{"id":"12b73483-e84e-4872-ac91-2949c177834a","arxiv_id":"2607.24217","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The sl(2)-weight system does not extend to a graph 4-invariant; only the specializations c=0, 3/8, 1, −3/32 remain candidates, with explicit or conjectural extensions constructed.","lead":"The sl(2)-weight system on chord diagrams does not extend to a graph 4-invariant in general, answering Lando's long-open question in the negative via an explicit certificate. Only four specializations (and some polynomial coefficients) can still extend, and the paper builds recurrences and formulae for several of them.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The negative answer rests entirely on one n=9 computational certificate; the logic around it is sound and the certificate is cheap to re-verify, so this is a correctness-risk concern only, not an internal gap.","rationale":"The reader identified the correct weakest assumption — computational correctness of the n=9 certificate, including graph-hash/isomorphism handling — and I find the same point, with no additional internal concern. The mathematical reduction (4T-span element with nonzero w_sl(2) value obstructs extension) is elementary and sound; the cited factorization through intersection graphs is established; the positive results (c=1 recurrence, leading-coefficient extensions via the unitriangular feature-matrix argument in §5) are proved in the text and do not lean on the certificate; conjectural material (oscillator representation, Conjectures 3.1/4.1/4.2) is clearly labeled and not load-bearing. Two points strengthen the ACCEPT beyond the reader's framing: (1) certificate verification is computationally trivial relative to generation, so the correctness risk is directly settleable by third parties; (2) the headline claim needs only a single-point evaluation at c∉{0,3/8,1,−3/32}, making confirmation exceptionally cheap. Kazarian's acknowledged independent verification adds further support, though it is not machine-checked. I recommend UNCHANGED (ACCEPT), with the concrete re-verification above as the natural step to convert correctness_risk from low to negligible.","tokens_in":17397,"tokens_out":2482,"duration_ms":78329,"concrete_test":"Write a small independent verifier for the posted sl2-cert artifact: (1) using canonical labeling (e.g., nauty), confirm each of the 3300 rows is a valid graph 4T-relation and sum them to recover the claimed combination of 5006 intersection graphs; (2) with a from-scratch implementation of the Chmutov–Varchenko chord-deletion recursion, evaluate that combination at the single rational point c=2 (exact integer/rational arithmetic). If the value is nonzero, Proposition 2.1(1) — the negative answer to Lando's question — is confirmed regardless of the exact polynomial. Then evaluate at four more points (e.g., c=3,4,5,−1) to confirm the full polynomial c(c−3/8)(c−1)(c+3/32) and hence part (2). If the c=2 value vanishes or the relation-sum mismatches, the certificate is unsound and the main claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The deductive skeleton of Proposition 2.1 is clean: any graph 4-invariant extending w_sl(2) must vanish on the span of graph 4T-relations, so an element C in that span with w_sl(2)(C) ≠ 0 is a complete obstruction. Factorization of w_sl(2) through intersection graphs is the established Chmutov–Lando theorem, and the Chmutov–Varchenko axioms (Def. 1.4) are standard. I find no internal inconsistency. The single load-bearing point is therefore exactly what the reader flagged: the certificate itself. Three things must all be right: (i) each of the 3300 rows is a genuine graph 4T-relation on correct isomorphism classes; (ii) their sum equals the claimed combination of 5006 intersection graphs; (iii) each intersection graph's value under the chord-deletion recursion (1) is exact. The failure modes are asymmetric: a hash collision merging non-isomorphic graphs could *fabricate* an inconsistency (a spurious certificate), while an evaluation bug could corrupt the polynomial. The custom hash (degree + 2-degree multisets) is not a complete invariant, so correctness depends on isomorphism checks being performed after hash matches, which the text asserts but the paper cannot show. Mitigations are real: the artifact is public, Krasilnikov's n≤8 table was reproduced, Kazarian is acknowledged as an independent verifier, and — decisively — verifying the certificate is vastly cheaper than discovering it, and even a single-point evaluation settles the headline claim. This is ordinary computational-correctness risk with unusually strong checkability, not a soft spot in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper answers Lando's long-standing question in the negative: the sl(2)-weight system, although its values depend only on intersection graphs (Chmutov–Lando), does not extend to a 4-invariant of graphs. The obstruction is an explicit n=9 certificate: a linear combination of 3300 graph 4T-relations expanding to 5006 intersection graphs whose Chmutov–Varchenko evaluation is the nonzero polynomial c(c−3/8)(c−1)(c+3/32). Its roots restrict possibly extendable specializations to c ∈ {0, 3/8, 1, −3/32}. On the positive side, the authors construct an explicit extension ψ at c=1 (the 3-dimensional representation) via the Z/2-corank η, prove its 4-invariance and a deletion-type recurrence (Theorem 3.6); conjecture an extension at c=−3/32 related to the oscillator representation, supported by a unique extension computed for graphs on ≤10 vertices; and completely settle the extension question for the polynomial coefficients of w_sl(2): Theorem 5.1 and Corollary 5.2 show [c^{n−k}] extends exactly for k≤4, while [c^k] for k≥1 does not, and Section 5.1 derives an exact 39-term closed formula for [c^{n−3}] by inverting a unitriangular subgraph-counting feature matrix. The deductive skeleton is clean throughout; the central result is computational, resting on the posted certificate.","tokens_in":17788,"tokens_out":7135,"duration_ms":210829,"significance":"This resolves a well-known open problem in finite-type knot invariant theory, posed by Lando and reviewed in [17]: even though the sl(2)-weight system factors through intersection graphs (Chmutov–Lando), it does not lift to a graph 4-invariant. The result is sharp — the certificate polynomial's roots are exactly the four Casimir eigenvalues at which extensions can exist, and the paper treats all four: trivial at c=0, known at c=3/8, a new proved extension at c=1, and a well-supported conjecture at c=−3/32. Strengths worth naming: the obstruction certificate is concrete, publicly posted, cheap to re-verify (a single-point evaluation settles the headline claim), and reportedly independently verified by M. Kazarian; the authors' code reproduces Krasilnikov's n≤8 table; the c=1 recurrence (Theorem 3.6) is proved by hand; the coefficient dichotomy (Corollary 5.2) and the exact 39-term formula for [c^{n−3}] are parameter-free and falsifiable; and the c=−3/32 conjecture comes with falsifiable n≤10 data and a conjectural recurrence. No free parameters or ad hoc axioms are introduced.","major_comments":[{"comment":"§2, proof of Proposition 2.1: the entire proof of the paper's central result is the sentence \"The certificate can be found at [GitHub URL].\" Since the negative answer to Lando's question rests on this object alone, the manuscript should (i) describe the certificate format and size in the text; (ii) state explicitly that the repository contains a self-contained verifier performing the three cheap checks — each of the 3300 rows is a valid graph 4T-relation on correctly identified isomorphism classes, the signed sum equals the claimed combination of 5006 intersection graphs, and its evaluation under recursion (1) is exactly c(c−3/8)(c−1)(c+3/32); and (iii) confirm that every hash match in the deduplication step is confirmed by an explicit isomorphism test, since the degree/2-degree hash of §2, step (1) is not a complete invariant and an unconfirmed collision merging non-isomorphic graphs co","section":"§2, Proposition 2.1"},{"comment":"§3.2, Propositions 3.4–3.5: the 4-invariance argument for ψ is compressed to four sentences. The claim that the inner sum \"splits into four framed 4T relations on 3^{η(G')}\" requires matching the sign (−1)^b in Definition 1.7 against the (−1)^{|V'|} weights and the framing flips on the two moving vertices; this sign bookkeeping is exactly where such arguments fail, and it is currently left to the reader. Likewise Proposition 3.5 asserts in one line the identification of ψ with the c=1 specialization via Theorem 3.1 and Proposition 3.2 (σ ↔ framing, |σ|−1 ↔ η, ∏σ(c) ↔ (−1)^{|V'|}). Please expand both steps; the c=1 extension is one of the paper's three positive results and deserves a fully written proof comparable to Theorem 3.6.","section":"§3.2, Propositions 3.4–3.5"},{"comment":"§5, Corollary 5.2 (the k>8 and [c^k], k≥1 cases): the proof applies the \"add an isolated vertex\" and \"add a leaf\" transforms \"to the original certificate,\" but only the effect on w_sl(2)-values (multiplication by c and by c−1/2) is argued. What is missing is one explicit justification that these operations, applied termwise to a linear combination of graph 4T-relations, again produce a combination of graph 4T-relations (locality of the 4T relation) whose expansion remains a combination of intersection graphs. A short lemma stating this formally would close the non-extension half of the coefficient dichotomy announced in the abstract; as written, that half relies on an unstated compatibility step.","section":"§5, Corollary 5.2"}],"minor_comments":[{"comment":"§2: the subscript notation is inconsistent — the algorithm is described \"for each n_i\" and terminates \"after all n_i = 0,…,n have been processed,\" but n_i is never defined; presumably these are just the integers 0,…,n.","section":"§2"},{"comment":"§2, step (1): in the definition of the hash, \"path of degree exactly 2\" should presumably read \"path of length exactly 2.\"","section":"§2, step (1)"},{"comment":"Theorem 5.7: the proof of the 39-term closed formula for [c^{n−3}] is again only a URL. Given that Lemmas 5.4–5.5 reduce it to an exact inversion of a 208-dimensional unitriangular matrix, please include in the text (or an appendix) the dimension and a spot check — e.g., evaluate both sides on several graphs on 7–8 vertices using the convolution of Theorem 5.1 — so the formula does not rest solely on the pipeline artifact.","section":"§5.1, Theorem 5.7"},{"comment":"The footnote in §5.1 reconciling the quadrangle conventions (qcd = −2×…) is hard to parse; a short displayed example matching one chord-diagram quadrangle to its three induced graph types would help.","section":"§5.1, footnote"},{"comment":"Conjecture 3.1: please state whether the verification on graphs with ≤9 vertices used exact rational arithmetic or floating point; the same question applies to the n≤10 table for Conjectures 4.1–4.2.","section":"§3.3, Conjecture 3.1"},{"comment":"§1.2, Remark: since 1T-relations are not imposed, state explicitly that Question 1.1 is answered in the framed sense (consistent with [9, 19]); a sentence on whether imposing 1T would affect the obstruction would be welcome.","section":"§1.2"},{"comment":"Table 1: add a source or method note for the values dim I_9 = 127954 and dim I_10 = 2165291, which to my knowledge are not tabulated elsewhere.","section":"Table 1"},{"comment":"Definition 1.8: clarify that the symbol \"a\" in \"preserved when a=0 and switched when a=1\" refers to the framing of vertex a, not the vertex itself.","section":"§1.4, Definition 1.8"}],"recommendation":"minor_revision","confidential_remarks":"The headline result is a large-scale computation; the certificate lives only on GitHub and the \"independent verification\" by Kazarian is an acknowledgment rather than a documented report. I do not doubt the claim, but I suggest the editor require the certificate and a verifier script as supplementary/ancillary material before publication, so the record does not depend on a personal repository. Otherwise the submission is well within scope and the negative answer will be of broad interest in the finite-type invariant community."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The headline result is real: sl(2) does not extend to a graph 4-invariant. They produce an explicit linear combination of graph 4T-relations on 9 vertices that evaluates, under the usual Chmutov–Varchenko recursion, to the nonzero polynomial c(c−3/8)(c−1)(c+3/32). That single obstruction settles Lando’s question in the negative and immediately pins the only possible specializations to the four Casimir eigenvalues 0, 3/8, 1, −3/32.\n\nWhat they do well beyond the no-go: they give a clean 4-invariant ψ at c=1 (the 3-dimensional rep), prove it via 2T-invariance of the corank η, and write a usable deletion recurrence. Leading coefficients through [c^{n−3}] are shown to extend; they recover an explicit closed formula for [c^{n−3}] by exact change-of-basis on the subgraph feature matrix up to 6 vertices, with code posted. Lower and mid coefficients are ruled out by the same certificate plus leaf/isolated-vertex transforms. The oscillator case at −3/32 is correctly labeled conjecture, with uniqueness checked through 10 vertices and a tentative recurrence for degree-2 vertices.\n\nThe soft spot is exactly the one you expect: the n=9 certificate is computational. Hash-plus-isomorphism handling and sparse elimination have to be right. Mitigations are good—public artifact, reproduction of Krasilnikov through 8, independent check by Kazarian, and verification is far cheaper than discovery. Ordinary computational risk, not a hole in the logic.\n\nThis is for people who work on weight systems, 4-invariants, and graph Hopf algebras. Citations are appropriate; the math is standard and the data trail is open. I would send it to referees without hesitation and would bring it to reading group. Worth citing if you touch extendability or coefficient formulas.","headline":"Lando’s question gets a clean negative answer via an explicit 9-vertex certificate, with solid constructive follow-ups for the surviving specializations and coefficients.","tokens_in":18782,"tokens_out":507,"would_cite":true,"duration_ms":14206,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C31","57M27","17B10"],"pacs":[],"model":"grok-4.5","headline":"The sl(2)-weight system does not extend to a graph 4-invariant in general.","keywords":["sl(2)-weight system","4-invariant","intersection graphs","chord diagrams","Casimir eigenvalues","Chmutov–Varchenko relations","graph 4T-relations"],"falsifier":"Re-run the nine-vertex 4T linear system (or inspect the published certificate) and check whether the combination of intersection graphs really evaluates to c(c−3/8)(c−1)(c+3/32) under the chord-deletion formula and is identically zero in the graph 4T quotient.","tokens_in":18548,"feed_emoji":"🔗","tokens_out":983,"duration_ms":16709,"temperature":0.7,"pith_summary":"Weight systems on chord diagrams are the algebraic engine behind finite-type knot invariants. A natural question is whether the famous sl(2) weight system, whose values depend only on the intersection graph of a chord diagram, can be rewritten as a function on all graphs that still obeys the same 4T relations. The paper answers no: there is an explicit linear combination of intersection graphs that evaluates to a nonzero polynomial under the weight system yet becomes zero once ordinary graph 4T relations are imposed. The same calculation isolates four special values of the Casimir parameter at which an extension might still exist; two of those extensions were already known, a third is given by a new recurrence, and the fourth is verified computationally up to ten vertices. The same methods settle which coefficients of the weight-system polynomial extend and which do not. The result shows that factoring through intersection graphs is not enough to guarantee a 4-invariant on the whole graph category.","feed_headline":"sl(2) weight system fails to extend to graph 4-invariants","feed_subtitle":"A nine-vertex certificate kills the general lift; only four Casimir values remain possible","key_machinery":"The computer-generated certificate C in the span of intersection graphs: a sparse dependence among 4T-relations on nine-vertex graphs whose evaluation by the Chmutov–Varchenko recurrence is the nonzero polynomial above. That single algebraic identity simultaneously disproves general extendability and pins down the only admissible specializations.","core_discovery":"The sl(2)-weight system does not extend to any 4-invariant of graphs. An explicit certificate—a linear combination of 3300 graph 4T-relations that expands to 5006 intersection graphs—evaluates under the weight system to the nonzero polynomial c(c−3/8)(c−1)(c+3/32). Consequently only the four roots of that polynomial remain candidates for specializations that could extend, and the paper constructs or verifies extensions at three of them while proving that all but the leading few polynomial coefficients fail to extend.","pith_inferences":["The same certificate technique can be applied verbatim to other Lie-algebra weight systems to decide extendability without first constructing candidate formulae.","A closed combinatorial formula for the oscillator specialization, if it exists, would complete the dictionary between Casimir eigenvalues and graph invariants.","Uniqueness of the three known extensions remains open and may require a separate generating-function or Hopf-algebra argument beyond the range of finite computation."],"forward_implications":["Only the four Casimir eigenvalues 0, 3/8, 1 and −3/32 can possibly admit graph 4-invariant extensions of the sl(2) weight system.","The specializations at c=0, 3/8 and 1 now possess explicit or recurrent graph formulae; the oscillator value c=−3/32 is unique at least through ten vertices.","All polynomial coefficients [c^{n−k}] for k\nge5 and all constant-term coefficients [c^k] for k\nge1 fail to extend to 4-invariants.","Even when a weight system factors completely through intersection graphs it need not lift to a 4-invariant on the larger graph category."],"fun_headline_variants":["sl(2)-weight system blocks any full graph 4-invariant","Nine-vertex certificate kills general sl(2) 4-invariant lift","Only four Casimir values left for sl(2) weight extensions","sl(2) weight polynomial obstructs unique graph 4-invariant","Explicit 4T-certificate rules out sl(2) as graph 4-invariant"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The computer certificate for nine-vertex graphs is complete and free of isomorphism or arithmetic error; if the sparse linear algebra missed a relation or mis-evaluated a graph, the non-extendability claim would collapse.","fun_headline_variants_meta":{"raw":{"variants":["sl(2)-weight system blocks any full graph 4-invariant","Nine-vertex certificate kills general sl(2) 4-invariant lift","Only four Casimir values left for sl(2) weight extensions","sl(2) weight polynomial obstructs unique graph 4-invariant","Explicit 4T-certificate rules out sl(2) as graph 4-invariant"]},"model":"grok-4.5","effort":"low","cost_usd":0.00407,"raw_usage":{"total_tokens":1184,"prompt_tokens":704,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":40704000,"prompt_tokens_details":{"text_tokens":704,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":393,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":704,"tokens_out":87,"duration_ms":7378,"temperature":1.0,"reasoning_tokens":393,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T20:43:15.573353+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Re-run the nine-vertex 4T linear system (or inspect the published certificate) and check whether the combination of intersection graphs really evaluates to c(c−3/8)(c−1)(c+3/32) under the chord-deletion formula and is identically zero in the graph 4T quotient.","supporting_citations":[],"review_version":1}