{"id":"dc0b9808-a3f4-4268-9f0b-83130ba78cd0","arxiv_id":"2607.26364","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A 12-vertex claw-free graph with chromatic symmetric function coefficient [s_(3,3,3,3)] = −64 disproves the Gasharov–Stanley Schur-positivity conjecture.","lead":"This paper reports a counterexample to a 25-year-old conjecture about symmetric functions of claw-free graphs: a specific 12-vertex graph whose chromatic symmetric function has a negative coefficient in the Schur basis. An exhaustive computer search shows that this is the smallest such graph, and exactly two exist at that size.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hand proof of the key coefficient in §2.3 unjustifiably drops all but three monomial coefficients; the Schur coefficient −64 is not derived.","rationale":"The reader's accepted verdict relied on the hand-checkable proof in §2 being internally consistent. However, the Kostka inversion in §2.3 contains a fundamental gap: it ignores the many monomial coefficients for partitions with more than four parts and assumes, without proof, that all Schur coefficients indexed by partitions with a part >4 vanish. This is demonstrably false—m_(3,3,3,3) alone forces [s_(5,4,3)] ≥ 768. The computational verification may still establish the result, but the paper's central proof is not sound as written. Since the repository and three exact implementations provide independent evidence, the appropriate verdict is CONDITIONAL: the counterexample may be accepted only if the proof of the coefficient is corrected or the computational verification is explicitly elevated to the primary evidence. The reader's weakest-assumption was the census, but the census is not the most fragile part; the hand proof is.","tokens_in":3285,"tokens_out":25339,"duration_ms":182042,"concrete_test":"Use the repository's exact programs to print the full Schur expansion of X_G and list all nonzero [s_ν]. If any [s_ν] with ν_1>4 or [s_(4,4,4)] or [s_(4,4,3,1)] is nonzero (e.g., [s_(5,4,3)] > 0), then the assumption in §2.3 is false. Then independently recompute [s_(3,3,3,3)] by applying the full inverse Kostka matrix to the complete list of [m_λ] for all λ with λ_1≤4; if the result differs from −64, the counterexample fails; if it equals −64, the numerical claim survives but the hand proof requires substantial correction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing step is in §2.3. The authors assert that because all monomial coefficients [m_λ]X_G with λ_1≥5 vanish, the only Schur coefficients that can be nonzero are indexed by partitions with parts at most 4, and then compute [s_(3,3,3,3)] by inverting only [m_(4,4,2,2)], [m_(4,3,3,2)], and [m_(3,3,3,3)]. Both assertions are unjustified. The vanishing of [m_λ] for λ_1≥5 does not restrict the Schur expansion: e.g., [m_(3,3,3,3)] = 768 is nonzero, and m_(3,3,3,3) has a nonzero Kostka coefficient in s_(4,4,4) and in s_(5,4,3) (explicit SSYT exist). Thus [s_(5,4,3)]X_G ≥ 768, so Schur functions with parts larger than 4 definitely occur. Nor do [s_(4,4,4)] and [s_(4,4,3,1)] vanish from [m_(4,4,4)] = [m_(4,4,3,1)] = 0, since they receive contributions from m_(3,3,3,3) as well. Consequently the equation 's_(3,3,3,3) = 768 − 2·128 − 3·192 = −64' is not a valid consequence of the counts in §2.2; terms from the many other monomial coefficients (partitions with parts ≤3 and more than four parts) are omitted. The hand proof of Theorem 1 is therefore incomplete.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a counterexample to the claw-free Schur-positivity conjecture of Stanley/Gasharov. The proposed graph is the line graph G = L(H) of a 10-vertex graph H built from a 4-cycle with two triangles and two pendant edges; H has 12 edges, so G has 12 vertices. The paper claims [s_(3,3,3,3)]X_G = −64, gives a hand proof based on counting stable partitions of G (matchings of H) and Kostka inversion, and reports three independent exact computations plus an exhaustive census showing that no counterexample exists on at most 11 vertices and that exactly two isomorphism classes on 12 vertices are non-Schur-positive. The result, if correct, disproves a 1998 conjecture and also its restriction to line graphs.","tokens_in":3642,"tokens_out":31474,"duration_ms":262394,"significance":"If the result is correct, it is a significant disproof of a well-known conjecture in chromatic symmetric function theory. The paper has notable strengths: the construction is explicit, the coefficient is reproducible by three independent exact implementations, the exhaustive census is feasible and transparently documented, and the provenance of the computation is public. These strengths make the result credible. However, the hand proof in §2.3 contains serious technical errors; as written, the proof of Theorem 1 does not establish the claimed coefficient. The computational verification may be sufficient to support the theorem, but the paper currently presents the hand proof as the derivation, and that derivation is invalid.","major_comments":[{"comment":"The 'triangular inversion' is not a valid inversion. From the displayed Kostka data, s_4422 = m_4422 + m_4332 + 2m_3333 + ...; with the paper's own counts this gives [s_4422] ≥ 128 + 320 + 2·768 = 1984, contradicting the claimed [s_4422] = 128. Similarly [s_4332] ≥ 320 + 3·768 = 2624, not 192. The coefficients 1, 2, 3 are Kostka numbers for expanding s_ν in the m basis; they are not the coefficients for the inverse expansion used here. The displayed subtraction is therefore unjustified and the proof of Theorem 1 fails at this step.","section":"§2.3, equations for [s_4422], [s_4332], [s_3333]"},{"comment":"The assertion that vanishing of all [m_λ] with λ_1 ≥ 5 implies that only Schur indices with parts ≤ 4 can be nonzero is false. For example s_(5,4,3) can receive a positive contribution from m_(3,3,3,2,1). Such stable partitions do occur: H admits the proper 4-edge-coloring {ab,cd,uv}, {ad,cx}, {au,bc,dm,xy}, {av,bℓ,cy}; splitting the third class gives partitions of type (3,3,3,2,1) and (3,3,2,2,2). These m-coefficients are positive, are dominated by (3,3,3,3), and contribute to [s_3333], but they are not counted in §2.2. (The specific stress-test example with s_(5,4,3) and m_(3,3,3,3) is itself impossible, but the objection is valid via m_(3,3,3,2,1).)","section":"§2.3, first sentence"},{"comment":"The claim that [s_444] vanishes because [m_444] = 0 is logically invalid. [s_444] receives contributions from all m_λ with λ ≤ (4,4,4), including [m_3333] = 768; indeed K_(444),(3333) ≥ 1, as shown by the SSYT with three rows 1 2 3 4. Thus [m_444] = 0 does not force [s_444] = 0. A separate proof or computation is required for this vanishing, and the same issue affects [s_(4,4,3,1)].","section":"§2.3, vanishing of [s_(4,4,4)] and [s_(4,4,3,1)]"}],"minor_comments":[{"comment":"Several places have missing spaces in rendered text (e.g., 'on12vertices', 's(3,3,3,3)]XG'); these are typesetting issues but should be corrected.","section":"Abstract/Introduction"},{"comment":"The notation switches between sν, s_ν, and displayed subscripts; please use one consistent subscripted notation throughout. Also, the phrase 'Using K ... :' should clearly specify whether the matrix direction is Kostka or inverse Kostka, since the current usage is ambiguous and, as noted, not the correct inverse direction.","section":"§2.3"},{"comment":"If the hand proof is revised to rely on the computational verification, the sentence 'The graph, short proof, exact verification programs, and census logs are available' should be updated to avoid calling the invalid hand derivation a 'short proof'.","section":"Computational provenance"}],"recommendation":"major_revision","confidential_remarks":"The reader's report overstates the soundness of the hand proof; the skeptical stress-test concern is valid in spirit, even though its particular example s_(5,4,3) with m_(3,3,3,3) is false. The computational verification is independent and appears strong, so the result is likely true and the manuscript can be repaired by either completing the monomial-count proof or explicitly framing Theorem 1 as a verified computational result. Given the importance of the claimed disproof, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result — a 12-vertex claw-free graph with a negative Schur coefficient — is likely true, and the computational work behind it is serious. The explicit line-graph construction is neat, the census is extensive, the code and logs are public, and the independent verification by Grinberg is meaningful. If I worked on chromatic symmetric functions, I would want to know about this example.\n\nBut the hand proof of Theorem 1 as written does not work. The stress-test is correct. The vanishing of [m_λ] for λ_1≥5 does not imply that only Schur functions with parts at most 4 can occur. For instance, s_(5,4,3) receives a positive contribution from m_(3,3,3,3), whose coefficient is 768 in this graph. So [s_(5,4,3)]X_G is at least 768, not zero. The inversion in §2.3 also drops stable partitions into more than four matchings, and it dismisses Schur functions like s_(4,4,3,1) on the false grounds that their monomial coefficients vanish. The equation [s_(3,3,3,3)] = −64 is therefore not derived by the argument given.\n\nWhat the paper really has is a reproducible exact computation: three independent routes plus external verification give the full Schur expansion, and the census supports minimality. That is real evidence — strong evidence, if the code is sound. But the manuscript misrepresents the basis of the result by presenting a flawed hand proof as the derivation. The authors should either fix the Kostka inversion, which I suspect cannot be patched without substantial additional counting, or reframe the main theorem as a computational result with a clear statement that the coefficient comes from verified exact computation, not from the short hand count.\n\nThe citation pattern is fine. The related work is acknowledged, and the independent Matherne–Morales paper is handled honestly. The flaw is internal to the proof, not a question of data fabrication.\n\nVerdict: this deserves a serious referee — the result is important enough — but it should not be accepted in its current form. I would send it back for major revision, asking for the hand proof to be removed or repaired and the computational verification to be presented as the primary evidence, ideally with a machine-checkable certificate or at least a fully auditable exact computation. A reader can then decide whether the computational proof is sufficient; as written, the paper's central proof is wrong.","headline":"The computational counterexample is probably real, but the hand proof in §2.3 is not; this paper needs a major rewrite before it should be accepted.","tokens_in":705,"tokens_out":1066,"would_cite":true,"duration_ms":136154,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05C15","05C75"],"pacs":[],"model":"deepseek-v4-flash","headline":"A 12-vertex claw-free graph whose chromatic symmetric function has Schur coefficient -64 at the partition (3,3,3,3) disproves the conjecture that every claw-free graph is Schur-positive.","keywords":["chromatic symmetric function","Schur-positivity","claw-free graph","line graph","counterexample","stable partition","Kostka inversion","Schur coefficient"],"falsifier":"Recompute the full Schur expansion of the 12-vertex line graph G and check whether the coefficient of s_(3,3,3,3) is -64; if it is nonnegative, the counterexample collapses. For the census claims, finding any connected claw-free graph on 11 or fewer vertices with a negative Schur coefficient, or a 12-vertex connected claw-free graph outside the two listed classes with a negative coefficient, would refute the minimality and completeness statements.","tokens_in":3152,"feed_emoji":"❌","tokens_out":5006,"duration_ms":40457,"temperature":0.7,"pith_summary":"The paper claims to disprove the long-standing conjecture that every claw-free graph has a Schur-positive chromatic symmetric function. It constructs a specific 12-vertex claw-free graph—the line graph of a 4-cycle with triangles at two opposite vertices and pendant edges at the other two—and proves by hand that the coefficient of s_(3,3,3,3) in its chromatic symmetric function is -64. Since Schur-positivity requires all coefficients to be nonnegative, this graph is a counterexample. An exhaustive computer census shows that no counterexample exists on fewer than 12 vertices and that exactly two non-Schur-positive isomorphism classes exist on 12 vertices. If correct, this settles the conjecture negatively and tightens the boundary between Schur-positive and non-positive claw-free graphs.","feed_headline":"12-vertex claw-free graph breaks Schur-positivity conjecture","feed_subtitle":"A line graph built from a 4-cycle with two triangles has a negative Schur coefficient, ending a 1998 conjecture.","key_machinery":"The argument rests on the stable-partition expansion of the chromatic symmetric function: for a line graph, stable sets of vertices correspond to matchings of the root graph. The root graph H has matching number 4, so only partition shapes with parts at most 4 contribute. Counting ordered stable partitions of type (4,4,2,2), (4,3,3,2), and (3,3,3,3) gives monomial coefficients 128, 320, and 768; unitriangular Kostka inversion then yields the Schur coefficient -64. The census relies on exhaustive generation of connected graphs, filtering claw-free graphs, and exact integer Schur-coefficient computation.","core_discovery":"The central discovery is a concrete counterexample: the line graph G of the graph H obtained from a 4-cycle by attaching a triangle at two opposite vertices and a pendant edge at each of the other two. G is connected and claw-free, and the Schur coefficient indexed by (3,3,3,3) in its chromatic symmetric function equals -64. The coefficient is computed by counting stable partitions of G (matchings of H) into four parts, then applying Kostka inversion. The paper further reports an exhaustive enumeration showing that every connected claw-free graph on at most 11 vertices is Schur-positive, so this graph is of minimum order, and that on 12 vertices exactly two isomorphism classes fail Schur-pos","pith_inferences":["If the census is correct, a refined conjecture might hold for claw-free graphs with maximum degree below some threshold, or with matching number below 4, since the counterexample requires matching number 4 in the root graph.","The two 12-vertex counterexamples share the same negative shape; comparing their root-graph structures may reveal a common substructure that forces Schur non-positivity, possibly generalizable to infinite families.","The extreme rarity of counterexamples (two among 1.7 million on 12 vertices) suggests that a characterization of Schur-positive claw-free graphs may still be feasible, perhaps by excluding a small set of induced subgraphs.","A non-computational proof of the census result would be valuable; until then, the minimality claim rests entirely on the correctness of the exhaustive search."],"forward_implications":["The claw-free Schur-positivity conjecture is false, so a graph being claw-free no longer guarantees that its chromatic symmetric function is Schur-positive.","The failure already occurs for a line graph whose root graph has maximum degree four, so the obstruction is not confined to exotic or high-degree constructions.","No counterexample exists on fewer than 12 vertices, and exactly two exist on 12 vertices, giving a precise minimum-order boundary for the conjecture.","The only negative coefficient in the counterexample is indexed by (3,3,3,3), so the failure of Schur-positivity can be witnessed by a single partition shape with four parts of size 3.","Because disconnected graphs reduce to connected components, the census claims cover all claw-free graphs of order at most 12, not just connected ones."],"fun_headline_variants":["12-vertex graph refutes claw-free Schur-positivity","Counterexample ends 1998 Schur-positivity conjecture","Only two 12-vertex graphs break Schur-positivity","Schur-positivity false for claw-free graphs at 12 vertices"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the exhaustive computer enumeration and exact arithmetic are error-free: a single misclassified graph or miscomputed Schur coefficient would invalidate the claims that 12 is the minimum order and that exactly two isomorphism classes exist on 12 vertices.","fun_headline_variants_meta":{"raw":{"variants":["12-vertex graph refutes claw-free Schur-positivity","Counterexample ends 1998 Schur-positivity conjecture","Only two 12-vertex graphs break Schur-positivity","Schur-positivity false for claw-free graphs at 12 vertices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000627,"raw_usage":{"total_tokens":2733,"prompt_tokens":734,"completion_tokens":1999,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":1924}},"tokens_in":478,"tokens_out":1999,"duration_ms":12812,"temperature":1.0,"reasoning_tokens":1924,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T17:20:38.100091+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the full Schur expansion of the 12-vertex line graph G and check whether the coefficient of s_(3,3,3,3) is -64; if it is nonnegative, the counterexample collapses. For the census claims, finding any connected claw-free graph on 11 or fewer vertices with a negative Schur coefficient, or a 12-vertex connected claw-free graph outside the two listed classes with a negative coefficient, would refute the minimality and completeness statements.","supporting_citations":[],"review_version":1}