{"id":"084af51a-7bc3-4b9f-84c9-9ff2b146b5fc","arxiv_id":"1908.11596","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A six-dimensional semidirect product example shows Proposition 3.1 of Burde-Wagemann (arXiv:1908.05963) is false, invalidating their proposed proof of the weak Leibniz homology conjecture.","lead":"This note constructs a six-dimensional Lie algebra that disproves a key proposition in a recent claimed proof of Pirashvili's conjecture on Leibniz homology. The result shows that the existing proof is invalid and the conjecture remains open.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The counterexample depends on an unproven duality bridging Lemma 1 to Proposition 3.1; a direct low-degree computation of the example would settle the issue.","rationale":"The reader's weakest assumption and my stress-test converge on the same point: the note's only bridge between the six-dimensional example and Proposition 3.1 is the unproven duality HL_{*+1}(g) ≅ HL_*(g, g^♯). I checked the other ingredients: Lemma 1's equivalences use standard long exact sequences and published results from [9] and [4], and the Killing-form argument for condition (v) is correct. The duality is plausible and likely true, so the concern is an omitted justification rather than a mathematical falsehood. Because the example is concrete and finite-dimensional, a direct computation of HL_2 and HL_1(g, r^♯) would confirm the counterexample without needing the duality. This is exactly the kind of check that would turn the conditional verdict into a definitive one. Therefore I do not propose changing the verdict.","tokens_in":2772,"tokens_out":21503,"duration_ms":202149,"concrete_test":"Compute the explicit chain complex for g = sl2 ⋉ sl2_ab and determine dim HL_2(g) and dim HL_1(g, r^♯). If these dimensions differ, Proposition 3.1 is false for p = 2 without invoking the unproven duality, and the counterexample stands; if they agree, the paper's stated route is insufficient and a higher-degree difference or a proof of the duality is needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the six-dimensional example refutes Proposition 3.1 of [1]. The proof route is: the example violates condition (v) of Lemma 1, so conditions (i)-(iv) fail, and then the unproven assertion in Section 3 that HL_{*+1}(g) ≅ HL_*(g, g^♯) 'holds always' converts failure of condition (i) into failure of Proposition 3.1. This duality is the load-bearing bridge. If it is false or has extra hypotheses, the example only shows that HL_*(g, s) ≠ 0 and that the coefficient map HL_*(g, r) → HL_*(g, g) is not an isomorphism; it does not by itself show HL_p(g) ≇ HL_{p-1}(g, r^♯). The note neither proves nor cites the duality, even though it is nontrivial: it identifies absolute Leibniz homology with Leibniz homology in adjoint coefficients. The rest of Lemma 1 rests on published theorems from [9] and [4] that are likely correct, so the unproven duality is the weakest point in the chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to refute Proposition 3.1 of Burde–Wagemann [1] (and consequently Theorem 4.7 and Corollary 4.8 of [3]) by constructing a six-dimensional Lie algebra g = r ⋊ s, where s = sl2 and r is the abelian radical given by the adjoint representation of s. The author proves a lemma giving equivalent conditions for a certain Leibniz-homology isomorphism to hold, shows that the example violates condition (v) of that lemma via the Killing form, and then uses the asserted duality HL_{*+1}(g) ≅ HL_*(g, g^♯) to conclude that Proposition 3.1 of [1] is false. If correct, the note invalidates the claimed proof of the weak Leibniz-homology conjecture, which remains open.","tokens_in":2991,"tokens_out":11750,"duration_ms":107148,"significance":"If the counterexample is valid, it is a significant contribution: it shows that the published proof of the weak conjecture in [1] and [3] is flawed and that the conjecture remains open. The example itself is simple and elegant, and the use of the Killing form to detect the invariant map H_1(r) ⊗ s → C is convincing. The proof of Lemma 1 is largely supported by standard spectral sequence results and the author's earlier work [9]. The main load-bearing gap is the unproven duality in Section 3, which, if supplied, makes the argument sound.","major_comments":[{"comment":"The assertion 'Since HL_{*+1}(g) ≅ HL_*(g,g^♯) holds always' is the key link between failure of condition (i) of Lemma 1 and falsity of Proposition 3.1 of [1], but no proof or reference is given. If this duality is false or has additional hypotheses, the example only shows that H_0(s, H_*(r) ⊗ s) ≠ 0 and does not by itself contradict Proposition 3.1. Please supply a proof or a precise citation for this duality, or alternatively verify directly for the stated example that HL_p(g) ≇ HL_{p-1}(g,r^♯) for some p (for instance p = 1).","section":"Section 3, first paragraph"},{"comment":"The sentence 'the cohomologies are dual vector spaces of homologies' is too terse and appears to conflate Leibniz cohomology with Leibniz homology with dual coefficients. Condition (i) is a statement about HL_*(g,g^♯) → HL_*(g,r^♯) while condition (ii) concerns HL_*(g,r) → HL_*(g,g); the claimed duality needs a precise formulation, including the relevant module structures, or an explicit reference.","section":"Section 2, Lemma 1 proof, (i)⇔(ii)"}],"minor_comments":[{"comment":"There are several typos: 'the the Cheva lley-Eilenberg', 'S. Bunde' in reference [1] should be 'S. Burde', and 'sc F. W agemann' in reference [3] should be 'F. Wagemann'.","section":"Section 1 and references"},{"comment":"The commutative diagram is difficult to read as typeset; please ensure all arrows and labels are clearly printed.","section":"Section 1, displayed diagram"},{"comment":"'This also imply' should be 'This also implies'.","section":"Section 3, final sentence"},{"comment":"It would be helpful to state explicitly that H_*(r) is regarded as an s-module via the action induced by the section, not merely as a graded vector space.","section":"Section 2, condition (v)"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the unsupported duality HL_{*+1}(g) ≅ HL_*(g,g^♯). If the author can point to a theorem in his own earlier paper [9] or another standard reference, the gap becomes a minor citation issue and the note is essentially correct. Otherwise, the counterexample is incomplete as written. I recommend asking for a proof or reference, or a direct low-degree computation of the example, before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this note looks right to me, and it does what it claims. The six-dimensional example—sl2 acting on an abelian radical by the adjoint representation—is a clean falsification of Proposition 3.1 in the Burde-Wagemann preprint. If it stands, the claimed proof of the weak Leibniz homology conjecture is invalid, and the conjecture is open again. That is a meaningful thing to publish, even though the note proves no positive theorem.\n\nWhat the paper does well: Lemma 1 is a neat reduction. The proof uses the Hochschild-Serre spectral sequence and the author's earlier theorems, and I did not find a gap there. The Killing form argument for condition (v) is correct and short: H1(r) is isomorphic to s, and the Killing form gives a nonzero s-invariant map s⊗s → C, so the coinvariants do not vanish.\n\nThe soft spot is the bridge. The line 'Since HL_{*+1}(g) ≅ HL_*(g,g♯) holds always' appears without proof or reference. That isomorphism is what converts the violation of condition (v) into a counterexample to Proposition 3.1. If it is true, everything works. If it has unstated hypotheses, the example only shows conditions (i)–(iv) of Lemma 1 fail, which is not directly a contradiction. I believe this duality is a known result, possibly in [9] or Loday-Pirashvili's papers, but the note should have cited it. A referee should ask for that citation or a one-paragraph proof. This is a presentation gap, not a fatal mathematical flaw.\n\nTwo smaller comments. The paper says Theorem 4.7 and Corollary 4.8 of [3] are wrong, but it never shows the dependency; that is worth a sentence. And the reference list has a typo ('Bunde' for 'Burde'), which is harmless but sloppy.\n\nWho this is for: anyone working on Leibniz homology or on the Pirashvili conjecture. The note is short, readable, and the counterexample is easy to verify. It deserves a serious referee and, once the duality question is resolved, publication. I would send it to review.","headline":"Pirashvili's six-dimensional counterexample likely refutes the key proposition in Burde-Wagemann's proof, but the note must pin down the unproven duality that makes the example bite.","tokens_in":3475,"tokens_out":3418,"would_cite":true,"duration_ms":33402,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A32","17B56","17B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs a six-dimensional Lie algebra for which a key isomorphism in a recent proof of the weak Leibniz homology conjecture fails, invalidating that proof.","keywords":["Leibniz homology","Lie algebra cohomology","counterexample","weak conjecture","adjoint representation","Levi decomposition","Hochschild-Serre spectral sequence","semi-simple Lie algebra"],"falsifier":"One could compute the Leibniz homology groups $HL_2(\\mathfrak{g})$ and $HL_1(\\mathfrak{g},\\mathfrak{r}^\\sharp)$ directly for $\\mathfrak{g}=\\mathfrak{sl}_2\\ltimes\\mathfrak{sl}_2$; if they turn out to be isomorphic, the note's central claim that Proposition 3.1 fails would be incorrect.","tokens_in":2562,"feed_emoji":"🧮","tokens_out":11616,"duration_ms":88007,"temperature":0.7,"pith_summary":"This note targets a recent proof of the weak Leibniz-homology conjecture, which says a finite-dimensional complex Lie algebra is semi-simple exactly when its Leibniz homology vanishes in all positive degrees. The author shows that the proof's key proposition—the claimed isomorphism between $HL_p(\\mathfrak{g})$ and $HL_{p-1}(\\mathfrak{g},\\mathfrak{r}^\\sharp)$ for every $p\\ge 1$—is false. The counterexample is the six-dimensional Lie algebra $\\mathfrak{g}=\\mathfrak{sl}_2\\ltimes\\mathfrak{r}$, where $\\mathfrak{r}$ is abelian and is the adjoint representation of $\\mathfrak{sl}_2$. The failure is detected by the nontrivial coinvariant module $H_0(\\mathfrak{s}, H_*(\\mathfrak{r})\\otimes\\mathfrak{s})$, forced by the Killing form. If the note is correct, the claimed proof collapses, two theorems in a companion preprint are wrong, and the weak conjecture remains open.","feed_headline":"Counterexample falsifies key step in Leibniz homology proof","feed_subtitle":"The six-dimensional example shows the weak conjecture on Leibniz homology is still open.","key_machinery":"The load-bearing device is Lemma 1, which lists five equivalent conditions under which the restriction map $\\mathfrak{g}^\\sharp\\to\\mathfrak{r}^\\sharp$ induces an isomorphism $HL_*(\\mathfrak{g},\\mathfrak{g}^\\sharp)\\cong HL_*(\\mathfrak{g},\\mathfrak{r}^\\sharp)$. Among these is condition (v), $H_0(\\mathfrak{s}, H_*(\\mathfrak{r})\\otimes\\mathfrak{s})=0$, which is computed with the homological Hochschild-Serre spectral sequence for the extension $0\\to\\mathfrak{r}\\to\\mathfrak{g}\\to\\mathfrak{s}\\to 0$. The counterexample works by showing condition (v) fails: the Killing form produces a nonzero element in $H_0(\\mathfrak{sl}_2,\\mathfrak{sl}_2\\otimes\\mathfrak{sl}_2)$, so the equivalent conditions cannot hold, and the claimed isomorphism of Proposition 3.1 fails.","core_discovery":"The central claim is that Proposition 3.1 of [1] does not hold for all Lie algebras. The author exhibits the Lie algebra $\\mathfrak{g}=\\mathfrak{sl}_2\\ltimes\\mathfrak{r}$ with $\\mathfrak{r}$ abelian and isomorphic to the adjoint representation of $\\mathfrak{sl}_2$, and shows that condition (v) of Lemma 1 fails: $H_0(\\mathfrak{s}, H_1(\\mathfrak{r})\\otimes\\mathfrak{s})\\neq 0$ because the Killing form gives a nonzero invariant pairing $H_1(\\mathfrak{r})\\otimes\\mathfrak{s}\\to\\mathbb{C}$. Since Lemma 1 asserts that condition (v) is equivalent to the isomorphism $HL_p(\\mathfrak{g})\\cong HL_{p-1}(\\mathfrak{g},\\mathfrak{r}^\\sharp)$ for all $p\\ge 1$, this example refutes Proposition 3.1. Consequently the proof of the weak conjecture in [1] is invalid, Theorem 4.7 and Corollary 4.8 of [3] are false, and the weak conjecture stays open.","pith_inferences":["A natural next step is to classify the Lie algebras for which condition (v) holds; the example suggests that the vanishing of $H_0(\\mathfrak{s},H_*(\\mathfrak{r})\\otimes\\mathfrak{s})$ is a genuine restriction, not an automatic consequence of semisimplicity.","The same construction should work for any semisimple $\\mathfrak{s}$ with a nondegenerate invariant bilinear form and $\\mathfrak{r}$ a non-trivial module: as long as an invariant map $\\mathfrak{r}\\otimes\\mathfrak{s}\\to\\mathbb{C}$ exists, condition (v) fails.","A direct computation of the Leibniz homology of the six-dimensional algebra would provide an independent check of the paper's conclusion and would clarify whether the duality $HL_{*+1}(-)\\cong HL_*(-,\\mathfrak{g}^\\sharp)$ itself holds in this case."],"forward_implications":["The proof of the weak Leibniz-homology conjecture given in [1] is invalid; the conjecture itself is not settled by that argument.","The isomorphism $HL_p(\\mathfrak{g})\\cong HL_{p-1}(\\mathfrak{g},\\mathfrak{r}^\\sharp)$ is not a general fact; any future proof must handle the radicals that admit invariant pairings with the semisimple part.","Theorems 4.7 and 4.8 of [3] are false as stated, since they rely on the same proposition.","For the six-dimensional example, the nontrivial action of $\\mathfrak{sl}_2$ on $H_1(\\mathfrak{r})$ through the adjoint representation is the obstruction; trivial or suitably constrained actions would satisfy condition (v)."],"supporting_citations":[{"why":"The preprint whose Proposition 3.1 is shown to be false; it supplies the claimed isomorphism under attack.","marker":"[1]"},{"why":"The preprint whose Theorem 4.7 and Corollary 4.8 are consequences of the false proposition.","marker":"[3]"},{"why":"Hochschild-Serre spectral sequence used to compute the homology H_*(g,s) in condition (v).","marker":"[4]"},{"why":"Supplies Theorem A and Proposition 1 used in Lemma 1 to pass between Leibniz and Lie homology.","marker":"[9]"},{"why":"States the weak conjecture that the counterexample shows remains open.","marker":"[10]"}],"fun_headline_variants":["Six-dimensional Lie algebra refutes Leibniz homology proposition","Burde-Wagemann proof fails; weak Leibniz conjecture open","Counterexample exposes error in Leibniz homology proof","Weak Leibniz conjecture remains unproven after counterexample","Feldvoss-Wagemann proposition refuted by explicit example"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument relies on the standing duality $HL_{*+1}(\\mathfrak{g})\\cong HL_*(\\mathfrak{g},\\mathfrak{g}^\\sharp)$, which translates Proposition 3.1 into condition (i) of Lemma 1; if that duality has hidden hypotheses, the counterexample would only refute condition (v) without directly contradicting Proposition 3.1.","fun_headline_variants_meta":{"raw":{"variants":["Six-dimensional Lie algebra refutes Leibniz homology proposition","Burde-Wagemann proof fails; weak Leibniz conjecture open","Counterexample exposes error in Leibniz homology proof","Weak Leibniz conjecture remains unproven after counterexample","Feldvoss-Wagemann proposition refuted by explicit example"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000767,"raw_usage":{"total_tokens":3330,"prompt_tokens":804,"completion_tokens":2526,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":420,"completion_tokens_details":{"reasoning_tokens":2448}},"tokens_in":420,"tokens_out":2526,"duration_ms":18302,"temperature":1.0,"reasoning_tokens":2448,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:11:00.150285+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One could compute the Leibniz homology groups $HL_2(\\mathfrak{g})$ and $HL_1(\\mathfrak{g},\\mathfrak{r}^\\sharp)$ directly for $\\mathfrak{g}=\\mathfrak{sl}_2\\ltimes\\mathfrak{sl}_2$; if they turn out to be isomorphic, the note's central claim that Proposition 3.1 fails would be incorrect.","supporting_citations":[{"cited_title":"Sympathetic Lie algebras and adjoint cohomology for Lie algebras","cited_arxiv_id":"1908.05963","evidence_quote":"The preprint whose Proposition 3.1 is shown to be false; it supplies the claimed isomorphism under attack."},{"cited_title":"Feldvoss and sc F","cited_arxiv_id":null,"evidence_quote":"The preprint whose Theorem 4.7 and Corollary 4.8 are consequences of the false proposition."},{"cited_title":"Hochschild and J.-P","cited_arxiv_id":null,"evidence_quote":"Hochschild-Serre spectral sequence used to compute the homology H_*(g,s) in condition (v)."},{"cited_title":"Pirashvili","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem A and Proposition 1 used in Lemma 1 to pass between Leibniz and Lie homology."},{"cited_title":"Pirashvili","cited_arxiv_id":null,"evidence_quote":"States the weak conjecture that the counterexample shows remains open."}],"review_version":1}