > s0 depend on it, and its stability clause goes beyond the cited result.","rationale":"The reader's verdict is CONDITIONAL and identifies Proposition 4.7 as the weakest assumption. My stress-test agrees: Proposition 4.7 is not proved in the paper, and every theorem that goes beyond the base case s0 (Theorems 1.15 and 1.17 for larger s) depends on it. I additionally note that the stability transfer inside Proposition 4.7 is not present in the cited [35, Theorem 3.11], so even if the density step-up is routine, the stability part needed for Theorem 1.17 is a new assertion. The paper's own Section 5 acknowledges that the step-up is not optimal (a1 = a0 is conjectured), which supports the view that the threshold behaviour is delicate. I also flag that the base proof of Theorem 4.8 relies on deferred calculations in a master's thesis for Proposition 4.2 and Corollaries 4.3-4.6; those are less central than the step-up but still unverified. None of this shows the theorems are false; it shows the proof is not self-contained at a load-bearing point. I would keep the CONDITIONAL verdict rather than move to ACCEPT or REJECT, because the missing pieces are plausibly routine and the rest of the argument is detailed and coherent.","tokens_in":47552,"tokens_out":31687,"duration_ms":291234,"concrete_test":"Write out the proof of Proposition 4.7 for the full family r = (r0,0,...,0,rd), following [35, Theorem 3.11] line by line, and verify that (i) the only strengthening of 'a sufficiently large' is to make (4.4) strict, and (ii) the 'Further' stability clause follows from the same argument or from an explicitly stated additional lemma. A concrete instance to test is r0 = 3, rd = 2, d = 2, s = s0 + 1: symbolically check the step-up inequalities and construct the claimed close (s, Sigma_s(T))-graph from a near-extremal (s+1, Sigma_{s+1}(T))-graph.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.2 states Proposition 4.7 as the bridge from the base case s0 to all larger s, then gives as proof 'essentially identical [35, Theorem 3.11]'. This is the only argument supporting Theorems 1.15 and 1.17 for s > s0. The cited [35, Theorem 3.11] was proved for r = (r-1,0,...,0,1) with rd = 1; the present patterns have general rd, so the step-up is a genuine generalisation, not a restatement. More importantly, the 'Further' clause of Proposition 4.7 — near-extremal (s+1)-graphs are o(n^2)-close to near-extremal (s)-graphs — is the exact stability transfer needed to induct Theorem 1.17, and that clause is not part of [35, Theorem 3.11] as cited. If the step-up needs hypotheses beyond 'a sufficiently large' (e.g. an admissibility assumption on (s+1)-sets that is not established by Corollary 4.4), then the equality and stability results for s > s0 are unsupported. The paper also defers the calculations behind Proposition 4.2 and Corollaries 4.3-4.6 to a master's thesis, so the base Theorem 4.8 itself has a smaller but similar verification gap. These are missing proofs, not contradictions, but they are load-bearing.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-07T15:38:10.502979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}