[ { "slug": "1-17353-planar-lower-bound-and-exact-local-envelopes-for-cost-preserving-single-", "name": "1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow", "shortName": "1.17353 planar bound and exact local envelopes for SSUFs", "problemNumber": null, "field": null, "fieldGroup": "Combinatorics", "resolution": "partial", "statement": "For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\\top y \\le c^\\top x$ and $y_a \\le x_a + C \\cdot D$ on every arc. We provide several new results on $C$: \n\n(1) record lower bound for planar instances (against known ceiling 2):\n$$C \\ge \\frac{58676765987259}{50000000000000} = 1.17353531974518;$$\n\n(2) local envelope ladder (proved): $E(2) = 1$, $E(3) = 9/8$, $E(4) = (299 - 41\\cdot\\sqrt{41})/32 = 1.13974707\\ldots$, attained by the counterexamples from our previous work; record constants of our previous work are now exact local envelopes of the general theory;\n\n(3) global results: every exact-two-path instance with rows touching at most three terminals satisfies $C \\le 2$ (first unconditional constant for an unbounded class); interaction arity m gives $\\lceil\\lfloor 3m/2\\rfloor /2\\rceil \\cdot D$;\n\n(4) classes closed exactly: out-trees 0; two-layer hubs 1; outerplanar two-exit interval spines 1 (sharp); series-parallel $\\le 1$;\n\n(5) band merger constant $K^* \\ge 2.5652\\ldots$ (twice the general lower bound $1.2826\\ldots$).", "posedBy": "Dinitz, Garg, Goemans", "yearPosed": 1999, "solveType": "proved", "solveDate": "2026-08-13", "model": "GPT-5.6 Sol, Claude Fable 5, Claude Opus 5", "modelMaker": "OpenAI, Anthropic", "humanCollaborators": [], "aiRole": "Like the previous parts (https://vibemathed.com/problem/1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u), this work was written in close collaboration with GPT 5.6 Sol, Claude Fable 5, and Claude Opus 5, which contributed proofs, failed routes, adversarial reviews, code for the verification campaign, and more. This time, I cannot claim that it was a single tour de force by the models like the original counterexample by Rybin; it was a long journey with a lot of human involvement, but the key ideas were provided by the LLMs. I have personally verified and edited this work in its entirety, and all errors are mine.", "verification": "site-confirmed", "verificationNote": "Reproduced by this site on 13 August 2026 from a clean clone, in two parts. First the repository's own suite: all fourteen verifiers in verify/run_all.sh pass, exit 0. Those cover Parts I and II only, so the headline planar record was rebuilt here independently. Reading only the raw arc list, a depth-first search rediscovers exactly two source-to-terminal paths for each of the six terminals; the fractional arc loads recompute exactly on all 21 arcs; all 64 routing overloads recompute exactly in rational arithmetic; the cost rule fits all 64 of the certificate's own cost deltas; 42 routings come out cost-preserving as claimed; and the minimum overload over those 42 is $58676765987259/50000000000000$, exactly the record. Planarity was checked independently too, by Euler ($V=16$, $E=21$, $F=7$) and by networkx. The envelope constant was derived symbolically from the stated quartic rather than read off: $t^*=(7-\\sqrt{41})/4$ is the unique critical point in $(0,2-\\sqrt3)$, giving $E(4)=(299-41\\sqrt{41})/32$. The 2,015-cell closure ledger is internally complete: five forms of 403, family counts summing to 2,015, every cell on one of nine solver-free lemmas. Not checked: the mixture characterization, the network-matrix total-unimodularity theorem and the tree-path four-colouring theorem, conventional proofs in an unreviewed preprint with no independent expert review. The tier records this site's reproduction of the certificates; the structural theory remains unreviewed.", "publication": "preprint", "resolutionMethod": "argument", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://github.com/snikolenko/unsplittable-flows", "sourceName": "Unsplittable flows repository", "resultNote": "Part III of a series, and the first unconditional positive results in it. Settled exactly: the local envelope ladder $E(2)=1$, $E(3)=9/8$ and $E(4)=(299-41\\sqrt{41})/32=1.13974707\\ldots$, which recasts the earlier record constants as exact envelopes of the general theory rather than isolated instances, plus exact constants for four classes - out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (sharp), series-parallel at most 1. Improved but not settled: the planar lower bound rises to $1.17353531974518$ against the known ceiling 2, and every exact-two-path instance whose rows touch at most three terminals satisfies $C\\le2$, the first unconditional constant for an unbounded class. The universal question is untouched - it reduces here to a single factor-two merger statement with certified wall $K^*\\ge2.5652\\ldots$, twice the refined general lower bound $1.28260069\\ldots$.", "aiContribution": "ai-co-developed", "significance": 15, "significanceNote": "The residual optimal-constant question left open by the Dinitz-Garg-Goemans disproof, scored level with Part II of the same series. Specialist, but rooted in a well-known 1999 conjecture, and this instalment adds the first unconditional positive results rather than another record. Held at 15 because the universal constant is still open and the exact answers are for restricted classes.", "links": [ { "label": "Zenodo preprint", "url": "https://zenodo.org/records/21922203" } ], "relations": [ { "to": "1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u", "kind": "continues", "note": "Part III of the same programme. Part II's 1.28249 record lower bound becomes an exact local envelope of the general theory built here." } ] }, { "slug": "1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u", "name": "1.28249... Lower Bound and Partial Upper Bounds for Cost-Preserving Single-Source Unsplittable Flows", "shortName": "Common-Point Interval Systems for SSUF", "problemNumber": null, "field": null, "fieldGroup": "Combinatorics", "resolution": "partial", "statement": "For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\\top y \\le c^\\top x$ and $y_a \\le x_a + C \\cdot d_{\\max}$ on every arc. Goemans conjectured $C=1$; this was disproved in July 2026 by a separate seven-vertex counterexample with critical constant $16/15$ (see the Dinitz–Garg–Goemans entry), leaving the optimal $C$ open.\n\nLower bound: a seventeen-terminal common-point interval instance certifies\n$$ C\\ \\ge\\ \\frac{1282494797984843521}{10^{18}}=1.28249\\ldots $$\n\nUpper bounds: the paper proves the first unconditional ceiling below 2, but for the codimension-two case only, at complement mass $q=2$. The record cells lie outside it, the $k=17$ instance having $q=11$, so that ceiling does not bound the record ladder. Two figures are conjectures rather than results: $4/3$ as the supremum of critical constants over common-point cells, approached but not attained and not an extrapolation from the ladder (Conjecture 1.1, Theorem 5.1), and $2$ for the universal constant itself (Conjecture 1.2). The proved gap remains $[1.28249\\ldots,\\ 2]$.", "posedBy": "Dinitz, Garg, Goemans", "yearPosed": 1999, "solveType": "proved", "solveDate": "2026-07-31", "model": "GPT-5.6 Sol, Claude Fable 5, Claude Opus 5", "modelMaker": "OpenAI, Anthropic", "humanCollaborators": [ "Sergey Nikolenko" ], "aiRole": "GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 carried out the search for constructions, symbolic envelope derivations, proofs, and the exact-verifier development; the human author framed the program, directed the search, set the claim scope, and verified all results independently by hand.", "verification": "site-confirmed", "verificationNote": "The k=17 lower bound is a finite certificate: the verifier rebuilds the 67-arc instance from raw interval data, rediscovers all paths by DFS, and enumerates all 2^17 routings in exact rational arithmetic. Re-run by the site from a clean clone on 2026-08-01; the exact constant, 15 minimizers, and 18-atom hull certificate reproduce. The deletion-star ceiling theorems are conventional proofs in an unreviewed preprint, checked by the author only, with no independent expert review and no formalization. Tier reflects the site's confirmation of the certificate; the structural results remain unreviewed.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://github.com/snikolenko/unsplittable-flows/", "sourceName": "Unsplittable flows repository", "resultNote": "Record lower bound only. The sub-2 ceiling is the codimension-two case and does not bound the record ladder (k=17 has complement mass 11). 4/3 and 2 are conjectures; the proved gap is [1.28249, 2].", "aiContribution": "ai-co-developed", "significance": 15, "significanceNote": "The residual optimal-constant question left open by the Dinitz-Garg-Goemans disproof; specialist, but rooted in a well-known 1999 conjecture.", "links": [ { "label": "Zenodo preprint", "url": "https://zenodo.org/records/21716713" }, { "label": "Paper I's Zenodo record", "url": "https://zenodo.org/records/21701162" } ], "relations": [ { "to": "dinitz-garg-goemans-unsplittable-flow", "kind": "builds-on", "note": "The optimal-constant question this entry attacks is exactly what the Dinitz-Garg-Goemans disproof left open." } ] }, { "slug": "a-counterexample-to-han-s-conjecture", "name": "Han's Conjecture", "shortName": "Han's conjecture", "problemNumber": null, "field": "Homological algebra", "fieldGroup": "Algebra", "resolution": "resolved", "statement": "For a finite-dimensional algebra $A$, finite global dimension forces $\\mathrm{HH}_n(A) = 0$ for all large $n$. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional $\\mathbb{C}$-algebra with $\\mathrm{HH}_n(A) = 0$ for every $n \\geq 1$ and $\\mathrm{gldim}\\, A = \\infty$, built by transporting Krah's phantom into a singularity category via one-periodic folding.", "posedBy": "Yang Han", "yearPosed": 2006, "solveType": "disproved", "solveDate": "2026-07-31", "model": "GPT-5.6 Sol Ultra", "modelMaker": "OpenAI", "humanCollaborators": [ "Bochao Kong", "Yeqin Liu", "Yu Shen" ], "aiRole": "The paper's acknowledgment in full: \"The counterexample presented in this paper was discovered with the assistance of OpenAI's GPT-5.6 Sol Ultra model. All mathematical arguments and references were independently verified by the authors.\" In a counterexample paper the algebra is the whole result, so crediting the model with its discovery is a claim about the central object, not about support work. The hedge \"with the assistance of\" keeps this below the top tier.", "verification": "unreviewed", "verificationNote": "A preprint days old with no independent review. The surrounding evidence is unusually strong for something this new: two of the three authors disproved the differential-graded analogue of the same conjecture in December 2025, this paper extends that program, and the construction runs on named recent machinery (Krah's phantom from Inventiones 2024, Chen's partial-resolution theorem, the Wang-Arunachalam-Keller identification) rather than novel unpublished tools. That is provenance, not verification, and the tier reflects the difference.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2608.00177", "sourceName": "arXiv", "resultNote": "The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.", "ageNote": "Posed by Han in the J. London Math. Soc. in 2006 and open since, with a dedicated survey and positive results for commutative, monomial, graded, Koszul, local and cellular algebras. Happel asked the same for Hochschild cohomology; Buchweitz, Green, Madsen and Solberg answered that negatively in 2005, but their examples have nonzero homology in infinitely many degrees, so Han's survived.", "aiContribution": "ai-co-developed", "significance": 25, "significanceNote": "A named conjecture of the representation theory of finite-dimensional algebras, open for twenty years, with its own survey article, and the surviving half of the Happel-Han pair after the cohomology version fell in 2005.", "links": [ { "label": "Liu and Shen's dg-version counterexample (December 2025, no AI involvement)", "url": "https://arxiv.org/abs/2512.12460" } ] }, { "slug": "a-counterexample-to-the-howland-kato-problem-for-positive-commutators", "name": "A counterexample to the Howland-Kato problem for positive commutators", "shortName": "Howland-Kato problem", "problemNumber": null, "field": "Operator theory", "fieldGroup": "Analysis", "resolution": "resolved", "statement": "The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\\arctan(P),\\arctan(Q)]$ is nonzero and nonnegative.", "posedBy": "James Howland; Tosio Kato", "yearPosed": 1991, "solveType": "disproved", "solveDate": "2026-08-07", "model": "Grok 4.5", "modelMaker": "xAI", "humanCollaborators": [ "Rupert L. Frank", "Paata Ivanisvili" ], "aiRole": "The paper discloses only \"The authors acknowledge the use of AI tools. All mathematical arguments and proofs in the final manuscript were checked and written by the authors.\" Co-author Paata Ivanisvili (@PI010101, Professor of Mathematics at UC Irvine) has since said publicly that \"AI deserves a fair amount of credit for finding\" the key identity, and, asked which model: \"Grok 4.5 in Cursor with an agent found a non-symmetric counterexample f(x) = arctan(x/2) and g(x) = tanh(x)/2 + tanh(3x)/2 which works and is correct. However, in the final manuscript we implemented symmetric example.\" So the model found a valid counterexample, but not the symmetric one the paper is built around, and the positivity proof is the authors' own.", "verification": "unreviewed", "verificationNote": null, "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/pdf/2608.07805", "sourceName": "arXiv", "aiContribution": "ai-co-developed", "significance": 25, "significanceNote": "A named open problem of Howland and Kato in operator/spectral theory, open since the 1980s and carrying Kato's name, but tracked within one community rather than across mathematics. Placed level with Simon's extendable shellability (25), another decades-old specialist named conjecture settled by counterexample, and below the record instances of household conjectures such as Borsuk N=63 and Hadamard 668 (30). Torn between 25 and 30; rule 3 takes the lower.", "links": [ { "label": "Author's announcement (X)", "url": "https://x.com/PI010101/status/2087250049000734961" }, { "label": "Author on Grok 4.5's role (X)", "url": "https://x.com/PI010101/status/2087263984433111395" } ] }, { "slug": "a-counterexample-to-the-inverse-generator-problem-and-related-questions", "name": "The Inverse Generator Problem on Hilbert Spaces", "shortName": "Inverse generator problem", "problemNumber": null, "field": "Semigroup theory", "fieldGroup": "Analysis", "resolution": "resolved", "statement": "If $A$ generates a bounded $C_0$-semigroup on a Hilbert space and has dense range, does $A^{-1}$ also generate a bounded $C_0$-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator with dense range generating a bounded, strongly stable semigroup whose inverse generates no $C_0$-semigroup at all. The counterexamples come from one explicit finite-dimensional construction, using bases of $\\mathbb{C}^{2n}$ with uniformly bounded partial-sum projections but unconditionality constants growing like $n^\\alpha$.", "posedBy": "Ralph deLaubenfels", "yearPosed": 1988, "solveType": "disproved", "solveDate": "2026-08-06", "model": "ChatGPT 5.6 Pro, Claude Fable", "modelMaker": "OpenAI, Anthropic", "humanCollaborators": [ "Emiel Lorist", "Martin Meyries", "Mark Veraar" ], "aiRole": "ChatGPT 5.6 Pro by OpenAI was used to explore Schauder basis counterexamples to the inverse generator problem. It also assisted the adaptation of Ansorena's construction used to obtain the explicit Schauder basis in Proposition 2.1, and was used to optimize the explicit constants in Theorem 1.1 and Proposition 2.1 by repeatedly dissecting the estimates and searching for numerical improvements.\n\nIn version 3, the authors additionally provide a complete Lean 4 formalization of Theorem 1.1. This formalization was produced using Claude Fable by Anthropic. The formalized theorem is the finite-dimensional construction underlying the subsequent counterexamples. The paper attributes exploration, assistance, optimization and formalization to the models rather than the central mathematical construction itself, so the AI contribution remains classified as AI-assisted.", "verification": "unreviewed", "verificationNote": "Version 3 has a complete Lean 4 machine-checked proof of Theorem 1.1, the explicit finite-dimensional construction underlying the counterexamples. The accompanying repository states that the development is self-contained on top of Mathlib, contains no sorry, no native_decide and no numerical or floating-point proof steps, and that the final theorem depends only on Lean's standard axioms propext, Classical.choice and Quot.sound.\n\nHowever, the Lean development formalizes Theorem 1.1 rather than the infinite-dimensional inverse-generator conclusion of Theorem 1.2. In particular, the passage assembling the finite-dimensional blocks into the Hilbert-space counterexample has not been formally verified or independently expert-reviewed. The entry should therefore remain Unreviewed rather than being upgraded to Lean-checked or Lean-verified.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2608.06272", "sourceName": "arXiv:2608.06272 - A solution to the inverse generator problem and related questions", "resultNote": "One finite-dimensional construction settles several related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time. Version 2 adds Theorem 1.4, whose part (i) solves Question 6.1 of Chalmoukis, Tsikalas and Yakubovich; that question is tracked as its own entry.", "ageNote": "Posed by deLaubenfels in 1988, with a documented ladder since: positive for sectorial operators of angle below π/2, positive on Hilbert spaces for contractive semigroups by Lumer-Phillips, counterexamples on Banach spaces back to Komatsu, and Gomilko-Zwart-Tomilov producing Hilbert-space generators with slow unbounded growth. The bounded-semigroup Hilbert case stayed open until this paper.", "aiContribution": "ai-assisted", "significance": 22, "significanceNote": "A named 1988 problem of semigroup theory with a 38-year ladder of partial results, motivated from numerical analysis, control theory and functional calculus, but read within those communities only.", "links": [ { "label": "Lean 4 formalization repository", "url": "https://github.com/zoowirt/inverse-generator-lean" } ] }, { "slug": "a-counterexample-to-the-stable-forking-conjecture", "name": "A Counterexample to the Stable Forking Conjecture", "shortName": "Stable Forking Conjecture", "problemNumber": null, "field": "Model theory", "fieldGroup": "Logic & foundations", "resolution": "resolved", "statement": "Using ChatGPT 5.6, we find a counterexample to the stable forking conjecture. This answers a long-standing open question of Hart, Kim and Pillay (1996).", "posedBy": "Bradd Hart, Byunghan Kim, and Anand Pillay", "yearPosed": 1996, "solveType": "disproved", "solveDate": "2026-08-31", "model": "GPT 5.6 Sol", "modelMaker": "OpenAI", "humanCollaborators": [ "James Freitag; Scott Mutchnik" ], "aiRole": "GPT-5.6 Sol generated the counterexample through an interactive, human-guided search. Freitag and Mutchnik prompted it with structural restrictions any counterexample would have to satisfy, including using the Kim-Pillay abstract-independence criterion to prove simplicity and characterise forking, and directed the search toward infinite rank - earlier work having placed severe obstructions on a finite-rank example. They independently wrote the proofs and the manuscript.\n\nThe paper says it plainly, in the abstract and again in the text: \"This is an AI-generated result proven with the help of GPT-5.6 Sol. Specifically, we prompted ChatGPT to construct a counterexample to the stable forking conjecture with a detailed series of prompts which took into account the likely restrictions such a counterexample would have to satisfy.\" They add that \"in retrospect, it seems unlikely that this counterexample would have been found in the near term without the use of generative AI\".", "verification": "unreviewed", "verificationNote": "Unreviewed. A preprint three days old. The construction is explicit and the authors give the abstract-independence argument in full, so it is checkable by a model theorist, but none has done so on the record.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2609.00436", "sourceName": "arXiv", "resultNote": "The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if $a \\not\\downarrow_C b$ then there is $\\varphi(x,\\bar b) \\in \\mathrm{tp}(a/Cb)$ forking over $C$ whose parameter-free form $\\varphi(x,y)$ is stable.\n\nThe counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised by an abstract independence relation, and the random graph is encoded into that relation so that it has the order property; stable formulas therefore cannot determine all forking in simple theories.", "aiContribution": "ai-co-developed", "significance": 26, "significanceNote": "The stable forking conjecture is a founding problem of neostability theory, posed by Hart, Kim and Pillay in 1996 and worked on for thirty years; simple theories and forking are the subject's core machinery, so this is close to field-famous within model theory. Just below the Lonely Runner Conjecture and the Polynomial-Time Low-Degree Conjecture at 30, both of which are known well outside their home fields; clearly above Courtade-Kumar at 22." }, { "slug": "a-quantum-oracle-separation-between-mathsf-qma-2-and-mathsf-qma", "name": "A Quantum Oracle Separation Between $\\mathsf{QMA}(2)$ and $\\mathsf{QMA}$", "shortName": "Quantum oracle separation of QMA(2) from QMA", "problemNumber": null, "field": "Quantum complexity theory", "fieldGroup": "Quantum information & computing", "resolution": "resolved", "statement": "We find a quantum oracle relative to which $\\mathsf{QMA}\\neq\\mathsf{QMA}(2)$. As a consequence, we resolve the no-disentanglers conjecture of Watrous: for every $\\varepsilon+\\delta<1$, any $(\\varepsilon,\\delta)$-disentangler requires input size exponential in the number of output qubits.", "posedBy": "John Watrous (the no-disentanglers conjecture, as reported by Aaronson, Beigi, Drucker, Fefferman and Shor)", "yearPosed": 2009, "solveType": "proved", "solveDate": "2026-09-02", "model": "ChatGPT 5.6 Sol", "modelMaker": "OpenAI", "humanCollaborators": [ "John Bostanci; Sabee Grewal; Jonas Haferkamp; Andrew Huang; Yeongwoo Hwang; Anand Natarajan; Chinmay Nirkhe" ], "aiRole": "The authors explicitly state that the proof idea underlying the main theorem was generated using ChatGPT 5.6 Sol. They initially directed Sol to the unitary polynomial method of She and Yuen; with minimal further guidance, the model proposed the proof idea used in the paper. The authors then verified, simplified, and developed the argument. The key construction uses symmetric and antisymmetric subspace projectors to make the relevant local-unitary invariant polynomials collapse to a single univariate polynomial, reducing the lower bound to the approximate degree of $\\mathrm{OR}$.", "verification": "unreviewed", "verificationNote": "Unreviewed. A 25-page preprint two days old, no peer review, no formalisation. Seven authors, including several who work on exactly this, state that they \"verified, simplified, and developed\" the model's proof idea and take full responsibility. The argument reduces to the approximate degree of OR through the She-Yuen unitary polynomial method, so it is checkable by anyone who knows that toolkit. Nobody outside the author list has done so on the record.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 6, "sourceUrl": "https://arxiv.org/abs/2609.02865", "sourceName": "arXiv", "renownNote": "The QMA article exists in six Wikipedia languages; the QMA versus QMA(2) question has no article of its own and is known through it.", "resultNote": "The authors construct a unitary oracle $U$ such that\n$$\n\\mathsf{QMA}^{U}\\neq\\mathsf{QMA}(2)^{U}.\n$$\nTheir black-box problem is solvable by a $\\mathsf{QMA}(2)$ verifier with one oracle query and linear-size unentangled proofs, whereas any $\\mathsf{QMA}$ verifier must use either exponentially many queries or an exponentially large witness.\n\nAs a non-oracle consequence, they prove that for every fixed $\\varepsilon,\\delta\\ge 0$ with $\\varepsilon+\\delta<1$, any $(\\varepsilon,\\delta)$-disentangler requires exponentially many input qubits in the number of output qubits, resolving Watrous's no-disentanglers conjecture.", "aiContribution": "ai-discovered", "significance": 36, "significanceNote": "Whether two unentangled quantum proofs beat one, QMA versus QMA(2), has been one of the central open questions of quantum complexity since Aaronson, Beigi, Drucker, Fefferman and Shor posed it in 2009, and Watrous's no-disentanglers conjecture is its best-known sharpening. This gives the first oracle separation, relative to a unitary oracle, and settles the conjecture; the unrelativized and classical-oracle questions stay open. Just above the Werner two-copy distillability entry at 35, the catalog's top quantum entry, because this question is followed across the whole field.", "links": [ { "label": "Aaronson, Beigi, Drucker, Fefferman and Shor, The Power of Unentanglement, where the question is posed", "url": "https://arxiv.org/abs/0804.0802" } ] }, { "slug": "a-smooth-counterexample-to-the-trautman-conjecture", "name": "A Smooth Counterexample to the Trautman Conjecture", "shortName": "Trautman Conjecture", "problemNumber": null, "field": "CR geometry", "fieldGroup": "Analysis", "resolution": "resolved", "statement": "The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in $\\mathbb{C}^2$. We modify a standard construction of nonembeddable smooth strongly pseudoconvex CR 3-manifolds so that the condition of having a nowhere-zero closed section of the canonical bundle is preserved, thus providing a strongly pseudoconvex counterexample to the Trautman conjecture.", "posedBy": "Andrzej Trautman", "yearPosed": 1998, "solveType": "disproved", "solveDate": "2026-09-02", "model": "ChatGPT (model not named; the paper says ChatGPT Plus)", "modelMaker": "OpenAI", "humanCollaborators": [ "Sean N. Curry" ], "aiRole": "Sean Curry states that the counterexample construction itself was discovered through experimentation with ChatGPT Plus on August 18, 2026. ChatGPT was also used to produce an initial rough draft and later for proofreading. Curry independently checked all calculations and arguments and takes responsibility for the paper.", "verification": "unreviewed", "verificationNote": "Unreviewed. A single-author arXiv preprint two days old, labelled by its author \"Preliminary version. Comments welcome.\" The author states he independently checked all calculations and arguments. The construction is a modification of a standard nonembeddable strongly pseudoconvex CR 3-manifold, arranged so that the canonical bundle keeps a nowhere-zero closed section, so it is checkable by anyone who knows the Rosay-type examples; three CR specialists are thanked in the paper but none is on the record as having checked it. No peer review, no formalisation.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2609.03198", "sourceName": "arXiv", "renownNote": "No Wikipedia article for the conjecture in any language. Trautman himself has one in ten, and CR manifolds in three; the conjecture lives in the literature only.", "resultNote": "For every neighborhood $U$ of the origin in $\\mathbb{C}\\times\\mathbb{R}$ and every $\\varepsilon>0$, Curry constructs a smooth nonnegative perturbation $\\phi$, supported in $U$ with $\\|\\phi\\|_{C^2}<\\varepsilon$, such that\n$$\nT^{0,1}_\\phi=\\operatorname{span}_{\\mathbb C}\\{L_\\phi\\}\n$$\nis strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every $C^1$ CR function near the origin satisfies $dh(0)=0$. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to $S^3$ as an arbitrarily small $C^1$ perturbation of the standard spherical CR structure.", "aiContribution": "ai-discovered", "significance": 25, "significanceNote": "A named conjecture of Andrzej Trautman from 1998, at the meeting point of CR geometry and exact solutions of the Maxwell and Einstein equations, open for 28 years and settled outright by a strongly pseudoconvex counterexample. Real and old, but invisible outside CR geometry: no Wikipedia article, and the paper thanks three CR specialists for their scepticism about the conjecture, so this confirms a doubt rather than overturns a consensus. Level with the xz-conjecture counterexample at 25, another named conjecture refuted by an explicit witness; below the Hessian counterexample at 30." }, { "slug": "adaboost-cycling", "name": "Exhaustive AdaBoost Cycling Question", "shortName": "AdaBoost cycling", "problemNumber": null, "field": "Learning theory", "fieldGroup": "Theoretical computer science", "resolution": "resolved", "statement": "Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.", "posedBy": "Cynthia Rudin, Robert Schapire & Ingrid Daubechies", "yearPosed": 2012, "solveType": "disproved", "solveDate": "2026-04-08", "model": "GPT-5.4 Pro, Claude Opus 4.6", "modelMaker": "OpenAI / Anthropic", "humanCollaborators": [], "aiRole": "The block-product gadget - two factors sharing an exact period-2 orbit whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio - was developed with GPT-5.4 Pro and Claude Opus 4.6.", "verification": "unreviewed", "verificationNote": "All assertions certified by exact rational arithmetic; computer-assisted arXiv preprint, not yet peer-reviewed.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2604.07055", "sourceName": "arXiv:2604.07055 - AdaBoost does not always cycle", "aiContribution": "ai-co-developed", "significance": 15, "significanceNote": "Posed by Rudin, Schapire and Daubechies; cited across boosting theory." }, { "slug": "albertson-berman-induced-forest-conjecture", "name": "Albertson–Berman Induced Forest Conjecture", "shortName": "Albertson–Berman", "problemNumber": null, "field": "Graph theory", "fieldGroup": "Combinatorics", "resolution": "resolved", "statement": "Albertson and Berman conjectured that for every simple planar graph $G$ on $n$ vertices, the largest vertex set inducing a forest has size at least $n/2$. The standing lower bound since the same year has been Borodin's $2n/5$, from his acyclic five-colour theorem. False: there is an explicit $31$-vertex simple $3$-connected maximal planar graph $T$ whose largest induced forest has exactly $15$ vertices, and an infinite family $M_k$ on $31k$ vertices with induced-forest number exactly $15k$, giving the ratio $15/31 < 1/2$ even for triangulations of minimum degree five.", "posedBy": "Michael O. Albertson, David M. Berman", "yearPosed": 1979, "solveType": "disproved", "solveDate": "2026-08-11", "model": "GPT-5.6 Sol", "modelMaker": "OpenAI", "humanCollaborators": [ "Heejae Jung" ], "aiRole": "The paper's \"Acknowledgments and AI disclosure\" section states that the two-terminal gadget \"was discovered, and substantial parts of the proof strategy were developed, through interaction with OpenAI GPT-5.6 Sol\", while the author \"selected the research problem, directed the computational search and subsequent proof development, and checked the resulting mathematical arguments and computational certificates\". GPT-5.6 Sol also assisted in preparing the manuscript and the verification code.", "verification": "site-confirmed", "verificationNote": "Reproduced here on 12 August 2026. The refutation is a single finite object, so it is checkable outright rather than on trust. The 31-vertex seed $T$ was rebuilt from the paper's own definitions - the 14-vertex gadget's cyclic neighbour lists, the pentagonal-bipyramid base, the decorated rim edges, the stated labelling and the two completion edges - without running the author's code. That yields a simple 3-connected planar graph on 31 vertices with $87 = 3n-6$ edges, hence a triangulation, with the paper's degree multiset $4^1 5^{17} 6^6 7^7$. Its maximum induced forest was then computed exactly by two independent algorithms: an ILP with lazy cycle-elimination cuts, and a branch-and-bound minimum feedback vertex set with no LP involved. Both give $a(T) = 15$, equivalently a minimum feedback vertex set of exactly 16, against the 15.5 the conjecture requires. The two finite inputs to the symbolic argument were separately brute-forced - the terminal profile $(6,6,6,5)$ over all $2^{12}$ internal subsets, and $\\beta = 3$ over all $2^7$ subsets of the core - and $M_k$ for $k = 2..5$ confirmed planar on $31k$ vertices with $93k-6$ edges, minimum degree five, every seed induced. Worth noting what the shipped verifier does not do: it certifies the gadget embedding, the profile, $\\beta$ and the sphere certificates, but never computes $a(T)$ or $a(M_k)$, and says so. That computation is the one this site supplied. Not peer-reviewed, not on arXiv, no independent expert review.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2608.17350", "sourceName": "A 15/31 Counterexample Family to the Albertson-Berman Conjecture", "renownNote": "No Wikipedia article, but the conjecture appears on both Bojan Mohar's and Douglas West's open-problem pages for planar graphs and is cited by name in the current literature.", "resultNote": "The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.", "ageNote": "Posed in 1979 and open for 47 years. A paper on the same quantity, Makarov's Large induced forests in planar multigraphs, still described the conjecture as open in its revision of 9 August 2026, two days before this preprint appeared.", "aiContribution": "ai-co-developed", "significance": 30, "significanceNote": "A named 1979 conjecture carried on the standard open-problem pages for planar graphs (Bojan Mohar's list, Douglas West's list), with a continuing literature on partial cases - girth 4 and 5, triangle-free, bipartite, 2-outerplanar, multigraphs - and a gap between the conjectured n/2 and Borodin's 2n/5 that stood for 47 years. Placed above a specialist named conjecture such as Simon's extendable shellability (25) because it is older and more widely tracked, and well below a household problem such as the cycle double cover conjecture (55).", "links": [ { "label": "Preprint PDF", "url": "https://zenodo.org/records/21883880/files/Disproving_the_Albertson_Berman_Conjecture.pdf" }, { "label": "Author's verification script", "url": "https://zenodo.org/records/21883880/files/verify_stronger_ab_family.py" }, { "label": "Mohar's open problem list: induced forests in planar graphs", "url": "https://www.sfu.ca/~mohar/Problems/P0208InducedForestPlanar.html" }, { "label": "Makarov, Large induced forests in planar multigraphs - calls the conjecture open, revised 9 Aug 2026", "url": "https://arxiv.org/abs/2601.04637" }, { "label": "Makarov - a preprint claiming a somewhat different counterexample found independently without AI", "url": "https://zenodo.org/records/21927902" }, { "label": "Makarov - preprint PDF", "url": "https://zenodo.org/records/21927902/files/albertson_berman_conjecture_counterexample.pdf" } ] }, { "slug": "aluffi-chen-marcolli-real-rootedness", "name": "The Aluffi-Chen-Marcolli Real-Rootedness Conjecture", "shortName": "M(0,n) real-rootedness", "problemNumber": null, "field": "Algebraic geometry", "fieldGroup": "Geometry & topology", "resolution": "resolved", "statement": "Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space $\\overline{\\mathcal{M}}_{0,n}$ of stable $n$-pointed rational curves has only real roots. True, with simple roots and strict interlacing between consecutive $n$.", "posedBy": "Paolo Aluffi, Wenxuan Chen, Matilde Marcolli", "yearPosed": null, "solveType": "proved", "solveDate": "2026-05-27", "model": "Co-Mathematician", "modelMaker": null, "humanCollaborators": [ "Gergely Berczi", "Young-Hoon Kiem" ], "aiRole": "The paper says the proof was found with the assistance of Co-Mathematician, a frontier agentic LLM-based system for mathematical research described in a separate paper, and its title calls the result an AI-assisted proof.", "verification": "unreviewed", "verificationNote": "arXiv preprint; not yet peer-reviewed.", "publication": "preprint", "resolutionMethod": "argument", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2605.29151", "sourceName": "arXiv:2605.29151 - Real-rootedness of the Poincare polynomials of M(0,n): an AI-assisted proof", "aiContribution": "ai-co-developed", "significance": 20, "significanceNote": "A named conjecture about the topology of the Deligne-Mumford moduli spaces, objects at the foundation of modern enumerative geometry." }, { "slug": "amdeberhan-medina-moll-arctan-conjecture", "name": "Amdeberhan-Medina-Moll Arctangent Sum Conjecture", "shortName": "Arctangent sums", "problemNumber": null, "field": "Number theory", "fieldGroup": "Number theory", "resolution": "partial", "statement": "Let $x_n = \\tan\\left(\\sum_{k=1}^{n} \\arctan k\\right)$. Amdeberhan, Medina and Moll conjectured that $x_n \\notin \\mathbb{Z}$ for every $n \\ge 5$. Any integer value $x_n = m$ must satisfy $|m| \\ge e^{(1/2+o(1)) n \\log n}$, which forces $\\#\\{1 \\le n \\le N : x_n \\in \\mathbb{Z}\\} = O(\\log N)$. The conjecture therefore holds for a density-one set of $n$, improving on the previously known density of $120/817 \\approx 0.147$.", "posedBy": "Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll", "yearPosed": 2008, "solveType": "proved", "solveDate": "2026-07-07", "model": "AxiomProver", "modelMaker": "Axiom Math", "humanCollaborators": [ "Ken Ono" ], "aiRole": "The system was given only a natural-language statement of the definitions and the three target results, plus an instruction to formalize and prove them with no sorry. From that input AxiomProver autonomously produced both the Lean formalization of the problem and a complete Lean proof. The human author then wrote the paper's exposition using the formal development as his reference, which reverses the usual order: the Lean came first and the prose was derived from it.", "verification": "unreviewed", "verificationNote": "A public Lean 4.28.0 development accompanies the paper, containing a formalization of the problem and a proof the author states is sorry-free and adds no axioms. We attempted to compile it and did not complete the build, so the axiom claim here rests on the author's statement rather than on our own check.", "publication": "preprint", "resolutionMethod": "argument", "citations": null, "citationsPaper": null,Lines 1–500 of 25022. Load more
[ { "slug": "1-17353-planar-lower-bound-and-exact-local-envelopes-for-cost-preserving-single-", "name": "1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow", "shortName": "1.17353 planar bound and exact local envelopes for SSUFs", "problemNumber": null, "field": null, "fieldGroup": "Combinatorics", "resolution": "partial", "statement": "For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\\top y \\le c^\\top x$ and $y_a \\le x_a + C \\cdot D$ on every arc. We provide several new results on $C$: \n\n(1) record lower bound for planar instances (against known ceiling 2):\n$$C \\ge \\frac{58676765987259}{50000000000000} = 1.17353531974518;$$\n\n(2) local envelope ladder (proved): $E(2) = 1$, $E(3) = 9/8$, $E(4) = (299 - 41\\cdot\\sqrt{41})/32 = 1.13974707\\ldots$, attained by the counterexamples from our previous work; record constants of our previous work are now exact local envelopes of the general theory;\n\n(3) global results: every exact-two-path instance with rows touching at most three terminals satisfies $C \\le 2$ (first unconditional constant for an unbounded class); interaction arity m gives $\\lceil\\lfloor 3m/2\\rfloor /2\\rceil \\cdot D$;\n\n(4) classes closed exactly: out-trees 0; two-layer hubs 1; outerplanar two-exit interval spines 1 (sharp); series-parallel $\\le 1$;\n\n(5) band merger constant $K^* \\ge 2.5652\\ldots$ (twice the general lower bound $1.2826\\ldots$).", "posedBy": "Dinitz, Garg, Goemans", "yearPosed": 1999, "solveType": "proved", "solveDate": "2026-08-13", "model": "GPT-5.6 Sol, Claude Fable 5, Claude Opus 5", "modelMaker": "OpenAI, Anthropic", "humanCollaborators": [], "aiRole": "Like the previous parts (https://vibemathed.com/problem/1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u), this work was written in close collaboration with GPT 5.6 Sol, Claude Fable 5, and Claude Opus 5, which contributed proofs, failed routes, adversarial reviews, code for the verification campaign, and more. This time, I cannot claim that it was a single tour de force by the models like the original counterexample by Rybin; it was a long journey with a lot of human involvement, but the key ideas were provided by the LLMs. I have personally verified and edited this work in its entirety, and all errors are mine.", "verification": "site-confirmed", "verificationNote": "Reproduced by this site on 13 August 2026 from a clean clone, in two parts. First the repository's own suite: all fourteen verifiers in verify/run_all.sh pass, exit 0. Those cover Parts I and II only, so the headline planar record was rebuilt here independently. Reading only the raw arc list, a depth-first search rediscovers exactly two source-to-terminal paths for each of the six terminals; the fractional arc loads recompute exactly on all 21 arcs; all 64 routing overloads recompute exactly in rational arithmetic; the cost rule fits all 64 of the certificate's own cost deltas; 42 routings come out cost-preserving as claimed; and the minimum overload over those 42 is $58676765987259/50000000000000$, exactly the record. Planarity was checked independently too, by Euler ($V=16$, $E=21$, $F=7$) and by networkx. The envelope constant was derived symbolically from the stated quartic rather than read off: $t^*=(7-\\sqrt{41})/4$ is the unique critical point in $(0,2-\\sqrt3)$, giving $E(4)=(299-41\\sqrt{41})/32$. The 2,015-cell closure ledger is internally complete: five forms of 403, family counts summing to 2,015, every cell on one of nine solver-free lemmas. Not checked: the mixture characterization, the network-matrix total-unimodularity theorem and the tree-path four-colouring theorem, conventional proofs in an unreviewed preprint with no independent expert review. The tier records this site's reproduction of the certificates; the structural theory remains unreviewed.", "publication": "preprint", "resolutionMethod": "argument", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://github.com/snikolenko/unsplittable-flows", "sourceName": "Unsplittable flows repository", "resultNote": "Part III of a series, and the first unconditional positive results in it. Settled exactly: the local envelope ladder $E(2)=1$, $E(3)=9/8$ and $E(4)=(299-41\\sqrt{41})/32=1.13974707\\ldots$, which recasts the earlier record constants as exact envelopes of the general theory rather than isolated instances, plus exact constants for four classes - out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (sharp), series-parallel at most 1. Improved but not settled: the planar lower bound rises to $1.17353531974518$ against the known ceiling 2, and every exact-two-path instance whose rows touch at most three terminals satisfies $C\\le2$, the first unconditional constant for an unbounded class. The universal question is untouched - it reduces here to a single factor-two merger statement with certified wall $K^*\\ge2.5652\\ldots$, twice the refined general lower bound $1.28260069\\ldots$.", "aiContribution": "ai-co-developed", "significance": 15, "significanceNote": "The residual optimal-constant question left open by the Dinitz-Garg-Goemans disproof, scored level with Part II of the same series. Specialist, but rooted in a well-known 1999 conjecture, and this instalment adds the first unconditional positive results rather than another record. Held at 15 because the universal constant is still open and the exact answers are for restricted classes.", "links": [ { "label": "Zenodo preprint", "url": "https://zenodo.org/records/21922203" } ], "relations": [ { "to": "1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u", "kind": "continues", "note": "Part III of the same programme. Part II's 1.28249 record lower bound becomes an exact local envelope of the general theory built here." } ] }, { "slug": "1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u", "name": "1.28249... Lower Bound and Partial Upper Bounds for Cost-Preserving Single-Source Unsplittable Flows", "shortName": "Common-Point Interval Systems for SSUF", "problemNumber": null, "field": null, "fieldGroup": "Combinatorics", "resolution": "partial", "statement": "For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\\top y \\le c^\\top x$ and $y_a \\le x_a + C \\cdot d_{\\max}$ on every arc. Goemans conjectured $C=1$; this was disproved in July 2026 by a separate seven-vertex counterexample with critical constant $16/15$ (see the Dinitz–Garg–Goemans entry), leaving the optimal $C$ open.\n\nLower bound: a seventeen-terminal common-point interval instance certifies\n$$ C\\ \\ge\\ \\frac{1282494797984843521}{10^{18}}=1.28249\\ldots $$\n\nUpper bounds: the paper proves the first unconditional ceiling below 2, but for the codimension-two case only, at complement mass $q=2$. The record cells lie outside it, the $k=17$ instance having $q=11$, so that ceiling does not bound the record ladder. Two figures are conjectures rather than results: $4/3$ as the supremum of critical constants over common-point cells, approached but not attained and not an extrapolation from the ladder (Conjecture 1.1, Theorem 5.1), and $2$ for the universal constant itself (Conjecture 1.2). The proved gap remains $[1.28249\\ldots,\\ 2]$.", "posedBy": "Dinitz, Garg, Goemans", "yearPosed": 1999, "solveType": "proved", "solveDate": "2026-07-31", "model": "GPT-5.6 Sol, Claude Fable 5, Claude Opus 5", "modelMaker": "OpenAI, Anthropic", "humanCollaborators": [ "Sergey Nikolenko" ], "aiRole": "GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 carried out the search for constructions, symbolic envelope derivations, proofs, and the exact-verifier development; the human author framed the program, directed the search, set the claim scope, and verified all results independently by hand.", "verification": "site-confirmed", "verificationNote": "The k=17 lower bound is a finite certificate: the verifier rebuilds the 67-arc instance from raw interval data, rediscovers all paths by DFS, and enumerates all 2^17 routings in exact rational arithmetic. Re-run by the site from a clean clone on 2026-08-01; the exact constant, 15 minimizers, and 18-atom hull certificate reproduce. The deletion-star ceiling theorems are conventional proofs in an unreviewed preprint, checked by the author only, with no independent expert review and no formalization. Tier reflects the site's confirmation of the certificate; the structural results remain unreviewed.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://github.com/snikolenko/unsplittable-flows/", "sourceName": "Unsplittable flows repository", "resultNote": "Record lower bound only. The sub-2 ceiling is the codimension-two case and does not bound the record ladder (k=17 has complement mass 11). 4/3 and 2 are conjectures; the proved gap is [1.28249, 2].", "aiContribution": "ai-co-developed", "significance": 15, "significanceNote": "The residual optimal-constant question left open by the Dinitz-Garg-Goemans disproof; specialist, but rooted in a well-known 1999 conjecture.", "links": [ { "label": "Zenodo preprint", "url": "https://zenodo.org/records/21716713" }, { "label": "Paper I's Zenodo record", "url": "https://zenodo.org/records/21701162" } ], "relations": [ { "to": "dinitz-garg-goemans-unsplittable-flow", "kind": "builds-on", "note": "The optimal-constant question this entry attacks is exactly what the Dinitz-Garg-Goemans disproof left open." } ] }, { "slug": "a-counterexample-to-han-s-conjecture", "name": "Han's Conjecture", "shortName": "Han's conjecture", "problemNumber": null, "field": "Homological algebra", "fieldGroup": "Algebra", "resolution": "resolved", "statement": "For a finite-dimensional algebra $A$, finite global dimension forces $\\mathrm{HH}_n(A) = 0$ for all large $n$. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional $\\mathbb{C}$-algebra with $\\mathrm{HH}_n(A) = 0$ for every $n \\geq 1$ and $\\mathrm{gldim}\\, A = \\infty$, built by transporting Krah's phantom into a singularity category via one-periodic folding.", "posedBy": "Yang Han", "yearPosed": 2006, "solveType": "disproved", "solveDate": "2026-07-31", "model": "GPT-5.6 Sol Ultra", "modelMaker": "OpenAI", "humanCollaborators": [ "Bochao Kong", "Yeqin Liu", "Yu Shen" ], "aiRole": "The paper's acknowledgment in full: \"The counterexample presented in this paper was discovered with the assistance of OpenAI's GPT-5.6 Sol Ultra model. All mathematical arguments and references were independently verified by the authors.\" In a counterexample paper the algebra is the whole result, so crediting the model with its discovery is a claim about the central object, not about support work. The hedge \"with the assistance of\" keeps this below the top tier.", "verification": "unreviewed", "verificationNote": "A preprint days old with no independent review. The surrounding evidence is unusually strong for something this new: two of the three authors disproved the differential-graded analogue of the same conjecture in December 2025, this paper extends that program, and the construction runs on named recent machinery (Krah's phantom from Inventiones 2024, Chen's partial-resolution theorem, the Wang-Arunachalam-Keller identification) rather than novel unpublished tools. That is provenance, not verification, and the tier reflects the difference.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2608.00177", "sourceName": "arXiv", "resultNote": "The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.", "ageNote": "Posed by Han in the J. London Math. Soc. in 2006 and open since, with a dedicated survey and positive results for commutative, monomial, graded, Koszul, local and cellular algebras. Happel asked the same for Hochschild cohomology; Buchweitz, Green, Madsen and Solberg answered that negatively in 2005, but their examples have nonzero homology in infinitely many degrees, so Han's survived.", "aiContribution": "ai-co-developed", "significance": 25, "significanceNote": "A named conjecture of the representation theory of finite-dimensional algebras, open for twenty years, with its own survey article, and the surviving half of the Happel-Han pair after the cohomology version fell in 2005.", "links": [ { "label": "Liu and Shen's dg-version counterexample (December 2025, no AI involvement)", "url": "https://arxiv.org/abs/2512.12460" } ] }, { "slug": "a-counterexample-to-the-howland-kato-problem-for-positive-commutators", "name": "A counterexample to the Howland-Kato problem for positive commutators", "shortName": "Howland-Kato problem", "problemNumber": null, "field": "Operator theory", "fieldGroup": "Analysis", "resolution": "resolved", "statement": "The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\\arctan(P),\\arctan(Q)]$ is nonzero and nonnegative.", "posedBy": "James Howland; Tosio Kato", "yearPosed": 1991, "solveType": "disproved", "solveDate": "2026-08-07", "model": "Grok 4.5", "modelMaker": "xAI", "humanCollaborators": [ "Rupert L. Frank", "Paata Ivanisvili" ], "aiRole": "The paper discloses only \"The authors acknowledge the use of AI tools. All mathematical arguments and proofs in the final manuscript were checked and written by the authors.\" Co-author Paata Ivanisvili (@PI010101, Professor of Mathematics at UC Irvine) has since said publicly that \"AI deserves a fair amount of credit for finding\" the key identity, and, asked which model: \"Grok 4.5 in Cursor with an agent found a non-symmetric counterexample f(x) = arctan(x/2) and g(x) = tanh(x)/2 + tanh(3x)/2 which works and is correct. However, in the final manuscript we implemented symmetric example.\" So the model found a valid counterexample, but not the symmetric one the paper is built around, and the positivity proof is the authors' own.", "verification": "unreviewed", "verificationNote": null, "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/pdf/2608.07805", "sourceName": "arXiv", "aiContribution": "ai-co-developed", "significance": 25, "significanceNote": "A named open problem of Howland and Kato in operator/spectral theory, open since the 1980s and carrying Kato's name, but tracked within one community rather than across mathematics. Placed level with Simon's extendable shellability (25), another decades-old specialist named conjecture settled by counterexample, and below the record instances of household conjectures such as Borsuk N=63 and Hadamard 668 (30). Torn between 25 and 30; rule 3 takes the lower.", "links": [ { "label": "Author's announcement (X)", "url": "https://x.com/PI010101/status/2087250049000734961" }, { "label": "Author on Grok 4.5's role (X)", "url": "https://x.com/PI010101/status/2087263984433111395" } ] }, { "slug": "a-counterexample-to-the-inverse-generator-problem-and-related-questions", "name": "The Inverse Generator Problem on Hilbert Spaces", "shortName": "Inverse generator problem", "problemNumber": null, "field": "Semigroup theory", "fieldGroup": "Analysis", "resolution": "resolved", "statement": "If $A$ generates a bounded $C_0$-semigroup on a Hilbert space and has dense range, does $A^{-1}$ also generate a bounded $C_0$-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator with dense range generating a bounded, strongly stable semigroup whose inverse generates no $C_0$-semigroup at all. The counterexamples come from one explicit finite-dimensional construction, using bases of $\\mathbb{C}^{2n}$ with uniformly bounded partial-sum projections but unconditionality constants growing like $n^\\alpha$.", "posedBy": "Ralph deLaubenfels", "yearPosed": 1988, "solveType": "disproved", "solveDate": "2026-08-06", "model": "ChatGPT 5.6 Pro, Claude Fable", "modelMaker": "OpenAI, Anthropic", "humanCollaborators": [ "Emiel Lorist", "Martin Meyries", "Mark Veraar" ], "aiRole": "ChatGPT 5.6 Pro by OpenAI was used to explore Schauder basis counterexamples to the inverse generator problem. It also assisted the adaptation of Ansorena's construction used to obtain the explicit Schauder basis in Proposition 2.1, and was used to optimize the explicit constants in Theorem 1.1 and Proposition 2.1 by repeatedly dissecting the estimates and searching for numerical improvements.\n\nIn version 3, the authors additionally provide a complete Lean 4 formalization of Theorem 1.1. This formalization was produced using Claude Fable by Anthropic. The formalized theorem is the finite-dimensional construction underlying the subsequent counterexamples. The paper attributes exploration, assistance, optimization and formalization to the models rather than the central mathematical construction itself, so the AI contribution remains classified as AI-assisted.", "verification": "unreviewed", "verificationNote": "Version 3 has a complete Lean 4 machine-checked proof of Theorem 1.1, the explicit finite-dimensional construction underlying the counterexamples. The accompanying repository states that the development is self-contained on top of Mathlib, contains no sorry, no native_decide and no numerical or floating-point proof steps, and that the final theorem depends only on Lean's standard axioms propext, Classical.choice and Quot.sound.\n\nHowever, the Lean development formalizes Theorem 1.1 rather than the infinite-dimensional inverse-generator conclusion of Theorem 1.2. In particular, the passage assembling the finite-dimensional blocks into the Hilbert-space counterexample has not been formally verified or independently expert-reviewed. The entry should therefore remain Unreviewed rather than being upgraded to Lean-checked or Lean-verified.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2608.06272", "sourceName": "arXiv:2608.06272 - A solution to the inverse generator problem and related questions", "resultNote": "One finite-dimensional construction settles several related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time. Version 2 adds Theorem 1.4, whose part (i) solves Question 6.1 of Chalmoukis, Tsikalas and Yakubovich; that question is tracked as its own entry.", "ageNote": "Posed by deLaubenfels in 1988, with a documented ladder since: positive for sectorial operators of angle below π/2, positive on Hilbert spaces for contractive semigroups by Lumer-Phillips, counterexamples on Banach spaces back to Komatsu, and Gomilko-Zwart-Tomilov producing Hilbert-space generators with slow unbounded growth. The bounded-semigroup Hilbert case stayed open until this paper.", "aiContribution": "ai-assisted", "significance": 22, "significanceNote": "A named 1988 problem of semigroup theory with a 38-year ladder of partial results, motivated from numerical analysis, control theory and functional calculus, but read within those communities only.", "links": [ { "label": "Lean 4 formalization repository", "url": "https://github.com/zoowirt/inverse-generator-lean" } ] }, { "slug": "a-counterexample-to-the-stable-forking-conjecture", "name": "A Counterexample to the Stable Forking Conjecture", "shortName": "Stable Forking Conjecture", "problemNumber": null, "field": "Model theory", "fieldGroup": "Logic & foundations", "resolution": "resolved", "statement": "Using ChatGPT 5.6, we find a counterexample to the stable forking conjecture. This answers a long-standing open question of Hart, Kim and Pillay (1996).", "posedBy": "Bradd Hart, Byunghan Kim, and Anand Pillay", "yearPosed": 1996, "solveType": "disproved", "solveDate": "2026-08-31", "model": "GPT 5.6 Sol", "modelMaker": "OpenAI", "humanCollaborators": [ "James Freitag; Scott Mutchnik" ], "aiRole": "GPT-5.6 Sol generated the counterexample through an interactive, human-guided search. Freitag and Mutchnik prompted it with structural restrictions any counterexample would have to satisfy, including using the Kim-Pillay abstract-independence criterion to prove simplicity and characterise forking, and directed the search toward infinite rank - earlier work having placed severe obstructions on a finite-rank example. They independently wrote the proofs and the manuscript.\n\nThe paper says it plainly, in the abstract and again in the text: \"This is an AI-generated result proven with the help of GPT-5.6 Sol. Specifically, we prompted ChatGPT to construct a counterexample to the stable forking conjecture with a detailed series of prompts which took into account the likely restrictions such a counterexample would have to satisfy.\" They add that \"in retrospect, it seems unlikely that this counterexample would have been found in the near term without the use of generative AI\".", "verification": "unreviewed", "verificationNote": "Unreviewed. A preprint three days old. The construction is explicit and the authors give the abstract-independence argument in full, so it is checkable by a model theorist, but none has done so on the record.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2609.00436", "sourceName": "arXiv", "resultNote": "The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if $a \\not\\downarrow_C b$ then there is $\\varphi(x,\\bar b) \\in \\mathrm{tp}(a/Cb)$ forking over $C$ whose parameter-free form $\\varphi(x,y)$ is stable.\n\nThe counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised by an abstract independence relation, and the random graph is encoded into that relation so that it has the order property; stable formulas therefore cannot determine all forking in simple theories.", "aiContribution": "ai-co-developed", "significance": 26, "significanceNote": "The stable forking conjecture is a founding problem of neostability theory, posed by Hart, Kim and Pillay in 1996 and worked on for thirty years; simple theories and forking are the subject's core machinery, so this is close to field-famous within model theory. Just below the Lonely Runner Conjecture and the Polynomial-Time Low-Degree Conjecture at 30, both of which are known well outside their home fields; clearly above Courtade-Kumar at 22." }, { "slug": "a-quantum-oracle-separation-between-mathsf-qma-2-and-mathsf-qma", "name": "A Quantum Oracle Separation Between $\\mathsf{QMA}(2)$ and $\\mathsf{QMA}$", "shortName": "Quantum oracle separation of QMA(2) from QMA", "problemNumber": null, "field": "Quantum complexity theory", "fieldGroup": "Quantum information & computing", "resolution": "resolved", "statement": "We find a quantum oracle relative to which $\\mathsf{QMA}\\neq\\mathsf{QMA}(2)$. As a consequence, we resolve the no-disentanglers conjecture of Watrous: for every $\\varepsilon+\\delta<1$, any $(\\varepsilon,\\delta)$-disentangler requires input size exponential in the number of output qubits.", "posedBy": "John Watrous (the no-disentanglers conjecture, as reported by Aaronson, Beigi, Drucker, Fefferman and Shor)", "yearPosed": 2009, "solveType": "proved", "solveDate": "2026-09-02", "model": "ChatGPT 5.6 Sol", "modelMaker": "OpenAI", "humanCollaborators": [ "John Bostanci; Sabee Grewal; Jonas Haferkamp; Andrew Huang; Yeongwoo Hwang; Anand Natarajan; Chinmay Nirkhe" ], "aiRole": "The authors explicitly state that the proof idea underlying the main theorem was generated using ChatGPT 5.6 Sol. They initially directed Sol to the unitary polynomial method of She and Yuen; with minimal further guidance, the model proposed the proof idea used in the paper. The authors then verified, simplified, and developed the argument. The key construction uses symmetric and antisymmetric subspace projectors to make the relevant local-unitary invariant polynomials collapse to a single univariate polynomial, reducing the lower bound to the approximate degree of $\\mathrm{OR}$.", "verification": "unreviewed", "verificationNote": "Unreviewed. A 25-page preprint two days old, no peer review, no formalisation. Seven authors, including several who work on exactly this, state that they \"verified, simplified, and developed\" the model's proof idea and take full responsibility. The argument reduces to the approximate degree of OR through the She-Yuen unitary polynomial method, so it is checkable by anyone who knows that toolkit. Nobody outside the author list has done so on the record.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 6, "sourceUrl": "https://arxiv.org/abs/2609.02865", "sourceName": "arXiv", "renownNote": "The QMA article exists in six Wikipedia languages; the QMA versus QMA(2) question has no article of its own and is known through it.", "resultNote": "The authors construct a unitary oracle $U$ such that\n$$\n\\mathsf{QMA}^{U}\\neq\\mathsf{QMA}(2)^{U}.\n$$\nTheir black-box problem is solvable by a $\\mathsf{QMA}(2)$ verifier with one oracle query and linear-size unentangled proofs, whereas any $\\mathsf{QMA}$ verifier must use either exponentially many queries or an exponentially large witness.\n\nAs a non-oracle consequence, they prove that for every fixed $\\varepsilon,\\delta\\ge 0$ with $\\varepsilon+\\delta<1$, any $(\\varepsilon,\\delta)$-disentangler requires exponentially many input qubits in the number of output qubits, resolving Watrous's no-disentanglers conjecture.", "aiContribution": "ai-discovered", "significance": 36, "significanceNote": "Whether two unentangled quantum proofs beat one, QMA versus QMA(2), has been one of the central open questions of quantum complexity since Aaronson, Beigi, Drucker, Fefferman and Shor posed it in 2009, and Watrous's no-disentanglers conjecture is its best-known sharpening. This gives the first oracle separation, relative to a unitary oracle, and settles the conjecture; the unrelativized and classical-oracle questions stay open. Just above the Werner two-copy distillability entry at 35, the catalog's top quantum entry, because this question is followed across the whole field.", "links": [ { "label": "Aaronson, Beigi, Drucker, Fefferman and Shor, The Power of Unentanglement, where the question is posed", "url": "https://arxiv.org/abs/0804.0802" } ] }, { "slug": "a-smooth-counterexample-to-the-trautman-conjecture", "name": "A Smooth Counterexample to the Trautman Conjecture", "shortName": "Trautman Conjecture", "problemNumber": null, "field": "CR geometry", "fieldGroup": "Analysis", "resolution": "resolved", "statement": "The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in $\\mathbb{C}^2$. We modify a standard construction of nonembeddable smooth strongly pseudoconvex CR 3-manifolds so that the condition of having a nowhere-zero closed section of the canonical bundle is preserved, thus providing a strongly pseudoconvex counterexample to the Trautman conjecture.", "posedBy": "Andrzej Trautman", "yearPosed": 1998, "solveType": "disproved", "solveDate": "2026-09-02", "model": "ChatGPT (model not named; the paper says ChatGPT Plus)", "modelMaker": "OpenAI", "humanCollaborators": [ "Sean N. Curry" ], "aiRole": "Sean Curry states that the counterexample construction itself was discovered through experimentation with ChatGPT Plus on August 18, 2026. ChatGPT was also used to produce an initial rough draft and later for proofreading. Curry independently checked all calculations and arguments and takes responsibility for the paper.", "verification": "unreviewed", "verificationNote": "Unreviewed. A single-author arXiv preprint two days old, labelled by its author \"Preliminary version. Comments welcome.\" The author states he independently checked all calculations and arguments. The construction is a modification of a standard nonembeddable strongly pseudoconvex CR 3-manifold, arranged so that the canonical bundle keeps a nowhere-zero closed section, so it is checkable by anyone who knows the Rosay-type examples; three CR specialists are thanked in the paper but none is on the record as having checked it. No peer review, no formalisation.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2609.03198", "sourceName": "arXiv", "renownNote": "No Wikipedia article for the conjecture in any language. Trautman himself has one in ten, and CR manifolds in three; the conjecture lives in the literature only.", "resultNote": "For every neighborhood $U$ of the origin in $\\mathbb{C}\\times\\mathbb{R}$ and every $\\varepsilon>0$, Curry constructs a smooth nonnegative perturbation $\\phi$, supported in $U$ with $\\|\\phi\\|_{C^2}<\\varepsilon$, such that\n$$\nT^{0,1}_\\phi=\\operatorname{span}_{\\mathbb C}\\{L_\\phi\\}\n$$\nis strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every $C^1$ CR function near the origin satisfies $dh(0)=0$. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to $S^3$ as an arbitrarily small $C^1$ perturbation of the standard spherical CR structure.", "aiContribution": "ai-discovered", "significance": 25, "significanceNote": "A named conjecture of Andrzej Trautman from 1998, at the meeting point of CR geometry and exact solutions of the Maxwell and Einstein equations, open for 28 years and settled outright by a strongly pseudoconvex counterexample. Real and old, but invisible outside CR geometry: no Wikipedia article, and the paper thanks three CR specialists for their scepticism about the conjecture, so this confirms a doubt rather than overturns a consensus. Level with the xz-conjecture counterexample at 25, another named conjecture refuted by an explicit witness; below the Hessian counterexample at 30." }, { "slug": "adaboost-cycling", "name": "Exhaustive AdaBoost Cycling Question", "shortName": "AdaBoost cycling", "problemNumber": null, "field": "Learning theory", "fieldGroup": "Theoretical computer science", "resolution": "resolved", "statement": "Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.", "posedBy": "Cynthia Rudin, Robert Schapire & Ingrid Daubechies", "yearPosed": 2012, "solveType": "disproved", "solveDate": "2026-04-08", "model": "GPT-5.4 Pro, Claude Opus 4.6", "modelMaker": "OpenAI / Anthropic", "humanCollaborators": [], "aiRole": "The block-product gadget - two factors sharing an exact period-2 orbit whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio - was developed with GPT-5.4 Pro and Claude Opus 4.6.", "verification": "unreviewed", "verificationNote": "All assertions certified by exact rational arithmetic; computer-assisted arXiv preprint, not yet peer-reviewed.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2604.07055", "sourceName": "arXiv:2604.07055 - AdaBoost does not always cycle", "aiContribution": "ai-co-developed", "significance": 15, "significanceNote": "Posed by Rudin, Schapire and Daubechies; cited across boosting theory." }, { "slug": "albertson-berman-induced-forest-conjecture", "name": "Albertson–Berman Induced Forest Conjecture", "shortName": "Albertson–Berman", "problemNumber": null, "field": "Graph theory", "fieldGroup": "Combinatorics", "resolution": "resolved", "statement": "Albertson and Berman conjectured that for every simple planar graph $G$ on $n$ vertices, the largest vertex set inducing a forest has size at least $n/2$. The standing lower bound since the same year has been Borodin's $2n/5$, from his acyclic five-colour theorem. False: there is an explicit $31$-vertex simple $3$-connected maximal planar graph $T$ whose largest induced forest has exactly $15$ vertices, and an infinite family $M_k$ on $31k$ vertices with induced-forest number exactly $15k$, giving the ratio $15/31 < 1/2$ even for triangulations of minimum degree five.", "posedBy": "Michael O. Albertson, David M. Berman", "yearPosed": 1979, "solveType": "disproved", "solveDate": "2026-08-11", "model": "GPT-5.6 Sol", "modelMaker": "OpenAI", "humanCollaborators": [ "Heejae Jung" ], "aiRole": "The paper's \"Acknowledgments and AI disclosure\" section states that the two-terminal gadget \"was discovered, and substantial parts of the proof strategy were developed, through interaction with OpenAI GPT-5.6 Sol\", while the author \"selected the research problem, directed the computational search and subsequent proof development, and checked the resulting mathematical arguments and computational certificates\". GPT-5.6 Sol also assisted in preparing the manuscript and the verification code.", "verification": "site-confirmed", "verificationNote": "Reproduced here on 12 August 2026. The refutation is a single finite object, so it is checkable outright rather than on trust. The 31-vertex seed $T$ was rebuilt from the paper's own definitions - the 14-vertex gadget's cyclic neighbour lists, the pentagonal-bipyramid base, the decorated rim edges, the stated labelling and the two completion edges - without running the author's code. That yields a simple 3-connected planar graph on 31 vertices with $87 = 3n-6$ edges, hence a triangulation, with the paper's degree multiset $4^1 5^{17} 6^6 7^7$. Its maximum induced forest was then computed exactly by two independent algorithms: an ILP with lazy cycle-elimination cuts, and a branch-and-bound minimum feedback vertex set with no LP involved. Both give $a(T) = 15$, equivalently a minimum feedback vertex set of exactly 16, against the 15.5 the conjecture requires. The two finite inputs to the symbolic argument were separately brute-forced - the terminal profile $(6,6,6,5)$ over all $2^{12}$ internal subsets, and $\\beta = 3$ over all $2^7$ subsets of the core - and $M_k$ for $k = 2..5$ confirmed planar on $31k$ vertices with $93k-6$ edges, minimum degree five, every seed induced. Worth noting what the shipped verifier does not do: it certifies the gadget embedding, the profile, $\\beta$ and the sphere certificates, but never computes $a(T)$ or $a(M_k)$, and says so. That computation is the one this site supplied. Not peer-reviewed, not on arXiv, no independent expert review.", "publication": "preprint", "resolutionMethod": "construction", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2608.17350", "sourceName": "A 15/31 Counterexample Family to the Albertson-Berman Conjecture", "renownNote": "No Wikipedia article, but the conjecture appears on both Bojan Mohar's and Douglas West's open-problem pages for planar graphs and is cited by name in the current literature.", "resultNote": "The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.", "ageNote": "Posed in 1979 and open for 47 years. A paper on the same quantity, Makarov's Large induced forests in planar multigraphs, still described the conjecture as open in its revision of 9 August 2026, two days before this preprint appeared.", "aiContribution": "ai-co-developed", "significance": 30, "significanceNote": "A named 1979 conjecture carried on the standard open-problem pages for planar graphs (Bojan Mohar's list, Douglas West's list), with a continuing literature on partial cases - girth 4 and 5, triangle-free, bipartite, 2-outerplanar, multigraphs - and a gap between the conjectured n/2 and Borodin's 2n/5 that stood for 47 years. Placed above a specialist named conjecture such as Simon's extendable shellability (25) because it is older and more widely tracked, and well below a household problem such as the cycle double cover conjecture (55).", "links": [ { "label": "Preprint PDF", "url": "https://zenodo.org/records/21883880/files/Disproving_the_Albertson_Berman_Conjecture.pdf" }, { "label": "Author's verification script", "url": "https://zenodo.org/records/21883880/files/verify_stronger_ab_family.py" }, { "label": "Mohar's open problem list: induced forests in planar graphs", "url": "https://www.sfu.ca/~mohar/Problems/P0208InducedForestPlanar.html" }, { "label": "Makarov, Large induced forests in planar multigraphs - calls the conjecture open, revised 9 Aug 2026", "url": "https://arxiv.org/abs/2601.04637" }, { "label": "Makarov - a preprint claiming a somewhat different counterexample found independently without AI", "url": "https://zenodo.org/records/21927902" }, { "label": "Makarov - preprint PDF", "url": "https://zenodo.org/records/21927902/files/albertson_berman_conjecture_counterexample.pdf" } ] }, { "slug": "aluffi-chen-marcolli-real-rootedness", "name": "The Aluffi-Chen-Marcolli Real-Rootedness Conjecture", "shortName": "M(0,n) real-rootedness", "problemNumber": null, "field": "Algebraic geometry", "fieldGroup": "Geometry & topology", "resolution": "resolved", "statement": "Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space $\\overline{\\mathcal{M}}_{0,n}$ of stable $n$-pointed rational curves has only real roots. True, with simple roots and strict interlacing between consecutive $n$.", "posedBy": "Paolo Aluffi, Wenxuan Chen, Matilde Marcolli", "yearPosed": null, "solveType": "proved", "solveDate": "2026-05-27", "model": "Co-Mathematician", "modelMaker": null, "humanCollaborators": [ "Gergely Berczi", "Young-Hoon Kiem" ], "aiRole": "The paper says the proof was found with the assistance of Co-Mathematician, a frontier agentic LLM-based system for mathematical research described in a separate paper, and its title calls the result an AI-assisted proof.", "verification": "unreviewed", "verificationNote": "arXiv preprint; not yet peer-reviewed.", "publication": "preprint", "resolutionMethod": "argument", "citations": null, "citationsPaper": null, "citationsSource": null, "citationsUrl": null, "renownLangs": 0, "sourceUrl": "https://arxiv.org/abs/2605.29151", "sourceName": "arXiv:2605.29151 - Real-rootedness of the Poincare polynomials of M(0,n): an AI-assisted proof", "aiContribution": "ai-co-developed", "significance": 20, "significanceNote": "A named conjecture about the topology of the Deligne-Mumford moduli spaces, objects at the foundation of modern enumerative geometry." }, { "slug": "amdeberhan-medina-moll-arctan-conjecture", "name": "Amdeberhan-Medina-Moll Arctangent Sum Conjecture", "shortName": "Arctangent sums", "problemNumber": null, "field": "Number theory", "fieldGroup": "Number theory", "resolution": "partial", "statement": "Let $x_n = \\tan\\left(\\sum_{k=1}^{n} \\arctan k\\right)$. Amdeberhan, Medina and Moll conjectured that $x_n \\notin \\mathbb{Z}$ for every $n \\ge 5$. Any integer value $x_n = m$ must satisfy $|m| \\ge e^{(1/2+o(1)) n \\log n}$, which forces $\\#\\{1 \\le n \\le N : x_n \\in \\mathbb{Z}\\} = O(\\log N)$. The conjecture therefore holds for a density-one set of $n$, improving on the previously known density of $120/817 \\approx 0.147$.", "posedBy": "Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll", "yearPosed": 2008, "solveType": "proved", "solveDate": "2026-07-07", "model": "AxiomProver", "modelMaker": "Axiom Math", "humanCollaborators": [ "Ken Ono" ], "aiRole": "The system was given only a natural-language statement of the definitions and the three target results, plus an instruction to formalize and prove them with no sorry. From that input AxiomProver autonomously produced both the Lean formalization of the problem and a complete Lean proof. The human author then wrote the paper's exposition using the formal development as his reference, which reverses the usual order: the Lean came first and the prose was derived from it.", "verification": "unreviewed", "verificationNote": "A public Lean 4.28.0 development accompanies the paper, containing a formalization of the problem and a proof the author states is sorry-free and adds no axioms. We attempted to compile it and did not complete the build, so the axiom claim here rests on the author's statement rather than on our own check.", "publication": "preprint", "resolutionMethod": "argument", "citations": null, "citationsPaper": null,