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Only the catalogue is acquired, and every record carries the attribution the licence requires."},"snapshot_policy":{"mode":"content_root_only","retention":"none","reason":"The adapter roots each record on the catalogue entry it observed and binds the retrieval that served it; no second archive of the dataset is created."},"adapter":{"adapter_id":"problems-data/vibemathed","version":"1.0.0","mode":"networked_acquisition","acquisition_contract":"vela.source-adapter-bundle.v2","observation_contract":"vela.math-source-observation.v1","adapter_root":"sha256:f1c6dcffe525f93b822d6e4a4256992123cc3e013ea10412a6754ae37da10d4b"},"coverage":{"repository_slugs":["math"],"included":["Every catalogue entry the dataset endpoint serves, with its curatorial labels: resolution, verification rung, AI-contribution tier, and significance score","The problem statement text where the catalogue carries one, retained under CC BY 4.0 with attribution"],"omissions":[{"code":"curatorial_labels_not_vela_standing","description":"Verification rung, AI-contribution tier, significance score and moderator approval are source attributions. None of them creates Vela Standing, Verification, or acceptance."},{"code":"no_pinned_revision","description":"The endpoint serves no commit locator and re-renders its envelope on a cache cycle, so an observation names a retrieval rather than a revision."},{"code":"community_state_not_observed","description":"Votes, comment counts, submitter pseudonyms, edit changelogs, discussion threads and member profiles are mutable site state and are not projected."},{"code":"linked_evidence_not_acquired","description":"The preprints and Lean repositories an entry cites are retained as locators; their bytes and their separate rights are outside this source."}]},"declaration_root":"sha256:24b6a98a77247e3cb6c86107f19cc6da6e90a7a95466c9257f6029a0f9244781"},"declaration_row_root":"sha256:24b6a98a77247e3cb6c86107f19cc6da6e90a7a95466c9257f6029a0f9244781","observation":{"schema":"vela.math-source-observation.v1","source_id":"source:vibemathed","observation_id":"observation:vibemathed:7a86a741ec053c8f","declaration_root":"sha256:24b6a98a77247e3cb6c86107f19cc6da6e90a7a95466c9257f6029a0f9244781","acquisition_root":"sha256:7a86a741ec053c8f441fd74c2dfec3293fb87ad1c29e95a2ab9cbccdc16e7d94","observed_at":"2026-08-19T14:43:50-04:00","native_revision":{"kind":"observation","value":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","content_root":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","tree":null},"snapshot_root":null,"snapshot_state":"content_root_only","projected_record_count":599,"projected_records_root":"sha256:ab49490ecd2387398a0ad0a8cd7a263c0c9248924967c6daef732a9128b43f23","coverage":{"status":"complete","included":["Every catalogue entry served by the VibeMathed dataset endpoint at the observed retrieval."],"native_record_count":599,"projected_record_count":599},"omissions":[{"code":"community_state_not_projected","description":"Votes, comment counts, submitter pseudonyms, discussion threads and the field-level edit changelog are mutable site state and are not projected."},{"code":"link_labels_not_retained","description":"An entry's cited links are retained as record locators; their upstream labels and kind tags are not carried as fields."},{"code":"linked_evidence_not_acquired","description":"The preprints and Lean repositories an entry cites are located, never fetched; their bytes and their separate rights are outside this source."},{"code":"source_attributions_remain_attributed","description":"Resolution, verification rung, AI-contribution tier and significance are source attributions, not Vela verification or acceptance."},{"code":"retrieval_is_not_a_revision","description":"The endpoint serves no commit locator, so the recorded revision roots the observed catalogue and names a retrieval rather than an upstream revision."}],"observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264"},"native_record_count":599,"repository_binding_count":0}],"native_records":[{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:1-17353-planar-lower-bound-and-exact-local-envelopes-for-cost-preserving-single-","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow","summary":"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 $1.2826\\ldots$).","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/1-17353-planar-lower-bound-and-exact-local-envelopes-for-cost-preserving-single-"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/snikolenko/unsplittable-flows"},{"locator_id":"native-3","kind":"artifact","url":"https://zenodo.org/records/21922203"}],"metadata":{"name":"1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow","slug":"1-17353-planar-lower-bound-and-exact-local-envelopes-for-cost-preserving-single-","field":null,"model":"GPT-5.6 Sol, Claude Fable 5, Claude Opus 5","ai_role":"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.","posed_by":"Dinitz, Garg, Goemans","citations":null,"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 $1.2826\\ldots$).","resolution":"partial","short_name":"1.17353 planar bound and exact local envelopes for SSUFs","solve_date":"2026-08-13","solve_type":"proved","source_url":"https://github.com/snikolenko/unsplittable-flows","year_posed":1999,"field_group":"Combinatorics","model_maker":"OpenAI, Anthropic","publication":"preprint","result_note":"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 $1.28260069\\ldots$.","source_name":"Unsplittable flows repository","significance":15,"verification":"site-confirmed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"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.","verification_note":"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.","human_collaborators":"[]"},"metadata_root":"sha256:6d241b59ebfd7386b246318036ce650bf30619628d4eceaefb31375044053a44","content_root":"sha256:a9070bd6e52566ae2129d90cd037c2565bc78c73bb0a5f54c310878b948d9cdb","availability":"reference_only","row_root":"sha256:a661fcaa9d2d94d5b4db008926d3bb79f14889a2a98b5899297774af591d16a8"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"1.28249... Lower Bound and Partial Upper Bounds for Cost-Preserving Single-Source Unsplittable Flows","summary":"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 $[1.28249\\ldots,\\ 2]$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/snikolenko/unsplittable-flows/"},{"locator_id":"native-3","kind":"artifact","url":"https://zenodo.org/records/21716713"},{"locator_id":"native-4","kind":"artifact","url":"https://zenodo.org/records/21701162"}],"metadata":{"name":"1.28249... Lower Bound and Partial Upper Bounds for Cost-Preserving Single-Source Unsplittable Flows","slug":"1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u","field":null,"model":"GPT-5.6 Sol, Claude Fable 5, Claude Opus 5","ai_role":"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.","posed_by":"Dinitz, Garg, Goemans","citations":null,"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 $[1.28249\\ldots,\\ 2]$.","resolution":"partial","short_name":"Common-Point Interval Systems for SSUF","solve_date":"2026-07-31","solve_type":"proved","source_url":"https://github.com/snikolenko/unsplittable-flows/","year_posed":1999,"field_group":"Combinatorics","model_maker":"OpenAI, Anthropic","publication":"preprint","result_note":"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].","source_name":"Unsplittable flows repository","significance":15,"verification":"site-confirmed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"The residual optimal-constant question left open by the Dinitz-Garg-Goemans disproof; specialist, but rooted in a well-known 1999 conjecture.","verification_note":"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.","human_collaborators":"[\"Sergey Nikolenko\"]"},"metadata_root":"sha256:69ec9c434775ec6d1c9f816548d2d3f8878640bb1645cbcafeab27b9c6584922","content_root":"sha256:a6507a7272b41c1c130ffd1b09149eed78189ea2f5d6439fe5f1b9213d65ae34","availability":"reference_only","row_root":"sha256:acfe54bfc2250a21f116d6116c7c6082428de4796b46a82f62947801f2bb2d65"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:a-counterexample-to-han-s-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Han's Conjecture","summary":"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.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/a-counterexample-to-han-s-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.00177"},{"locator_id":"native-3","kind":"artifact","url":"https://arxiv.org/abs/2512.12460"}],"metadata":{"name":"Han's Conjecture","slug":"a-counterexample-to-han-s-conjecture","field":"Homological algebra","model":"GPT-5.6 Sol Ultra","ai_role":"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.","posed_by":"Yang Han","citations":null,"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.","resolution":"resolved","short_name":"Han's conjecture","solve_date":"2026-07-31","solve_type":"disproved","source_url":"https://arxiv.org/abs/2608.00177","year_posed":2006,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":"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.","source_name":"arXiv","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"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.","verification_note":"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.","human_collaborators":"[\"Bochao Kong\",\"Yeqin Liu\",\"Yu Shen\"]"},"metadata_root":"sha256:5f98355dabfabbfd5b595d51a872add3b57eaf31bdde1bd3bf3a9af96da11110","content_root":"sha256:d4ec7f1ec93d207d7b9e3ab8319ce4d73a9f307138a1eb8c8f1e4f4f8707e1a3","availability":"reference_only","row_root":"sha256:6d27eb539d18aef134228826f5f9cace450dde76688ed95047bfa5b6d344e99b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:a-counterexample-to-the-howland-kato-problem-for-positive-commutators","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"A counterexample to the Howland-Kato problem for positive commutators","summary":"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.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/a-counterexample-to-the-howland-kato-problem-for-positive-commutators"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/pdf/2608.07805"},{"locator_id":"native-3","kind":"artifact","url":"https://x.com/PI010101/status/2087250049000734961"},{"locator_id":"native-4","kind":"artifact","url":"https://x.com/PI010101/status/2087263984433111395"}],"metadata":{"name":"A counterexample to the Howland-Kato problem for positive commutators","slug":"a-counterexample-to-the-howland-kato-problem-for-positive-commutators","field":"Operator theory","model":"Grok 4.5","ai_role":"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.","posed_by":"James Howland; Tosio Kato","citations":null,"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.","resolution":"resolved","short_name":"Howland-Kato problem","solve_date":"2026-08-07","solve_type":"disproved","source_url":"https://arxiv.org/pdf/2608.07805","year_posed":1991,"field_group":"Analysis","model_maker":"xAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"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.","verification_note":null,"human_collaborators":"[\"Rupert L. Frank\",\"Paata Ivanisvili\"]"},"metadata_root":"sha256:314ecce0b067ffb4bf333f64acc709022525dcdf654bb46ea6444f06ab42bfac","content_root":"sha256:bc3e3d9fb5958a41f31d3fbe697f122408544e0762197fd8ccf22790351f06af","availability":"reference_only","row_root":"sha256:862024746e7acc73461b363dcfb5f5be93fede1fdf8677c1390c2ab92ec10f38"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:a-counterexample-to-the-inverse-generator-problem-and-related-questions","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Inverse Generator Problem on Hilbert Spaces","summary":"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 $n^\\alpha$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/a-counterexample-to-the-inverse-generator-problem-and-related-questions"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.06272"}],"metadata":{"name":"The Inverse Generator Problem on Hilbert Spaces","slug":"a-counterexample-to-the-inverse-generator-problem-and-related-questions","field":"Semigroup theory","model":"ChatGPT 5.6 Pro, Claude Fable 5","ai_role":"The paper's disclosure in full: \"ChatGPT 5.6 Pro by OpenAI was used to explore proof strategies and to check intermediate steps. Claude Fable 5 by Anthropic was used to check the arguments and detect mistakes. The authors take full responsibility for the content of this note.\" Strategy exploration and checking rather than an attributed step, so the lower tier applies.","posed_by":"Ralph deLaubenfels","citations":null,"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 $n^\\alpha$.","resolution":"resolved","short_name":"Inverse generator problem","solve_date":"2026-08-06","solve_type":"disproved","source_url":"https://arxiv.org/abs/2608.06272","year_posed":1988,"field_group":"Analysis","model_maker":"OpenAI, Anthropic","publication":"preprint","result_note":"One finite-dimensional construction settles three 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.","source_name":"arXiv","significance":22,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"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.","verification_note":"A preprint days old; nobody independent has checked it and there is no formalization. The construction is unusually checkable, though: every matrix entry has a closed formula, and a curator-run numerical check of Proposition 2.1 (the explicit finite-dimensional construction underlying Theorem 1.1) confirmed all of its claimed bounds - the Toeplitz inverse identity, uniform partial-sum projections, both multiplier bounds, and the n^alpha growth of the alternating multiplier - at sizes up to n = 256 for several alpha. That validates the engine of the counterexample, not the full operator-theoretic argument, so the entry stays unreviewed.","human_collaborators":"[\"Emiel Lorist\",\"Martin Meyries\",\"Mark Veraar\"]"},"metadata_root":"sha256:08bd07ec0cd23c05073ca4de145afdf5888d688e9675131729bcc0b97e079346","content_root":"sha256:9f192a555ef0f0715507e6a3c45cef20eb5e37d22c4f4175bdaeaf8e10fffbe4","availability":"reference_only","row_root":"sha256:2edf9dffaf040f6a0d7cf88a84320cad60f60fa9a7775226eb62299b354cbb23"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:adaboost-cycling","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Exhaustive AdaBoost Cycling Question","summary":"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.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/adaboost-cycling"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2604.07055"}],"metadata":{"name":"Exhaustive AdaBoost Cycling Question","slug":"adaboost-cycling","field":"Learning theory","model":"GPT-5.4 Pro, Claude Opus 4.6","ai_role":"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.","posed_by":"Cynthia Rudin, Robert Schapire & Ingrid Daubechies","citations":null,"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.","resolution":"resolved","short_name":"AdaBoost cycling","solve_date":"2026-04-08","solve_type":"disproved","source_url":"https://arxiv.org/abs/2604.07055","year_posed":2012,"field_group":"Theoretical computer science","model_maker":"OpenAI / Anthropic","publication":"preprint","result_note":null,"source_name":"arXiv:2604.07055 - AdaBoost does not always cycle","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Posed by Rudin, Schapire and Daubechies; cited across boosting theory.","verification_note":"All assertions certified by exact rational arithmetic; computer-assisted arXiv preprint, not yet peer-reviewed.","human_collaborators":"[]"},"metadata_root":"sha256:13c5c01b2bb39f8a1a97856e31b6b956fcda19c7ac32656bedc3ca43182deb27","content_root":"sha256:dff4c91770b93970f1d97996bac16499a1d536ed953efd59b1d7418c3078d600","availability":"reference_only","row_root":"sha256:04bc564160cc27b7418a24fa629db70b5eebbd9d9bcdaa5b3beda47b0e7cb5af"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:albertson-berman-induced-forest-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Albertson–Berman Induced Forest Conjecture","summary":"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.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/albertson-berman-induced-forest-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://zenodo.org/records/21883880"},{"locator_id":"native-3","kind":"artifact","url":"https://zenodo.org/records/21883880/files/Disproving_the_Albertson_Berman_Conjecture.pdf"},{"locator_id":"native-4","kind":"artifact","url":"https://zenodo.org/records/21883880/files/verify_stronger_ab_family.py"},{"locator_id":"native-5","kind":"artifact","url":"https://www.sfu.ca/~mohar/Problems/P0208InducedForestPlanar.html"},{"locator_id":"native-6","kind":"artifact","url":"https://arxiv.org/abs/2601.04637"},{"locator_id":"native-7","kind":"artifact","url":"https://zenodo.org/records/21927902"},{"locator_id":"native-8","kind":"artifact","url":"https://zenodo.org/records/21927902/files/albertson_berman_conjecture_counterexample.pdf"}],"metadata":{"name":"Albertson–Berman Induced Forest Conjecture","slug":"albertson-berman-induced-forest-conjecture","field":"Graph theory","model":"GPT-5.6 Sol","ai_role":"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.","posed_by":"Michael O. Albertson, David M. Berman","citations":null,"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.","resolution":"resolved","short_name":"Albertson–Berman","solve_date":"2026-08-11","solve_type":"disproved","source_url":"https://zenodo.org/records/21883880","year_posed":1979,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":"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.","source_name":"Zenodo - A 15/31 Counterexample Family to the Albertson–Berman Conjecture","significance":30,"verification":"site-confirmed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"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).","verification_note":"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.","human_collaborators":"[\"Heejae Jung\"]"},"metadata_root":"sha256:d893e140755b02fb86a6e7c491ad16199ac99f018afb14c1497149b47be7bee8","content_root":"sha256:d238525b7374152b54cc564d026e25892344e5f9e2403b92df2f026f4fb8f66b","availability":"reference_only","row_root":"sha256:b0c425f5ea10aeeae8d3c30feb5090df1df9c516a8e7f51596600e4de472433b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:aluffi-chen-marcolli-real-rootedness","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Aluffi-Chen-Marcolli Real-Rootedness Conjecture","summary":"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 $n$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/aluffi-chen-marcolli-real-rootedness"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.29151"}],"metadata":{"name":"The Aluffi-Chen-Marcolli Real-Rootedness Conjecture","slug":"aluffi-chen-marcolli-real-rootedness","field":"Algebraic geometry","model":"Co-Mathematician","ai_role":"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.","posed_by":"Paolo Aluffi, Wenxuan Chen, Matilde Marcolli","citations":null,"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 $n$.","resolution":"resolved","short_name":"M(0,n) real-rootedness","solve_date":"2026-05-27","solve_type":"proved","source_url":"https://arxiv.org/abs/2605.29151","year_posed":null,"field_group":"Geometry & topology","model_maker":null,"publication":"preprint","result_note":null,"source_name":"arXiv:2605.29151 - Real-rootedness of the Poincare polynomials of M(0,n): an AI-assisted proof","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A named conjecture about the topology of the Deligne-Mumford moduli spaces, objects at the foundation of modern enumerative geometry.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Gergely Berczi\",\"Young-Hoon Kiem\"]"},"metadata_root":"sha256:26278d447063da535b05240bfb21d0c687feb1aaecfae6781c2b3a9721869fc0","content_root":"sha256:ddc81fdf5e095f749a9b659159fd77bd02cb897e99da13359f4df6f6fcfc13f5","availability":"reference_only","row_root":"sha256:e5cd53be9e9120c51ba2466e340729f584b356568b0a55c1823c1985f2950a1d"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:amdeberhan-medina-moll-arctan-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Amdeberhan-Medina-Moll Arctangent Sum Conjecture","summary":"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 $120/817 \\approx 0.147$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/amdeberhan-medina-moll-arctan-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.05739"},{"locator_id":"native-3","kind":"artifact","url":"https://github.com/AxiomMath/TanArctan"}],"metadata":{"name":"Amdeberhan-Medina-Moll Arctangent Sum Conjecture","slug":"amdeberhan-medina-moll-arctan-conjecture","field":"Number theory","model":"AxiomProver","ai_role":"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.","posed_by":"Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll","citations":null,"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 $120/817 \\approx 0.147$.","resolution":"partial","short_name":"Arctangent sums","solve_date":"2026-07-07","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.05739","year_posed":2008,"field_group":"Number theory","model_maker":"Axiom Math","publication":"preprint","result_note":"density-one set of n; the conjecture itself remains open","source_name":"arXiv:2607.05739 - Integer values of tan(arctan 1 + arctan 2 + ... + arctan n) are rare","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A named conjecture from a 2008 Journal of Number Theory paper with a documented line of partial results, familiar within the arctangent-sums and Gaussian-integer literature but not beyond it.","verification_note":"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.","human_collaborators":"[\"Ken Ono\"]"},"metadata_root":"sha256:0b6a410fc98d1d6e745a095fdf4cce5df627fb0bbfa570daf108280bac875611","content_root":"sha256:656c4396ab0dc45e5106ef544e6c1d72a0aed3ed9d6a2cc13613438f6cd9b645","availability":"reference_only","row_root":"sha256:1c46c7e323f39ee35a80db039d7fb0f887dc8961c18b7a73f780b39f6613eab9"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:ame-states-five-open-cases","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Absolutely Maximally Entangled States in Five Open Cases","summary":"A pure state of $n$ parties with $q$ levels each is absolutely maximally entangled, written $\\mathrm{AME}(n,q)$, when every subsystem of at most $\\lfloor n/2 \\rfloor$ parties is maximally mixed. These are the perfect tensors, and existence is a parameter-by-parameter problem: some $(n,q)$ admit one, some provably do not, and a maintained table records which cells are still unknown.\n\nThis paper settles five of them. It exhibits Hermitian self-dual MDS codes $[12,6,7]_{25}$, $[18,9,10]_{121}$ and $[18,9,10]_{169}$, from which the stabilizer construction gives $\\mathrm{AME}(12,5)$, $\\mathrm{AME}(18,11)$ and $\\mathrm{AME}(18,13)$, and projecting one party gives $\\mathrm{AME}(17,11)$ and $\\mathrm{AME}(17,13)$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/ame-states-five-open-cases"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.05781"},{"locator_id":"native-3","kind":"artifact","url":"https://tp.nt.uni-siegen.de/ame/ame.html"},{"locator_id":"native-4","kind":"artifact","url":"https://www.codetables.de/"}],"metadata":{"name":"Absolutely Maximally Entangled States in Five Open Cases","slug":"ame-states-five-open-cases","field":"Quantum error correction / AME states","model":"Claude Fable 5, ChatGPT 5.6 Sol","ai_role":"From the paper's \"Author contributions and AI use\" section, in the authors' words: under their direction, Claude Fable 5 (Anthropic) and ChatGPT 5.6 Sol (OpenAI) were used extensively throughout the project, including the entire computational search, the development and implementation of search methods, exact verification, bibliographic checks, and manuscript preparation. Both authors independently reviewed the constructions, computations and full text.\n\nThe objects are the result here, and the search that produced them is attributed to the models in full, which is why this is classified AI-discovered rather than assisted. One of the authors reported the paper to this site, noting it was almost entirely AI-generated.","posed_by":null,"citations":null,"statement":"A pure state of $n$ parties with $q$ levels each is absolutely maximally entangled, written $\\mathrm{AME}(n,q)$, when every subsystem of at most $\\lfloor n/2 \\rfloor$ parties is maximally mixed. These are the perfect tensors, and existence is a parameter-by-parameter problem: some $(n,q)$ admit one, some provably do not, and a maintained table records which cells are still unknown.\n\nThis paper settles five of them. It exhibits Hermitian self-dual MDS codes $[12,6,7]_{25}$, $[18,9,10]_{121}$ and $[18,9,10]_{169}$, from which the stabilizer construction gives $\\mathrm{AME}(12,5)$, $\\mathrm{AME}(18,11)$ and $\\mathrm{AME}(18,13)$, and projecting one party gives $\\mathrm{AME}(17,11)$ and $\\mathrm{AME}(17,13)$.","resolution":"resolved","short_name":"AME states, five open cases","solve_date":"2026-08-06","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.05781","year_posed":null,"field_group":"Quantum information & computing","model_maker":"Anthropic, OpenAI","publication":"preprint","result_note":"Five existence statements, all by explicit construction: $\\mathrm{AME}(12,5)$, $\\mathrm{AME}(17,11)$, $\\mathrm{AME}(18,11)$, $\\mathrm{AME}(17,13)$ and $\\mathrm{AME}(18,13)$. The $[12,6,7]_{25}$ code came from a direct search with no symmetry imposed; its automorphism group turned out to have a regular $\\mathbb{Z}_3^2$ coordinate orbit, and imposing that translation symmetry on two nine-coordinate orbits collapses an unrestricted $9 \\times 9$ block to a nine-element kernel, which is what made the length-eighteen searches feasible.\n\nThe symmetry is search scaffolding, not part of the proof: the three printed matrices and the two checks suffice on their own. The paper is explicit that the searches were not exhaustive, so it proves existence and classifies nothing - equivalence and classification for these parameters stay open. The length-twelve code is also shown not to be monomially equivalent to a generalized Reed-Solomon code.","source_name":"Symmetry-guided constructions of AME states in five open cases","significance":12,"verification":"site-confirmed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Individual cells of the AME existence table, not the general existence problem. That problem is genuinely well known in quantum information - perfect tensors underpin holographic codes and quantum secret sharing, and it appears on published open-problem lists - but any one parameter pair is an ordinary table entry. Above the anchor at 10 for a typical numbered Erdos problem, because the table is actively maintained and consulted and its maintainer engaged with this manuscript. Below the named-conjecture band at 15.","verification_note":"This site re-ran the certificate. The paper's logical content is three printed matrices and two checks on each, so verification means doing the checks again - here with field arithmetic built from the printed minimal polynomials and an independent determinant routine, nothing taken from the authors' code.\n\nAll three matrices satisfy $A\\overline{A}^{\\mathsf T} = -I_k$. Every nonempty square minor is nonzero: 923 for the $6 \\times 6$ block and 48,619 for each $9 \\times 9$, 98,161 in total, matching the counts the paper states. The nine convolution equations that the paper says are equivalent to self-duality for the group-circulant blocks hold. And the Schur square of the length-twelve code has dimension 12, the paper's own argument that it is not monomially equivalent to a generalized Reed-Solomon code, since every GRS $[12,6]$ code has Schur-square dimension at most 11. The field conventions were checked first: Frobenius is an involution and norms land in the base field.\n\nWhat this does not settle is whether the five cases were open. The reachable copy of the Huber-Wyderka table (last updated February 2024) covers local dimensions up to 10, and $\\mathrm{AME}(12,5)$ does sit in its unknown region, but it has no $q=11$ or $q=13$ axis, and the URL the paper cites for a newer version is dead. The paper's prior-art comparisons stand unchecked here.","human_collaborators":"[\"Samuel Bevins\",\"Yunus Bidav\"]"},"metadata_root":"sha256:39a041d5a934cc984c0abe303ab1d2266e74b0c47dd75300e59cfb837ad63524","content_root":"sha256:6c05b9e2746d7fa1fa67bc56113ab64233ea287c2d275f0ade2ec06722c71f57","availability":"reference_only","row_root":"sha256:913e4e7b80ccd3fb2862de02081cb9f5b2bfb42dedabd1ab2a9608582ef6686f"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:amenability-base-field-independence","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Whether Amenability of an Algebra Depends on the Ground Field","summary":"Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/amenability-base-field-independence"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.08161"}],"metadata":{"name":"Whether Amenability of an Algebra Depends on the Ground Field","slug":"amenability-base-field-independence","field":"Noncommutative algebra","model":"ChatGPT 5.6 Sol","ai_role":"The abstract states it plainly: \"A significant part of the argument is based on ideas of ChatGPT 5.6 Sol.\" The author goes further in a footnote on the title page: \"While it is currently prohibited by arXiv policy to list AI as a coauthor, the (human) coauthor is confident that ChatGPT's contribution merits an author credit.\"","posed_by":"Yves Cornulier","citations":null,"statement":"Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.","resolution":"resolved","short_name":"Amenability and the base field","solve_date":"2026-08-08","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.08161","year_posed":2015,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":"The footnote is worth reading on its own: an author saying in print that the model earned coauthorship and that policy is what prevents it.","source_name":"arXiv","significance":10,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A MathOverflow question from a well-known mathematician: documented and real, but never a programme with a literature behind it.","verification_note":"A preprint days old, with no independent review.","human_collaborators":"[\"Be'eri Greenfeld\"]"},"metadata_root":"sha256:b1ca81bad8e8c5fe10531ad1c5dc949a2e07fb3ae330a3d39c764cdc04b1df40","content_root":"sha256:4377ee4e7b59f926c7c2ca18f4f9d8cbe7da976310129987035acba0b2ce7f8d","availability":"reference_only","row_root":"sha256:ece1c991d67d73ae256332e4139725cb9872b171c436339b55e3905ba03c5e36"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:anari-charikar-thin-matching","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Thin Matching Problem","summary":"Anari, Charikar and Ramakrishnan asked whether every fractional perfect matching admits a perfect matching that is $\\alpha$-thin with respect to it, meaning it crosses every cut at most $\\alpha$ times the fractional amount. Resolved up to polylogarithmic factors.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/anari-charikar-thin-matching"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2606.01330"}],"metadata":{"name":"The Thin Matching Problem","slug":"anari-charikar-thin-matching","field":"Graph algorithms","model":"GPT-5.5 Pro","ai_role":"The acknowledgement says the authors used GPT-5.5 Pro during the research, and points at the connection to cut-tree sparsification as where it mattered.","posed_by":"Nima Anari, Moses Charikar, Prasanna Ramakrishnan","citations":null,"statement":"Anari, Charikar and Ramakrishnan asked whether every fractional perfect matching admits a perfect matching that is $\\alpha$-thin with respect to it, meaning it crosses every cut at most $\\alpha$ times the fractional amount. Resolved up to polylogarithmic factors.","resolution":"partial","short_name":"Thin matchings","solve_date":"2026-05-31","solve_type":"proved","source_url":"https://arxiv.org/abs/2606.01330","year_posed":2023,"field_group":"Algorithms & optimization","model_maker":"OpenAI","publication":"preprint","result_note":"up to polylogarithmic factors","source_name":"arXiv:2606.01330 - On Thin Perfect Matchings up to Polylogarithmic Factors","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A problem posed at STOC-level in 2023, connected to thin-tree questions and the asymmetric travelling salesman literature.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Alireza Haqi\",\"Shayan Oveis Gharan\"]"},"metadata_root":"sha256:a2e150795495497484137e236a9f633d6954d98aa7b784c5a2bce85fe1704924","content_root":"sha256:54786be8a64732bd0d13bb39adc5856970b29448f1f833661c16e5e5e449dd12","availability":"reference_only","row_root":"sha256:a729529e2db06fca6eb95bd85b7913e602b4c4b0553c7cc859bd26d6e30cb785"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:anchored-gda-last-iterate","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Last-Iterate Rate for Anchored Gradient Descent-Ascent","summary":"For smooth convex-concave min-max problems, can anchored gradient descent-ascent be scheduled so that its exact last-iterate squared-gradient residual is $O(1/t)$, closing the gap left by the 2019 analysis?","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/anchored-gda-last-iterate"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2604.03782"}],"metadata":{"name":"Last-Iterate Rate for Anchored Gradient Descent-Ascent","slug":"anchored-gda-last-iterate","field":"Convex optimization","model":"AlphaProof Nexus","ai_role":"The agent searched for the anchoring schedule and its proof simultaneously, discovering a parameter choice yielding the stronger guarantee via a discrete-time recurrence argument rather than the usual continuous-time ODE analysis.","posed_by":null,"citations":null,"statement":"For smooth convex-concave min-max problems, can anchored gradient descent-ascent be scheduled so that its exact last-iterate squared-gradient residual is $O(1/t)$, closing the gap left by the 2019 analysis?","resolution":"resolved","short_name":"Anchored GDA rate","solve_date":"2026-04-04","solve_type":"proved","source_url":"https://arxiv.org/abs/2604.03782","year_posed":2019,"field_group":"Algorithms & optimization","model_maker":"Google DeepMind","publication":"preprint","result_note":null,"source_name":"arXiv:2604.03782 - An improved last-iterate convergence rate for anchored gradient descent ascent","significance":10,"verification":"lean-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A specialist rate question in min-max optimization.","verification_note":"Lean-checked; accompanying arXiv preprint by the DeepMind team.","human_collaborators":"[]"},"metadata_root":"sha256:67eb5879bab09a1c10e60cd7ba5333276f87ba152445bb718f2112cf49c1255d","content_root":"sha256:4633499607164173733e18c91a953cd985f23788c51f06af5f8bb4e4984ca658","availability":"reference_only","row_root":"sha256:1dd6275fb1d771cda1769c0b87fbb5d129a6cbaa5ceeb73121a9e3a0971783dd"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:anderson-quasi-completeness","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Anderson's Quasi-Completeness Question","summary":"Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring $A = k^p[[X, Y]][k]$ with $k = \\mathbb{F}_p(u_1, u_2, \\dots)$ is weakly quasi-complete but not quasi-complete.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/anderson-quasi-completeness"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2604.03789"}],"metadata":{"name":"Anderson's Quasi-Completeness Question","slug":"anderson-quasi-completeness","field":"Commutative algebra","model":"Rethlas + Archon (GPT-5.4 Pro)","ai_role":"The dual-agent framework (Rethlas for informal reasoning, Archon for formal verification) ran roughly 80 hours; the decisive example is a classical ring going back to Nagata, which the system recognized as answering Anderson's question.","posed_by":"D. D. Anderson","citations":null,"statement":"Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring $A = k^p[[X, Y]][k]$ with $k = \\mathbb{F}_p(u_1, u_2, \\dots)$ is weakly quasi-complete but not quasi-complete.","resolution":"resolved","short_name":"Anderson quasi-complete","solve_date":"2026-04-04","solve_type":"disproved","source_url":"https://arxiv.org/abs/2604.03789","year_posed":2014,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2604.03789 - Automated conjecture resolution with formal verification","significance":10,"verification":"lean-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A named but specialist question in commutative ring theory.","verification_note":"Lean-checked with a statement comparator guarding against misformalization; arXiv preprint documents the pipeline.","human_collaborators":"[]"},"metadata_root":"sha256:0376280f898e4f5d6ead172aa3c273ae943982bd01e440a2aa9f3ca10ccc3c13","content_root":"sha256:c11a95092cac87734954eb7848061a32eb4b14fac0a28c6cf07ca4b9fa776006","availability":"reference_only","row_root":"sha256:746384812387460e5f6e4c140eca84552802c18158e3e97ce76430e38db96aed"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:anstee-sali-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Anstee-Sali Conjecture on Forbidden Configurations","summary":"For a forbidden configuration F, the Anstee-Sali conjecture predicts that forb(m, F) is Theta(m^(X(F)-1)), where X(F) comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core joined to the edges of a 4-cycle has X(F) = 4, so the conjecture predicts Theta(m^3), while a random-alteration argument gives Omega(m^(10/3)).","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/anstee-sali-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.07646"}],"metadata":{"name":"The Anstee-Sali Conjecture on Forbidden Configurations","slug":"anstee-sali-conjecture","field":"Extremal set theory","model":"GPT-5.6 Sol","ai_role":"The abstract ends \"The example was found by GPT-5.6 Sol\", and a dedicated disclosure section repeats it: \"The authors used GPT-5.6 Sol for finding the example. The authors reviewed and revised all outputs, verified results, and take full responsibility for the final manuscript.\"","posed_by":"Richard Anstee, Attila Sali","citations":null,"statement":"For a forbidden configuration F, the Anstee-Sali conjecture predicts that forb(m, F) is Theta(m^(X(F)-1)), where X(F) comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core joined to the edges of a 4-cycle has X(F) = 4, so the conjecture predicts Theta(m^3), while a random-alteration argument gives Omega(m^(10/3)).","resolution":"resolved","short_name":"Anstee-Sali conjecture","solve_date":"2026-08-07","solve_type":"disproved","source_url":"https://arxiv.org/abs/2608.07646","year_posed":2005,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"The central conjecture of a named research programme with its own long-running dynamic survey, well known inside extremal set theory and invisible outside it.","verification_note":"A preprint days old, with no independent review.","human_collaborators":"[\"Pei Wu\"]"},"metadata_root":"sha256:f9c16262164df0f4dec7fa6c093ac2dce0efd18765f0ec923137dbd28537eede","content_root":"sha256:b54a19f6ac673c005d7399350a1483abb56a57703bbfc381e7e0ec6d8ddeea7d","availability":"reference_only","row_root":"sha256:ccb7b9f9995f2f0b636b4e30a630dcba98050a9dea9834dc06dec5a93ae845ad"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:araujo-piga-schacht-tight-hamilton","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Araujo-Piga-Schacht Question on Tight Hamilton Cycles","summary":"Araujo, Piga and Schacht asked whether density and codegree both above $1/4$ force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is $p_0 = \\max_{0 \\le x \\le 1}\\min\\{x^3, 1-x\\} \\approx 0.3177$, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every $p > 1/3$ the asymptotically sharp minimum-codegree threshold is determined.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/araujo-piga-schacht-tight-hamilton"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.21568"}],"metadata":{"name":"Araujo-Piga-Schacht Question on Tight Hamilton Cycles","slug":"araujo-piga-schacht-tight-hamilton","field":"Hypergraph theory","model":"ChatGPT","ai_role":"The author acknowledges using ChatGPT in the early stage of the project and during manuscript preparation, and states specifically that an initial idea leading to the first construction in the paper arose during an interaction with it.","posed_by":"Araujo, Piga, Schacht","citations":null,"statement":"Araujo, Piga and Schacht asked whether density and codegree both above $1/4$ force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is $p_0 = \\max_{0 \\le x \\le 1}\\min\\{x^3, 1-x\\} \\approx 0.3177$, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every $p > 1/3$ the asymptotically sharp minimum-codegree threshold is determined.","resolution":"resolved","short_name":"Tight Hamilton cycles","solve_date":"2026-07-23","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.21568","year_posed":null,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":"the question is answered negatively and the correct threshold is determined","source_name":"arXiv:2607.21568 - Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A named question in quasirandom hypergraph Hamiltonicity, a well-worked corner of extremal combinatorics.","verification_note":"Single-author arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Xichao Shu\"]"},"metadata_root":"sha256:97d739c66e6fc5666861f06007cf4338add0e0659de4531e1e14e50139663b26","content_root":"sha256:7dd72f0bf967e037fa66a607324400ec76c64f8a4f930ac2e9e8902780f62574","availability":"reference_only","row_root":"sha256:ca7139b6d979b28f0a90e3015a001b71a15c789f0f6895e57264e67fcc072c90"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:arithmetic-kakeya-bounded-slopes","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Sum-Difference Exponents for Boundedly Many Slopes","summary":"The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate controlled by a new notion of rational complexity - mapping where the conjectured route cannot succeed.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/arithmetic-kakeya-bounded-slopes"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2511.15135"}],"metadata":{"name":"Sum-Difference Exponents for Boundedly Many Slopes","slug":"arithmetic-kakeya-bounded-slopes","field":"Arithmetic combinatorics","model":"AlphaEvolve","ai_role":"\"Inspired by numerical explorations from the tool AlphaEvolve\" - the tool's experiments pointed at the bounded-slope regime and its convergence behaviour; the theorems are Tao's.","posed_by":"Nets Katz, Terence Tao (arithmetic Kakeya program)","citations":null,"statement":"The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate controlled by a new notion of rational complexity - mapping where the conjectured route cannot succeed.","resolution":"partial","short_name":"Arithmetic Kakeya, bounded slopes","solve_date":"2025-11-19","solve_type":"proved","source_url":"https://arxiv.org/abs/2511.15135","year_posed":2002,"field_group":"Analysis","model_maker":"Google DeepMind","publication":"preprint","result_note":"Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.","source_name":"arXiv","significance":22,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"The arithmetic Kakeya route (Katz-Tao 2002) is a known pathway to the full Kakeya conjecture, one of harmonic analysis's central open problems.","verification_note":null,"human_collaborators":"[\"Terence Tao\"]"},"metadata_root":"sha256:b8bd87001c94667b3b4d45226dc20f558ceb6ae86aa55098347bcf97c961ed89","content_root":"sha256:0319198abb9f04fdfd4480d3e5e43d608ab9692dd7bffaa85a93264209e4d006","availability":"reference_only","row_root":"sha256:d2f34c224edb20a6c2dfb94f8a9b008d882d1f4f3b56e82a8b4b1b716590c2cf"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:asymptotically-attaining-the-moore-bound","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Asymptotically attaining the Moore bound","summary":"For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. It is shown that\n$$\\lim_{d \\to \\infty}\\frac{n_k(d)}{d^k} = 1$$\nfor every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter. \n\nAlso proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/asymptotically-attaining-the-moore-bound"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.03965"},{"locator_id":"native-3","kind":"artifact","url":"https://github.com/woutercvb/wewantmoore"},{"locator_id":"native-4","kind":"artifact","url":"https://en.wikipedia.org/wiki/Degree_diameter_problem"}],"metadata":{"name":"Asymptotically attaining the Moore bound","slug":"asymptotically-attaining-the-moore-bound","field":null,"model":"GPT-5.6","ai_role":"The paper's tool disclosure states that GPT-5.6 Pro, used in exploratory brainstorming directed by the authors, suggested splitting complete flags into their odd- and even-rank subflags. That suggestion arose in connection with the edge problem but became the halved-flag construction carrying Theorem 1.1 itself, the graph being named for it. The authors developed it, formulated and verified every argument, and take full responsibility. Generative AI also assisted the Lean 4 formalization.","posed_by":"Béla Bollobás","citations":null,"statement":"For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. It is shown that\n$$\\lim_{d \\to \\infty}\\frac{n_k(d)}{d^k} = 1$$\nfor every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter. \n\nAlso proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.","resolution":"resolved","short_name":"Degree-Diameter Problem","solve_date":"2026-08-04","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.03965","year_posed":1978,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":"Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.","source_name":"Asymptotically attaining the Moore bound","significance":38,"verification":"lean-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"The degree-diameter problem carries its own Electronic Journal of Combinatorics dynamic survey (DS14, Miller and Siran), and the asymptotic form is Bollobas's own conjecture, recorded in Extremal Graph Theory and Random Graphs. It stood 48 years with the asymptotic known only for k in {2, 3, 5}, through generalized polygons, and the best uniform coefficient for large k was 0.629.","verification_note":"Lean 4 formalization at github.com/woutercvb/wewantmoore, checked by the site on 2026-08-06 at commit 32beb227. `DegreeDiameter.theorem_1_1` states Theorem 1.1 itself, as a limit of nKD k d / d^k, and `corollary_1_2` states Corollary 1.2; neither is a weakened lemma, and a second independent route is proved alongside each under `_via_big_cell`. No sorry or admit appears in the sources, and the committed axiom audit shows both final theorems resting only on propext, Classical.choice and Quot.sound. That audit is not taken on trust: the repository's CI builds the project from the pinned toolchain and manifest, regenerates the axiom and dependency reports, and fails if they differ from the committed ones. It passes on this commit. The formalization was itself AI-assisted, per the paper's disclosure, and the authors note it is not a line-by-line transcription: k = 1 is handled by the same construction rather than by complete graphs, and the order and cap asymptotics go through leading terms rather than the displayed O_k(q^-1) estimates.","human_collaborators":"[\"Wouter Cames van Batenburg\",\"Samuel Korsky\"]"},"metadata_root":"sha256:67b0a566cfd2c383dbf075da047e70c82234b78e5bb605f552c2e8905ad85748","content_root":"sha256:d24576b386c77e4a74ff3daee39007aefbdcd6f818876f6cae5ea6b55fca15cf","availability":"reference_only","row_root":"sha256:a3da80e14cd0571486ad41fc008798a3d88b25ee46cdaeaa03473bf7f0c6cce3"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:atom-centered-descriptor-completeness","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Completeness of Fixed-Order Atom-Centered Descriptors","summary":"Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since was dissolved by going to a higher correlation order, leaving open whether the trispectrum (5-body correlations), or any fixed finite order, separates all noncongruent environments. It does not. There are noncongruent three-dimensional environments agreeing on every cluster of up to seven neighbours, and for each finite correlation order and angular cutoff there are continuous families of noncongruent environments with identical retained features.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/atom-centered-descriptor-completeness"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.26984"},{"locator_id":"native-3","kind":"artifact","url":"https://arxiv.org/abs/2001.11696"}],"metadata":{"name":"Completeness of Fixed-Order Atom-Centered Descriptors","slug":"atom-centered-descriptor-completeness","field":"Invariants of point configurations","model":"Claude Opus 4.8 and Fable, Codex 5.5 and Sol 5.6","ai_role":"The paper is as much about the usage pattern as the result, and says so in its title. The authors had been stuck on these questions for years and could not get past the known Pozdnyakov examples. Coding agents built on Claude (Opus 4.8 and Fable) and Codex (5.5 and Sol 5.6) were given a summary of the field's literature and, across unstructured iteration, produced the degenerate configurations by locating results in unrelated communities and recognising what they implied here, among them the homometric structures studied in musical rhythm theory. The authors verified the constructions manually and with model-generated code, and lifted the cyclic degeneracies to three dimensions themselves. They also ran a reproducibility experiment, eight runs per model under each of two prompt conditions, to test how reliably a query of this kind lands.","posed_by":"Sergey N. Pozdnyakov, Michael J. Willatt, Albert P. Bartók, Christoph Ortner, Gábor Csányi, Michele Ceriotti","citations":null,"statement":"Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since was dissolved by going to a higher correlation order, leaving open whether the trispectrum (5-body correlations), or any fixed finite order, separates all noncongruent environments. It does not. There are noncongruent three-dimensional environments agreeing on every cluster of up to seven neighbours, and for each finite correlation order and angular cutoff there are continuous families of noncongruent environments with identical retained features.","resolution":"resolved","short_name":"Fixed-order descriptor completeness","solve_date":"2026-07-29","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.26984","year_posed":2020,"field_group":"Geometry & topology","model_maker":null,"publication":"preprint","result_note":"The key ingredients were already in the literature, decades old and in distant fields; what was missing was anyone connecting them to this question.","source_name":"arXiv:2607.26984 - Using large language models to probe the limits of atom-centered structural descriptors","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A six-year-old completeness question from the atomistic machine-learning literature, settled negatively at every fixed order. The authors themselves note the degenerate structures are far from plausible chemistry and largely reinforce a view the field had already reached.","verification_note":"arXiv preprint, not peer-reviewed. The validation scripts and input structures are promised only on publication, so the constructions cannot yet be re-run from the public record; the authors state they checked them by hand and with model-generated code. The counterexamples are explicit point configurations, so they are checkable once released.","human_collaborators":"[\"Michelangelo Domina\",\"Michele Ceriotti\"]"},"metadata_root":"sha256:3212f5ea21a2983109604303558ff2ba5bd2b2adcecde8045c2460f7c5b88471","content_root":"sha256:a7db8becfc7711f1cf7db1a2e2acd40a7a8c8d96fa3471895a91291cdbe553ac","availability":"reference_only","row_root":"sha256:f5ef75d6775b59e1a4350af60db7cf4584508485f6b2ef134d433d6cfc9ae3de"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:auslander-reiten-smalo-perfect-fields","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Infinitely Many Components in Auslander–Reiten Quivers over Perfect Fields","summary":"For a perfect field $k$ and a representation-infinite finite-dimensional $k$-algebra $A$, the Auslander–Reiten quiver of $A$ has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for finite-dimensional algebras over perfect fields.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/auslander-reiten-smalo-perfect-fields"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.24466"}],"metadata":{"name":"Infinitely Many Components in Auslander–Reiten Quivers over Perfect Fields","slug":"auslander-reiten-smalo-perfect-fields","field":"Representation Theory of Algebras","model":"ChatGPT","ai_role":"The authors credit ChatGPT with the technical construction and verification of the semilinear twist argument in Lemma 4, and say it was used more substantially in extending the result from algebraically closed fields to arbitrary perfect fields - the step that gives the paper its stated generality.","posed_by":"Auslander, Reiten and Smalø","citations":null,"statement":"For a perfect field $k$ and a representation-infinite finite-dimensional $k$-algebra $A$, the Auslander–Reiten quiver of $A$ has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for finite-dimensional algebras over perfect fields.","resolution":"resolved","short_name":"Auslander–Reiten–Smalø","solve_date":"2026-07-27","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.24466","year_posed":null,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A named conjecture of Auslander, Reiten and Smalø, established over perfect fields rather than only in special cases.","verification_note":"No independent review. The disclosure names the specific lemma and the extension the model contributed. Preprint, not refereed.","human_collaborators":"[\"Wen Chang\",\"Quanyu Tang\"]"},"metadata_root":"sha256:60f1ffaeaeef0001d10e11fa8e2dcdee4197a5a7247e7124f4ca266d37be61fb","content_root":"sha256:6962db438fba85fe567e1ed520c097b13fa93ccac963251217f53bd4fffcf29d","availability":"reference_only","row_root":"sha256:f6a8019ee0a2683f0c3338b05310253c1f898df7ce53a03073b53d30f0ce6d7d"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:autonomous-lipschitz-fast-dynamo-on-the-three-torus","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Autonomous Lipschitz Fast Dynamo on the Three-Torus","summary":"Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field $u\\in W^{1,\\infty}(\\mathbb T^3;\\mathbb R^3)$, chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat three-torus? The author constructs such a field and constants $\\varepsilon_0,\\gamma_0>0$ such that, for every $0<\\varepsilon\\le\\varepsilon_0$, the induction operator has an eigenvalue $\\lambda_\\varepsilon$ with $\\operatorname{Re}\\lambda_\\varepsilon\\ge\\gamma_0$. Thus every sufficiently small diffusivity admits a nonzero real divergence-free magnetic field with exact exponential $L^2$ growth. The velocity is Lipschitz but not $C^1$, so this settles only the Lipschitz regularity variant; Arnold's smooth autonomous fast-dynamo problem on $\\mathbb T^3$ remains open.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/autonomous-lipschitz-fast-dynamo-on-the-three-torus"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.02586"}],"metadata":{"name":"Autonomous Lipschitz Fast Dynamo on the Three-Torus","slug":"autonomous-lipschitz-fast-dynamo-on-the-three-torus","field":"Dynamo theory; spectral PDE","model":"GPT-5.5 Pro, GPT-5.6 Sol","ai_role":"During exploration, GPT-5.5 Pro and GPT-5.6 Sol helped identify a candidate fast-dynamo construction. They were later used to check calculations; identify errors, inconsistencies, and gaps in preliminary arguments; support development of some arguments; and assist with drafting and revision. The author states that he critically reviewed and verified every mathematical claim, calculation, and AI-generated suggestion and takes full responsibility for the manuscript.","posed_by":"V. I. Arnold","citations":null,"statement":"Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field $u\\in W^{1,\\infty}(\\mathbb T^3;\\mathbb R^3)$, chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat three-torus? The author constructs such a field and constants $\\varepsilon_0,\\gamma_0>0$ such that, for every $0<\\varepsilon\\le\\varepsilon_0$, the induction operator has an eigenvalue $\\lambda_\\varepsilon$ with $\\operatorname{Re}\\lambda_\\varepsilon\\ge\\gamma_0$. Thus every sufficiently small diffusivity admits a nonzero real divergence-free magnetic field with exact exponential $L^2$ growth. The velocity is Lipschitz but not $C^1$, so this settles only the Lipschitz regularity variant; Arnold's smooth autonomous fast-dynamo problem on $\\mathbb T^3$ remains open.","resolution":"variant","short_name":"Lipschitz fast dynamo","solve_date":"2026-08-03","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.02586","year_posed":1994,"field_group":"Mathematical physics","model_maker":"OpenAI","publication":"preprint","result_note":"Lipschitz velocity; the smooth autonomous fast-dynamo conjecture on T^3 remains open.","source_name":"arXiv:2608.02586 - An autonomous Lipschitz fast dynamo on the three-torus","significance":30,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Arnold's fast dynamo problem has organized mathematical MHD for decades, with a Springer monograph and a sustained literature, while staying invisible outside that community.","verification_note":"Single-author arXiv preprint; not yet peer-reviewed, Lean-verified, or independently expert-checked.","human_collaborators":"[\"Lukas Niebel\"]"},"metadata_root":"sha256:1520674f2dfb1b4212c098d7d4ac208e0e20b28e90729e139097dc1f637ea443","content_root":"sha256:b204c2a825fa739ee99f096c8749e65c75f6f897a4731fa9a0a710707e7278fc","availability":"reference_only","row_root":"sha256:d9ab737140b562b94022f176ce152ca8e9e6c1db051c516b6a5953df1597f787"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:averkov-hofscheier-nill-h-star-real-rootedness","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Real-Rootedness of Ehrhart h*-Polynomials at Large Width","summary":"A question of Averkov, Hofscheier and Nill on whether the Ehrhart $h^*$-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and unimodality of the $h^*$-vector, with the analogous statement for the local $h^*$-polynomial of a lattice simplex.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/averkov-hofscheier-nill-h-star-real-rootedness"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.03635"}],"metadata":{"name":"Real-Rootedness of Ehrhart h*-Polynomials at Large Width","slug":"averkov-hofscheier-nill-h-star-real-rootedness","field":"Polyhedral combinatorics","model":"ChatGPT 5.6 Sol","ai_role":"The acknowledgments state that the proofs were found using ChatGPT 5.6 Sol, which also produced a first draft of the paper, with the author solely responsible for the final version.","posed_by":"Gennadiy Averkov, Johannes Hofscheier, Benjamin Nill","citations":null,"statement":"A question of Averkov, Hofscheier and Nill on whether the Ehrhart $h^*$-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and unimodality of the $h^*$-vector, with the analogous statement for the local $h^*$-polynomial of a lattice simplex.","resolution":"resolved","short_name":"h* real-rootedness","solve_date":"2026-08-04","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.03635","year_posed":null,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2608.03635 - Lattice polytopes of large width have real-rooted Ehrhart h*-polynomials","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A stated question in Ehrhart theory, where unimodality of the h*-vector has been a recurring target; the answer follows from a result of Basu and Oertel once the right reduction is seen.","verification_note":"arXiv preprint, not yet peer-reviewed.","human_collaborators":"[\"Benjamin Nill\"]"},"metadata_root":"sha256:e54ae889fe1afec4c0eea49bca076c15e0842b822772b04378342736479fc238","content_root":"sha256:3ff7addda45c3149952ec342be550d50356239aadb3a240150b9f4ed66f28ce5","availability":"reference_only","row_root":"sha256:0b88090616add49fff9504d5b62b25084e76981f4be123a38abe82768b9f88e3"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:avidor-zwick-max-cut-sdp","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Avidor-Zwick Question on Low-Dimensional Max-Cut SDP","summary":"For fixed $d$, can every $d$-dimensional feasible solution of the triangle-strengthened Max-Cut SDP be rounded in polynomial time with ratio strictly larger than $\\alpha_{GW}$? A rounding achieving $\\alpha_{GW} + 2^{-O(d)}$ answers yes.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/avidor-zwick-max-cut-sdp"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2604.13971"}],"metadata":{"name":"Avidor-Zwick Question on Low-Dimensional Max-Cut SDP","slug":"avidor-zwick-max-cut-sdp","field":"Approximation algorithms","model":"Gemini (internal), ChatGPT-5.2 Extended Pro, Gemini 3.0 Pro DeepThink","ai_role":"The key anti-concentration lemma for signs of low-dimensional Gaussian projections was first proved by Google's internal Gemini model with a weaker bound; the optimal $2^{-\\Theta(d)}$ form was then obtained with ChatGPT-5.2 Extended Pro and Gemini 3.0 Pro DeepThink, with proofs edited by the authors.","posed_by":"Adi Avidor & Uri Zwick","citations":null,"statement":"For fixed $d$, can every $d$-dimensional feasible solution of the triangle-strengthened Max-Cut SDP be rounded in polynomial time with ratio strictly larger than $\\alpha_{GW}$? A rounding achieving $\\alpha_{GW} + 2^{-O(d)}$ answers yes.","resolution":"resolved","short_name":"Avidor-Zwick Max-Cut","solve_date":"2026-04-16","solve_type":"proved","source_url":"https://arxiv.org/abs/2604.13971","year_posed":2005,"field_group":"Theoretical computer science","model_maker":"Google DeepMind / OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2604.13971 - Max Cut with small-dimensional SDP solutions","significance":10,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A specialist question from the Max-Cut SDP literature.","verification_note":"Author-edited and checked arXiv preprint. Not yet peer-reviewed.","human_collaborators":"[]"},"metadata_root":"sha256:9ab305712a7eb9db01f727f78786ac798f4c331c500addaa3ef14c967feacb7e","content_root":"sha256:36e03d4026660401e3e39badf0045e1454ac3629e7331d6d6bed6707474ec974","availability":"reference_only","row_root":"sha256:cedf58c89d33bacc038e4f2aa6901efcf64d5974975e9db48ff784964bd14a0b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:axiotis-sviridenko-condition-number-barrier","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Axiotis-Sviridenko Condition-Number Conjecture","summary":"Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. Their conjectured lower bound is established for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer and Tulsiani.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/axiotis-sviridenko-condition-number-barrier"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.02588"}],"metadata":{"name":"The Axiotis-Sviridenko Condition-Number Conjecture","slug":"axiotis-sviridenko-condition-number-barrier","field":"Approximation algorithms","model":"Gemini-based agentic system (internal)","ai_role":"The acknowledgements state that the proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google, with the authors verifying it and editing for presentation.","posed_by":"Kyriakos Axiotis, Maxim Sviridenko","citations":null,"statement":"Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. Their conjectured lower bound is established for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer and Tulsiani.","resolution":"partial","short_name":"Condition-number barrier","solve_date":"2026-08-03","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.02588","year_posed":2021,"field_group":"Algorithms & optimization","model_maker":"Google","publication":"preprint","result_note":"Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.","source_name":"arXiv:2608.02588 - The Condition-Number Barrier in Sparse Least Squares","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A 2021 conjecture on the price of sparsity in least squares, sitting in the hardness-of-approximation literature that grew from the Small-Set Expansion Hypothesis.","verification_note":"arXiv preprint, not yet peer-reviewed.","human_collaborators":"[\"Honghao Lin\",\"Vahab Mirrokni\",\"David P. Woodruff\"]"},"metadata_root":"sha256:9af414ee4cfbc4a892d4ae3d165c648cf77e05da23264b73edce3075a30c7369","content_root":"sha256:42ab129fb201c8b018a2558147be125ecbbb71ca7d93c102aeb71e02f1a43079","availability":"reference_only","row_root":"sha256:9ce5b3e03b9d9f86dd9bdba60eb4b691a0de30cce3105deee7658671760996d5"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:babai-frankl-oddtown-composite","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Babai and Frankl's Oddtown Question for Composite Moduli","summary":"An $\\ell$-Oddtown is a family of subsets of an $n$-element set whose set sizes are not divisible by $\\ell$ while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is $n$ for prime $\\ell$, Babai and Frankl extended this to prime powers and asked whether $n$ still holds for other moduli, a question open even for $\\ell = 6$. Bukh, Chao and Zheng answer it negatively with an explicit superlinear construction, complemented by new upper bounds.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/babai-frankl-oddtown-composite"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2509.00586"}],"metadata":{"name":"Babai and Frankl's Oddtown Question for Composite Moduli","slug":"babai-frankl-oddtown-composite","field":"Extremal set theory","model":"GPT-5.6 Sol","ai_role":"The lower-bound construction in Section 2 was first proposed by GPT-5.6 Sol in response to prompts from the authors; the upper-bound results were obtained without AI assistance. (The disclosure was added in the paper's second version.)","posed_by":"László Babai, Péter Frankl","citations":null,"statement":"An $\\ell$-Oddtown is a family of subsets of an $n$-element set whose set sizes are not divisible by $\\ell$ while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is $n$ for prime $\\ell$, Babai and Frankl extended this to prime powers and asked whether $n$ still holds for other moduli, a question open even for $\\ell = 6$. Bukh, Chao and Zheng answer it negatively with an explicit superlinear construction, complemented by new upper bounds.","resolution":"resolved","short_name":"Oddtown mod composite","solve_date":"2025-08-30","solve_type":"disproved","source_url":"https://arxiv.org/abs/2509.00586","year_posed":1992,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A named question from Babai and Frankl's widely used linear-algebra-methods text, open for decades and known across extremal combinatorics.","verification_note":null,"human_collaborators":"[\"Boris Bukh\",\"Ting-Wei Chao\",\"Zeyu Zheng\"]"},"metadata_root":"sha256:5c16d57522e98374b541f221b31023393bdae0d0b3a8b3e929868ad0b4d7ca25","content_root":"sha256:02efab6699ab3e203a55180a0bf4d9f8a6f85e2732e9bf009d17401a14c7bfe0","availability":"reference_only","row_root":"sha256:e23b5584e56194427747e601e819715dcfd4f162e9d3552538fb4b233c3f37b7"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:babai-minimal-cayley-chromatic","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Babai's Minimal Cayley Graph Problem","summary":"A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily large chromatic number.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/babai-minimal-cayley-chromatic"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.06254"}],"metadata":{"name":"Babai's Minimal Cayley Graph Problem","slug":"babai-minimal-cayley-chromatic","field":"Algebraic graph theory","model":"ChatGPT 5.6 Sol","ai_role":"The paper's statement of AI use: \"An initial proof was found by ChatGPT 5.6 Sol given [DHY24] in the input. Substantial parts of Section 2 originate from an early draft created in interaction with ChatGPT 5.6 Sol, which was subsequently edited and improved by the authors.\" The model found the first proof, given one of the authors' own earlier papers as context.","posed_by":"László Babai","citations":null,"statement":"A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily large chromatic number.","resolution":"resolved","short_name":"Babai minimal Cayley","solve_date":"2026-08-06","solve_type":"disproved","source_url":"https://arxiv.org/abs/2608.06254","year_posed":1978,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":30,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A question of Babai's standing open since 1978, in the algebraic graph theory his name anchors, with recent partial results narrowing it just before it fell.","verification_note":"A preprint days old, with no independent review.","human_collaborators":"[\"James Davies\",\"Meike Hatzel\",\"Liana Yepremyan\"]"},"metadata_root":"sha256:63ff43bde3daedaabd67a5fc117901b94d06c70609605b9356da57dd2e54585f","content_root":"sha256:18e1ba40bfa0ca914d5f4a092476f2b8db1b6c4df3719d37d0fc705e54871921","availability":"reference_only","row_root":"sha256:0e3711b13f01ca11f38fb3198b298323ee1f9b338551d20cac73965bd28df316"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:baker-anti-bertini-hyperplane","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Baker's Question on Smooth Hyperplane Sections over Finite Fields","summary":"Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must acquire a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted no for every positive-dimensional variety, and that prediction is correct.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/baker-anti-bertini-hyperplane"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2606.23513"}],"metadata":{"name":"Baker's Question on Smooth Hyperplane Sections over Finite Fields","slug":"baker-anti-bertini-hyperplane","field":"Algebraic geometry","model":"GPT-based models, Rethlas","ai_role":"The acknowledgement says the author used GPT-based large language models to generate and compare possible proof strategies, some of which was incorporated into the final manuscript, and used the Rethlas agent as an auxiliary proof-checker.","posed_by":"Matthew Baker; prediction by Bjorn Poonen","citations":null,"statement":"Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must acquire a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted no for every positive-dimensional variety, and that prediction is correct.","resolution":"resolved","short_name":"Anti-Bertini embeddings","solve_date":"2026-06-22","solve_type":"disproved","source_url":"https://arxiv.org/abs/2606.23513","year_posed":null,"field_group":"Algebra","model_maker":"OpenAI / Frenzy Math","publication":"preprint","result_note":null,"source_name":"arXiv:2606.23513 - Hyperplane anti-Bertini embeddings over finite fields","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A question of Baker recorded by Poonen in the Bertini-over-finite-fields literature, with a stated prediction that this confirms.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Yutong Zhang\",\"Yaoran Yang\"]"},"metadata_root":"sha256:e38c2766003a7fab3149527511917047c337c4e8d14b820b48abec5b1f120fcc","content_root":"sha256:b2687d94c56172132a3731866a5bae1cc2d057ff33352343b5887d15b3edc1f4","availability":"reference_only","row_root":"sha256:d2ccb749b284d71fb51a83ea2509a5df475a30259fc2281afdcdae0e802aa62b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:ballantine-beck-feigon-maurischat-subsum","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials","summary":"Ballantine, Beck, Feigon and Maurischat introduced the subsum polynomial $\\mathrm{sp}(\\lambda,x) := \\prod_i (1+x^{\\lambda_i})$ attached to an integer partition $\\lambda$, studied rational functions built by summing reciprocals of these polynomials over natural classes of partitions, and posed ten conjectures. Six are now proved: the ordinary and binary coprimality and divisibility conjectures, and the odd and ternary special-value and recurrence conjectures.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/ballantine-beck-feigon-maurischat-subsum"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.21718"}],"metadata":{"name":"The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials","slug":"ballantine-beck-feigon-maurischat-subsum","field":"Partitions and q-series","model":"AxiomProver","ai_role":"AxiomProver autonomously produced Lean and mathlib formalizations and machine-checkable proofs of all six conjectures. It also discovered a counterexample to one of the conjectures as printed, so the same system both proved and refuted statements drawn from one list, which is a useful demonstration that it was reading the statements rather than pattern-matching toward the expected answer.","posed_by":"Ballantine, Beck, Feigon and Maurischat","citations":null,"statement":"Ballantine, Beck, Feigon and Maurischat introduced the subsum polynomial $\\mathrm{sp}(\\lambda,x) := \\prod_i (1+x^{\\lambda_i})$ attached to an integer partition $\\lambda$, studied rational functions built by summing reciprocals of these polynomials over natural classes of partitions, and posed ten conjectures. Six are now proved: the ordinary and binary coprimality and divisibility conjectures, and the odd and ternary special-value and recurrence conjectures.","resolution":"partial","short_name":"Subsum polynomial conjectures","solve_date":"2026-05-20","solve_type":"proved","source_url":"https://arxiv.org/abs/2605.21718","year_posed":null,"field_group":"Number theory","model_maker":null,"publication":"preprint","result_note":"six of the ten conjectures proved; one was found false as printed and its corrected form remains open","source_name":"arXiv:2605.21718 - Reciprocals of Partition Polynomials","significance":15,"verification":"lean-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Six conjectures from one recent paper on subsum polynomials: real, published, specialist, with no accumulated literature beyond the posing paper.","verification_note":"The six proofs are Lean and mathlib formalizations, so they are machine-checkable rather than dependent on refereeing. Note that the catalog has not compiled the artifact itself, so this records the authors' claim of formalization, not an independent build.","human_collaborators":"[\"Evan Chen\",\"Ken Ono\",\"Jujian Zhang\"]"},"metadata_root":"sha256:acebe80cf0d9a1f05b8d94ee964e4dc4000aa7373f20eda543da6766501dffba","content_root":"sha256:2db2dbbdb5c65cc142c9a510b771020a3924331d613d4c851dd7bf10e15d0c38","availability":"reference_only","row_root":"sha256:15dd7d3fdcc7e36fddf95aea759c7fec2fc0d5c978674146c7ddb207a3c7eac9"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:banach-s-isometric-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Banach's isometric conjecture","summary":"Banach asked in 1932 whether a real Banach space X whose ndimensional subspaces, for some fixed 1 < n < dim X, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov’s even-dimensional result, this completes Banach’s isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/banach-s-isometric-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.13536"}],"metadata":{"name":"Banach's isometric conjecture","slug":"banach-s-isometric-conjecture","field":"Functional analysis","model":"ChatGPT 5.6 Pro, ChatGPT 5.5 Pro","ai_role":"The authors state that they had already reduced the main problem to proving Theorem 3.10 before using generative AI. An approach to that theorem then emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro. GPT-5.6 Sol generated the initial draft of Section 3 and corresponding material in Section 2 following this approach; the authors subsequently checked and rewrote it. GPT-5.6 Sol was also used to improve the exposition.","posed_by":"Stefan Banach","citations":null,"statement":"Banach asked in 1932 whether a real Banach space X whose ndimensional subspaces, for some fixed 1 < n < dim X, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov’s even-dimensional result, this completes Banach’s isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.","resolution":"resolved","short_name":"Banach isometric conjecture","solve_date":"2026-08-13","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.13536","year_posed":1932,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":"The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.","source_name":"arXiv","significance":40,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Posed by Banach himself in 1932, open 94 years, with Gromov's even-dimensional theorem (1967) the landmark partial result and a literature of odd-dimensional cases since. A famous named conjecture recognisable well outside convex geometry - placed with Sendov at 40, below the household band.","verification_note":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.13536, Lu-Yang, 21 pages). The AI declaration is verbatim as this entry quotes it: the authors reduced the problem to one theorem before using AI, the approach to that theorem emerged through interactions with ChatGPT 5.5 Pro and 5.6 Pro, and GPT-5.6 Sol drafted Section 3 and parts of Section 2, subsequently checked and rewritten by the authors. The mathematics - bundle topology plus Brouwer degree - was not checked here and needs a geometer. One-day-old preprint, no independent review.","human_collaborators":"[\"Xinbao Lu\",\"Kaiwen Yang\"]"},"metadata_root":"sha256:0cb4c0af2025dd90261ba2b32feba0a0cc239eec022b41d525e012d14cbd988c","content_root":"sha256:8e92f78ae5aceeb1cb6d4191542762488bdf5f64c422c50b629a76f146165925","availability":"reference_only","row_root":"sha256:26f1001da661e6de0f3794e1817f62ace55dc60deba06dbd4290e6d65cf0d367"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bandelt-dress-quartet-distance","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Bandelt-Dress Quartet Distance Conjecture","summary":"The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on $n$ leaves. Proved: it is $(2/3 + o(1))\\binom{n}{4}$, by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bandelt-dress-quartet-distance"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.03542"}],"metadata":{"name":"The Bandelt-Dress Quartet Distance Conjecture","slug":"bandelt-dress-quartet-distance","field":"Combinatorics","model":"GPT-5.6","ai_role":"The paper states plainly that the author used GPT-5.6 to prove the theorem and to draft an initial version of the manuscript.","posed_by":"Hans-Jurgen Bandelt, Andreas Dress","citations":null,"statement":"The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on $n$ leaves. Proved: it is $(2/3 + o(1))\\binom{n}{4}$, by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.","resolution":"resolved","short_name":"Quartet distance","solve_date":"2026-08-04","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.03542","year_posed":1986,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2608.03542 - The maximum quartet distance between phylogenetic trees","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A 1986 conjecture calibrating the scale of the quartet distance, which is a standard metric for comparing phylogenetic trees in computational biology.","verification_note":"arXiv preprint, not yet peer-reviewed.","human_collaborators":"[\"Lior Pachter\"]"},"metadata_root":"sha256:a766bc3d559c9c4617dd330123a7d08d84ecf4d5c2377f6447770fe0d9d6daa9","content_root":"sha256:aff15f79890ee3c3a10f8d2415026c3f9d639d2aaa11b07dd951513afd702833","availability":"reference_only","row_root":"sha256:658af305b8f9bedf6bac6658d3947b58fad361b89894b6ff4cb33073b4065fb3"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:banks-martin-primitive-sets","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Banks-Martin Conjecture on Primitive Sets","summary":"Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem for the area, is proved here.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/banks-martin-primitive-sets"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.00301"}],"metadata":{"name":"The Banks-Martin Conjecture on Primitive Sets","slug":"banks-martin-primitive-sets","field":"Number theory","model":"GPT-5.5 Pro (early version)","ai_role":"This paper discloses per theorem rather than in a blanket statement, and this theorem is one of the more modest entries: an early version of GPT-5.5 Pro was used to assist with the initial proof. Elsewhere in the same paper the model's role is larger, with the proof of the Erdos #1196 theorem generated by an autonomous GPT-5.4 Pro run whose transcript is public. Across all of it the authors state that the final proofs were generated and reviewed by them, using the AI-generated proofs as starting points where appropriate. The whole method, Markov chains with von Mangoldt weights, was itself suggested by model output.","posed_by":"William D. Banks, Greg Martin; revised form proposed by Jared Duker Lichtman","citations":null,"statement":"Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem for the area, is proved here.","resolution":"resolved","short_name":"Odd Banks-Martin","solve_date":"2026-05-01","solve_type":"proved","source_url":"https://arxiv.org/abs/2605.00301","year_posed":2013,"field_group":"Number theory","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2605.00301 - Primitive sets and von Mangoldt chains: Erdos Problem #1196 and beyond","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Described in the paper as long viewed as a unifying master theorem for primitive sets, implying results in the area that had been proved separately. Conjectured 2013, revised after a counterexample at the prime 2.","verification_note":"arXiv preprint, not peer-reviewed. Other results in the same paper were formalized in Lean using Codex and Gauss, but this theorem was not among them.","human_collaborators":"[\"Boris Alexeev\",\"Kevin Barreto\",\"Yanyang Li\",\"Jared Duker Lichtman\",\"Liam Price\",\"Jibran Iqbal Shah\",\"Quanyu Tang\",\"Terence Tao\"]"},"metadata_root":"sha256:c7a44f7e2d970cd27793a4645c5e200132ba9c075c5bf5d1419cc8246706bd94","content_root":"sha256:4ddb4f26bdf07b93134935d3d8a6d4da9ea2c18242fc591f281f67b05a45d42a","availability":"reference_only","row_root":"sha256:2da4789c7103f462a9e18b64b673fc849323ca568a8c176bc0d81838ad28ce4f"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:batyrev-stringy-hodge-numbers","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Batyrev's Stringy Hodge Number Conjecture","summary":"For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/batyrev-stringy-hodge-numbers"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.19184"}],"metadata":{"name":"Batyrev's Stringy Hodge Number Conjecture","slug":"batyrev-stringy-hodge-numbers","field":"Algebraic Geometry","model":"GPT","ai_role":"Satriano and Usatine found the counterexample with the assistance of GPT: $X = M_0 \\times \\mathbb{P}^1$, where $M_0$ is the coarse moduli space of rank-2 semistable bundles with trivial determinant over a genus-3 curve. $X$ is a 7-dimensional projective variety with Gorenstein terminal singularities whose stringy $E$-function is a polynomial, yet its stringy Hodge number $h^{2,5}_{\\mathrm{st}}(X) = -1$ is negative.","posed_by":"Victor Batyrev","citations":null,"statement":"For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)","resolution":"resolved","short_name":"Batyrev's Conjecture","solve_date":"2026-07-21","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.19184","year_posed":1998,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.19184","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Batyrev's stringy invariants are foundational in birational geometry and mirror symmetry.","verification_note":"arXiv preprint 2607.19184 (21 Jul 2026) by Matthew Satriano and Jeremy Usatine. The proof is short and fully explicit: the stringy $E$-function is written out and its $u^2 v^5$ coefficient gives $h^{2,5}_{\\mathrm{st}} = -1$, so it is hand-verifiable. A domain-expert preprint, not yet peer-reviewed. Distinct from the unrelated Batyrev-Manin conjecture on rational points.","human_collaborators":"[\"Matthew Satriano\",\"Jeremy Usatine\"]"},"metadata_root":"sha256:83983ae50c4669e385a441adff08868aba8d30637ad65ad2502b51c8deaae336","content_root":"sha256:d7ba5704adaf0c365fac97e89e06c1f102749096c2f57b889a4654162f1b22ab","availability":"reference_only","row_root":"sha256:23e7194c66a3619da51f8dc67810443d1f9a7e5c4765d5d46de551e1fdf4e11c"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:belinskaya-theorem-measure-preserving-flows","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Belinskaya's Theorem for Measure-Preserving Flows","summary":"Two free ergodic measure-preserving flows whose $\\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This proves the flow analogue of Belinskaya's theorem, answering a question posed by François Le Maître and the author.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/belinskaya-theorem-measure-preserving-flows"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.14444"}],"metadata":{"name":"Belinskaya's Theorem for Measure-Preserving Flows","slug":"belinskaya-theorem-measure-preserving-flows","field":"Ergodic Theory, Orbit Equivalence","model":"GPT-5 + Claude Opus 4","ai_role":"The paper states that the criterion the proof turns on, and its application, were discovered autonomously by a two-agent AI system running GPT-5 and Claude Opus 4 in a research loop; Codex was used separately for proofreading and stylistic editing.","posed_by":"François Le Maître and Konstantin Slutsky","citations":null,"statement":"Two free ergodic measure-preserving flows whose $\\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This proves the flow analogue of Belinskaya's theorem, answering a question posed by François Le Maître and the author.","resolution":"resolved","short_name":"Belinskaya for flows","solve_date":"2026-07-16","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.14444","year_posed":null,"field_group":"Analysis","model_maker":null,"publication":"preprint","result_note":null,"source_name":"arXiv","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Answers a question the author had posed with Le Maître, transferring a classical rigidity theorem from transformations to flows.","verification_note":"No independent review. The paper attributes the key criterion to an autonomous agent loop rather than to prompted assistance. Preprint, not refereed.","human_collaborators":"[\"Konstantin Slutsky\"]"},"metadata_root":"sha256:af634154a2a548a6d66748f48ca5b216481de391b5c53c4e76f1360307caf8f1","content_root":"sha256:301de1beb915212cfbda7374690bd96ad5323ade6cec2b0a2f60580cc9dc358c","availability":"reference_only","row_root":"sha256:198de5301d5f568a73d1cc63f8884189ad65cbced17cf0c43e4d83d2040c0000"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bellman-lost-in-forest-golden-gnomon","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Bellman's Lost-in-a-Forest Problem for the Golden Gnomon","summary":"What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides $1$ and apex angle $108^\\circ$ - from an unknown starting position and heading? The optimum is a symmetric seven-piece path of segments, circular shoulders and tangents, of exactly determined transcendental length $C = 1.282676\\ldots$ - the first proved exact optimum for an isosceles triangle with base angle below $45^\\circ$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bellman-lost-in-forest-golden-gnomon"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.24483"},{"locator_id":"native-3","kind":"artifact","url":"https://github.com/atemerev/gnomon"}],"metadata":{"name":"Bellman's Lost-in-a-Forest Problem for the Golden Gnomon","slug":"bellman-lost-in-forest-golden-gnomon","field":"Convex geometry","model":"Claude Fable 5, GPT-5.6 Sol, Claude Opus 5","ai_role":"The models were used throughout: to search out the extremal curve, to draft the arguments, and to write the accompanying Lean 4 development. The paper states precisely which steps are machine-checked, and notes those checks hold regardless of how the statements were found.","posed_by":"Richard E. Bellman","citations":null,"statement":"What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides $1$ and apex angle $108^\\circ$ - from an unknown starting position and heading? The optimum is a symmetric seven-piece path of segments, circular shoulders and tangents, of exactly determined transcendental length $C = 1.282676\\ldots$ - the first proved exact optimum for an isosceles triangle with base angle below $45^\\circ$.","resolution":"resolved","short_name":"Lost in a forest (gnomon)","solve_date":"2026-07-27","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.24483","year_posed":1956,"field_group":"Geometry & topology","model_maker":"Anthropic / OpenAI","publication":"preprint","result_note":"Bellman's problem for general regions remains open","source_name":"arXiv:2607.24483 - The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon","significance":30,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Bellman's 1956 lost-in-a-forest problem, a fixture of unsolved-problem collections.","verification_note":"Lean 4 verifies the two finite algebraic certificate families and the discrete ledger identities, but not the full argument end to end; the paper's appendix states exactly which steps are machine-checked. arXiv preprint, not yet peer-reviewed.","human_collaborators":"[\"Alexander Temerev\",\"Alessio Doria\"]"},"metadata_root":"sha256:c03c73c6dcac82412125e64f8c9363740b5f63b4c30132a4db7722cdca5d63b8","content_root":"sha256:c91117541fa902d546f4cb8f4c3908a880de26d04d137d2d68f035688916f6e5","availability":"reference_only","row_root":"sha256:10e3fb1c6937c97ffc5e22bfc6811e79dc5d4da2422d1ae9535368ccaf41f933"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:benjamini-hochberg-correlated-gaussian","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Benjamini-Hochberg FDR Under Correlated Gaussian Tests","summary":"Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives $\\mathrm{FDR} > 0.0104$ at nominal level $\\alpha = 0.01$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/benjamini-hochberg-correlated-gaussian"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.12208"}],"metadata":{"name":"Benjamini-Hochberg FDR Under Correlated Gaussian Tests","slug":"benjamini-hochberg-correlated-gaussian","field":"Statistics","model":"GPT-5.6 Pro","ai_role":"The counterexample was obtained by GPT-5.6 Pro and carefully checked by the author, with a rigorous interval-arithmetic certificate valid for all sufficiently large numbers of hypotheses.","posed_by":null,"citations":null,"statement":"Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives $\\mathrm{FDR} > 0.0104$ at nominal level $\\alpha = 0.01$.","resolution":"resolved","short_name":"BH under correlation","solve_date":"2026-07-13","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.12208","year_posed":2006,"field_group":"Probability & statistics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.12208 - The Benjamini-Hochberg procedure can fail to control the FDR","significance":14,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"FDR under correlation is a widely felt applied-statistics question, but diffuse as a single problem.","verification_note":"Author-checked arXiv preprint with an interval-arithmetic certificate. Not yet peer-reviewed.","human_collaborators":"[\"Edgar Dobriban\"]"},"metadata_root":"sha256:4de049e36d1fae8e4fd92a193576d691408f194f27e946b3439c73679a4be05b","content_root":"sha256:7680506c54a000babd8ad42fb46572dbeb324063c1fb0c25d874ce1466fb67e8","availability":"reference_only","row_root":"sha256:330250810f757dd560a0de8e91bfa1b693c9a99160cd36a484a43d9f2367c687"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bertoin-yor-moment-determinacy","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Bertoin-Yor Moment Determinacy Conjecture","summary":"For an unkilled Levy process $\\xi$ drifting to $+\\infty$ with all positive exponential moments, let $I_\\xi = \\int_0^\\infty e^{-\\xi_t}\\,dt$ and $X_\\xi = 1/I_\\xi$. Bertoin and Yor proved $X_\\xi$ is moment-determinate when $\\xi$ has no positive jumps and conjectured that this condition is necessary. The conjecture is settled.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bertoin-yor-moment-determinacy"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.00132"}],"metadata":{"name":"Bertoin-Yor Moment Determinacy Conjecture","slug":"bertoin-yor-moment-determinacy","field":"Probability","model":"GPT-5.4 Thinking, GPT-5.5 Thinking and Pro","ai_role":"The paper has a dedicated Use of AI tools section stating the models were used during the exploratory and editorial stages of the work. Exploration is mathematical work rather than prose work, but no individual step is attributed, so the lowest tier applies.","posed_by":"Jean Bertoin, Marc Yor","citations":null,"statement":"For an unkilled Levy process $\\xi$ drifting to $+\\infty$ with all positive exponential moments, let $I_\\xi = \\int_0^\\infty e^{-\\xi_t}\\,dt$ and $X_\\xi = 1/I_\\xi$. Bertoin and Yor proved $X_\\xi$ is moment-determinate when $\\xi$ has no positive jumps and conjectured that this condition is necessary. The conjecture is settled.","resolution":"resolved","short_name":"Bertoin-Yor determinacy","solve_date":"2026-06-30","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.00132","year_posed":2002,"field_group":"Probability & statistics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.00132 - On a moment determinacy conjecture of Bertoin and Yor","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A named 2002 conjecture on exponential functionals of Levy processes, a well-worked corner of probability.","verification_note":"Single-author arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Martin Minchev\"]"},"metadata_root":"sha256:1876d047f1db22a7ba9db703730955609ae1efc209afd80184fa495a1af0091e","content_root":"sha256:83dc063522a94ecb53240db325b7339d729e84ffba442dcbd260661ad98f8beb","availability":"reference_only","row_root":"sha256:92f87f3fac5a29a97ebda14832d5456f5ebe82402861f4f214e876f9e970ba56"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bh-fdr-universal-multiplicative-bound","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Universal Multiplicative FDR Bound for Benjamini-Hochberg","summary":"The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \\downarrow 0$, with an explicit two-sided lower bound $q\\sqrt{\\log(1/q)}/(2\\sqrt{\\pi}) + 0.6493 q + o(q)$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bh-fdr-universal-multiplicative-bound"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.14812"}],"metadata":{"name":"Universal Multiplicative FDR Bound for Benjamini-Hochberg","slug":"bh-fdr-universal-multiplicative-bound","field":"Statistics","model":"GPT-5.6 Sol","ai_role":"The disclosure says only that the author used the model for assistance with proof exploration, exposition and editing. It does not attribute any specific step, so the lowest tier applies.","posed_by":"folklore in the FDR literature","citations":null,"statement":"The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \\downarrow 0$, with an explicit two-sided lower bound $q\\sqrt{\\log(1/q)}/(2\\sqrt{\\pi}) + 0.6493 q + o(q)$.","resolution":"resolved","short_name":"BH multiplicative bound","solve_date":"2026-07-20","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.14812","year_posed":null,"field_group":"Probability & statistics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.14812 - How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?","significance":30,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Benjamini-Hochberg is one of the most used procedures in applied statistics, and how badly dependence can break it is a question the multiple-testing community has tracked for two decades.","verification_note":"Single-author arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Lihua Lei\"]"},"metadata_root":"sha256:b2ebced534e631a92a74287cf4afd21759084b24ea093c8dd26701f31c7cc88f","content_root":"sha256:f7c850fcaf1adb8636c5d4d3bcb56d785229f94b3c6b9aa9e5d130fd12972819","availability":"reference_only","row_root":"sha256:ed2d304cabddd113a51d961e85fd1bfc8ff680c91838fcf0050e844b97fac796"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:binary-code-upper-bounds","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Upper Bounds for Binary and Spherical Codes","summary":"What is the maximum size of a binary code of given minimum distance? The linear-programming bounds of McEliece, Rodemich, Rumsey and Welch (1977) resisted improvement for half a century. The new upper bounds are exponentially stronger at every prescribed distance, with analogous results for high-dimensional spherical codes.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/binary-code-upper-bounds"},{"locator_id":"native-2","kind":"artifact","url":"https://openai.com/index/ten-advances-in-mathematics/"},{"locator_id":"native-3","kind":"artifact","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf"},{"locator_id":"native-4","kind":"artifact","url":"https://github.com/openai/ten-proofs/blob/main/MetricCodes.lean"},{"locator_id":"native-5","kind":"artifact","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"metadata":{"name":"Upper Bounds for Binary and Spherical Codes","slug":"binary-code-upper-bounds","field":"Coding theory","model":"Astra (internal preview)","ai_role":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","posed_by":null,"citations":null,"statement":"What is the maximum size of a binary code of given minimum distance? The linear-programming bounds of McEliece, Rodemich, Rumsey and Welch (1977) resisted improvement for half a century. The new upper bounds are exponentially stronger at every prescribed distance, with analogous results for high-dimensional spherical codes.","resolution":"partial","short_name":"Binary code bounds","solve_date":"2026-08-01","solve_type":"proved","source_url":"https://openai.com/index/ten-advances-in-mathematics/","year_posed":1977,"field_group":"Theoretical computer science","model_maker":"OpenAI","publication":"announcement","result_note":"exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open","source_name":"OpenAI: Ten advances in mathematics and theoretical computer science","significance":39,"verification":"lean-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"The rate-distance trade-off is coding theory's central asymptotic question; the MRRW barrier stood since 1977.","verification_note":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.","human_collaborators":"[]"},"metadata_root":"sha256:edf963165639b5e47d6161a92647d8d5d9d583c882e7bef216dae06cce0191d8","content_root":"sha256:3302cc579911809040f4e20ac36250b29d0f71e4b7eb9ff8a19694d88d496bea","availability":"reference_only","row_root":"sha256:1724e1dcb31786ccc98471e6aa094fbee61965ee436bbf70ed3d51cbbb86f3b5"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bipartite-bound-information","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Existence of Bipartite Bound Information","summary":"Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bipartite-bound-information"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.25838"}],"metadata":{"name":"Existence of Bipartite Bound Information","slug":"bipartite-bound-information","field":"Classical & quantum information theory","model":"GPT-5.6 Sol","ai_role":"The explicit example - a distribution on two bits and a trit with zero distillable key but positive secrecy cost - was found with GPT-5.6 Sol; the authors reconstructed the proof line by line.","posed_by":"Nicolas Gisin & Stefan Wolf","citations":null,"statement":"Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?","resolution":"resolved","short_name":"Bound information","solve_date":"2026-07-28","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.25838","year_posed":2000,"field_group":"Quantum information & computing","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.25838 - Bipartite bound information exists","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Gisin-Wolf 2000; a named open question of quantum key distillation for 25 years.","verification_note":"Authors reconstructed the proof line by line with exact ancillary checks; public arXiv preprint, not yet peer-reviewed. The paper also shows the distributions that originally motivated the conjecture are not themselves examples.","human_collaborators":"[]"},"metadata_root":"sha256:717b262b2f0e1fa3fb164474b62bb774d7251f151bab408bec38fc052ede2b26","content_root":"sha256:10d97a7cd588c6812569deef45f7e0519eac7170ff92bef05d961fa7205dfe7e","availability":"reference_only","row_root":"sha256:e802ed6ced5b7acc8e08949d4b418c2d8e663a9e572b83a1af8f20a5d1bf56c0"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bipartite-exact-matching","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Bipartite Exact Matching in P","summary":"The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time since Mulmuley-Vazirani-Vazirani (1987) while membership in P stayed open for four decades. The paper claims a deterministic polynomial-time algorithm, replacing probabilistic amplification with deterministic evaluations.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bipartite-exact-matching"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2604.01571"}],"metadata":{"name":"Bipartite Exact Matching in P","slug":"bipartite-exact-matching","field":"Algorithms; derandomization","model":"GPT-5.4 Pro, Claude Opus 4.6, Aristotle","ai_role":"GPT-5.4 Pro assisted with theoretical route selection and problem reduction: identifying viable proof strategies, formulating equivalent reformulations of the main conjecture, and narrowing the search space. Claude Opus 4.6 (via Claude Code) ran rapid iterative computational experiments that tested conjectures and produced counterexamples to failed approaches. Lean 4 with Mathlib served as the formal verification backend, with Harmonic's Aristotle discharging proof obligations during the formalization.","posed_by":"Christos Papadimitriou, Mihalis Yannakakis","citations":null,"statement":"The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time since Mulmuley-Vazirani-Vazirani (1987) while membership in P stayed open for four decades. The paper claims a deterministic polynomial-time algorithm, replacing probabilistic amplification with deterministic evaluations.","resolution":"candidate","short_name":"Bipartite exact matching","solve_date":"2026-04-02","solve_type":"proved","source_url":"https://arxiv.org/abs/2604.01571","year_posed":1982,"field_group":"Theoretical computer science","model_maker":"OpenAI, Anthropic, Harmonic","publication":"preprint","result_note":null,"source_name":"arXiv","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A 1982 Papadimitriou-Yannakakis question known across theoretical computer science through the Mulmuley-Vazirani-Vazirani RNC algorithm; one of the standard examples of a problem in RNC not known to be in P.","verification_note":"A single-author preprint claiming a forty-year-open result. The paper reports a Lean 4/Mathlib formalization with Aristotle assisting, but no independent expert has reviewed the claim; entered as a candidate pending community scrutiny.","human_collaborators":"[\"Yuefeng Du\"]"},"metadata_root":"sha256:3155f12d641cb510e45bbafcf83e507d0d8e9e17d91cf38790492976c53a734c","content_root":"sha256:e8b37ea40dd5115d81089d4850327fe49ab323787219b5b2dbecedd9d6f90f39","availability":"reference_only","row_root":"sha256:89fe40169952a51e01f3d33c2719594a0493485b186616bd64ea01e6c899fb9d"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bombari-sign-quantization-question","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Bombari's Question on Sign-Quantized Linear Maps","summary":"A dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear maps, valid for any bounded function under a well-conditioned covariance, which answers a question of Simone Bombari on sign quantization.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bombari-sign-quantization-question"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.27563"}],"metadata":{"name":"Bombari's Question on Sign-Quantized Linear Maps","slug":"bombari-sign-quantization-question","field":"Probability","model":"Gemini 3.5 Flash","ai_role":"The abstract states the result was discovered by Gemini 3.5 Flash, and the title calls the paper an AI-assisted note. Worth recording that a small fast model, not a frontier reasoning tier, produced it.","posed_by":"Simone Bombari","citations":null,"statement":"A dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear maps, valid for any bounded function under a well-conditioned covariance, which answers a question of Simone Bombari on sign quantization.","resolution":"resolved","short_name":"Quantized subgaussianity","solve_date":"2026-05-26","solve_type":"proved","source_url":"https://arxiv.org/abs/2605.27563","year_posed":null,"field_group":"Probability & statistics","model_maker":"Google","publication":"preprint","result_note":null,"source_name":"arXiv:2605.27563 - On the Subgaussianity of Quantized Linear Maps: An AI-Assisted Note","significance":10,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A question from the high-dimensional probability literature on what survives sign quantization, narrow but explicitly posed.","verification_note":"Short arXiv note; not yet peer-reviewed.","human_collaborators":"[\"Guangyi Zou\",\"Roman Vershynin\"]"},"metadata_root":"sha256:51b4c2787ff88d0175c30b9c1eacda62111ba5c164d7d4567eeaa31d7b0625d4","content_root":"sha256:a8be3385bc2c63acfd3c661d29352ed34d2b0759de8db23529f40baa138afd07","availability":"reference_only","row_root":"sha256:fa64a30af130be4963c042fac7247bb734bb3594faefb3eadfe6a0ff3e5ae589"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bondal-polishchuk-smooth-projective-counterexample","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Bondal-Polishchuk Conjecture for a Smooth Projective Variety","summary":"Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form $D^b(X)$ for a smooth projective variety was known. A particular weak Fano threefold provides one.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bondal-polishchuk-smooth-projective-counterexample"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.25391"}],"metadata":{"name":"Bondal-Polishchuk Conjecture for a Smooth Projective Variety","slug":"bondal-polishchuk-smooth-projective-counterexample","field":"Algebraic geometry","model":"ChatGPT 5.6","ai_role":"The AI disclosure is carefully bounded and worth quoting in full: the model most significantly found numerous mistakes in earlier attempted constructions, leading to the successful modifications, and it located the reference containing the threefold used. The author states that it did not produce the actual counterexample and was not used to generate any of the text.","posed_by":"Alexei Bondal, Alexander Polishchuk","citations":null,"statement":"Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form $D^b(X)$ for a smooth projective variety was known. A particular weak Fano threefold provides one.","resolution":"resolved","short_name":"Bondal-Polishchuk","solve_date":"2026-07-28","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.25391","year_posed":1993,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":"first counterexample of the form D^b(X) for X smooth projective","source_name":"arXiv:2607.25391 - A smooth projective counterexample to Bondal-Polishchuk's conjecture","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A 1993 conjecture about exceptional collections that is standard background in derived categories of coherent sheaves and had resisted a smooth projective counterexample.","verification_note":"Single-author arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Anya Nordskova\"]"},"metadata_root":"sha256:2b230b14e74ba67e6fd064030549fef465117cc237c3ce6514f870978db9dc14","content_root":"sha256:0e3940f5c0bd7823c8c66206c61fbf83c43fec238de65d62fa6fb53edeafbb37","availability":"reference_only","row_root":"sha256:6809525ba0e43c742f9d4cb068f7a65164ffbcd524a042498491d2f47f89506f"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:boolean-max-k-csp-approximation","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Approximation Ratio for Boolean Max-k-CSP","summary":"How well can an arbitrary boolean constraint satisfaction problem of arity $k$ be approximated in polynomial time? The paper gives a $(k/2^k)$-approximation, improving the previous best constant of $0.626612\\,k/2^k$ due to Makarychev and Makarychev.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/boolean-max-k-csp-approximation"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.05331"}],"metadata":{"name":"The Approximation Ratio for Boolean Max-k-CSP","slug":"boolean-max-k-csp-approximation","field":"Approximation algorithms","model":"GPT-5.6 Sol Max","ai_role":"\"GPT 5.6 Sol Max assisted in the lengthy computations that appear in the proof.\" Computational support inside a human-led argument.","posed_by":null,"citations":null,"statement":"How well can an arbitrary boolean constraint satisfaction problem of arity $k$ be approximated in polynomial time? The paper gives a $(k/2^k)$-approximation, improving the previous best constant of $0.626612\\,k/2^k$ due to Makarychev and Makarychev.","resolution":"partial","short_name":"Boolean Max-k-CSP","solve_date":"2026-08-05","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.05331","year_posed":null,"field_group":"Algorithms & optimization","model_maker":"OpenAI","publication":"preprint","result_note":"Removes the constant factor from the previous best guarantee; whether $k/2^k$ is optimal is not settled here.","source_name":"arXiv","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"The Max-k-CSP approximation constant is a tracked quantity of the approximation-algorithms literature, with a documented ladder running through Makarychev and Makarychev.","verification_note":"A preprint days old, with no independent review.","human_collaborators":"[\"Ainesh Bakshi\"]"},"metadata_root":"sha256:7bc7d8b9d590727d28217424662c0c77caea713d2cff597233ce2529f00982b9","content_root":"sha256:13cccd19a522e739e24065d25b5bd10385d5594a613b8618e7cdac1f556fef1e","availability":"reference_only","row_root":"sha256:fef64432f3640f89bd27e75fcd60298f707d7b84e70c671ba6618f31ce70434e"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:boots-royle-cao-vince-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Boots-Royle/Cao-Vince Conjecture on Planar Spectral Radius","summary":"Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge with a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for every $n \\ge 9$. Tait and Tobin proved it for sufficiently large $n$ in 2017; the conjecture now holds for all $n \\ge 9$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/boots-royle-cao-vince-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.19268"}],"metadata":{"name":"Boots-Royle/Cao-Vince Conjecture on Planar Spectral Radius","slug":"boots-royle-cao-vince-conjecture","field":"Spectral graph theory","model":"ChatGPT","ai_role":"The declaration credits the model with generating the code that searched for extremal planar graphs, and with assisting in several computations and symbolic derivations, alongside language polishing.","posed_by":"Barry Boots, Gordon Royle; Dasong Cao, Andrew Vince","citations":null,"statement":"Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge with a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for every $n \\ge 9$. Tait and Tobin proved it for sufficiently large $n$ in 2017; the conjecture now holds for all $n \\ge 9$.","resolution":"resolved","short_name":"Boots-Royle/Cao-Vince","solve_date":"2026-07-21","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.19268","year_posed":1991,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.19268 - A complete solution to the Boots-Royle/Cao-Vince conjecture","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A conjecture posed independently by two groups around 1991 and a standard reference point in spectral extremal graph theory, with Tait and Tobin's asymptotic solution in JCTB.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Lele Liu\",\"Bo Ning\",\"Yi Wang\"]"},"metadata_root":"sha256:f2c06ad9ce9c6bd192d46fd7e2d5a377ca40b5062e4244f583507f9c8b748498","content_root":"sha256:3053c0fc49f4490f70a9f406482751982695f495634e2ec04dffe43a81f0dcd7","availability":"reference_only","row_root":"sha256:2bc40fed310cce113fec7a058c3e1064c7ee5270fe8448a8cbccfeaeeb32ce6a"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:boppana-entropy-generalization","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"A Generalization of Boppana's Entropy Inequality","summary":"A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/boppana-entropy-generalization"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2601.19327"},{"locator_id":"native-3","kind":"artifact","url":"https://github.com/boonsuan/entropy-inequality"}],"metadata":{"name":"A Generalization of Boppana's Entropy Inequality","slug":"boppana-entropy-generalization","field":"Entropy inequalities","model":"GPT-5.2 Pro, Harmonic Aristotle, Gemini 3 Pro","ai_role":"\"Key steps in some proofs were generated with the assistance of GPT-5.2 pro. The result has also been formalized in Lean 4 using Harmonic Aristotle and Gemini 3 Pro Preview\" - with the formalization public.","posed_by":null,"citations":null,"statement":"A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.","resolution":"resolved","short_name":"Boppana entropy","solve_date":"2026-01-27","solve_type":"proved","source_url":"https://arxiv.org/abs/2601.19327","year_posed":null,"field_group":"Combinatorics","model_maker":"OpenAI, Harmonic, Google DeepMind","publication":"preprint","result_note":null,"source_name":"arXiv","significance":8,"verification":"lean-checked","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Extends a named inequality from the union-closed-sets toolbox; specialist, but in a lineage other results depend on.","verification_note":"Formalized in Lean 4 (Aristotle plus Gemini); code public on GitHub. No independent review. Tier: the formalization was produced by the assisting systems and checked by the author; no independent statement audit.","human_collaborators":"[\"Boon Suan Ho\"]"},"metadata_root":"sha256:810714b9e232fcd2243da1707ce1c11281e07e0eafc57d51c06ef742056c1fa8","content_root":"sha256:248020c1c172504d31ee76bbaef4bb38984ed84bcee3914e58be5f57c4c3df0d","availability":"reference_only","row_root":"sha256:d16909e8d07ace2a2ec509c409c622660b019c0ec7318e45f9134b92fd7d54fd"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Borsuk Conjecture lowest-ever counterexample (N=63)","summary":"Borsuk's conjecture asked whether every bounded set in $\\mathbb{R}^n$ can be partitioned into $n+1$ subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in $\\mathbb{R}^{63}$ whose smaller-diameter subsets have at most 5 points, so at least $\\lceil 321/5\\rceil = 65 > 64$ parts are required. The previous record dimension was 64 (Jenrich-Brouwer, 2014), and the first failing dimension remains open for $4 \\le n \\le 62$. The construction modifies Bondarenko's $G_2(4)$ two-distance set: a 320-point rank-63 subconfiguration plus one added scaled projected point, which makes the set three-distance - precisely why it was not reachable inside the two-distance framework in which all previous work took place.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/borsuk-conjecture-lowest-ever-counterexample-n-63"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/maaxgrin/borsuk-63-counterexample"},{"locator_id":"native-3","kind":"artifact","url":"https://teorth.github.io/optimizationproblems/constants/28a.html"},{"locator_id":"native-4","kind":"artifact","url":"https://nickk124.github.io/borsuk/"},{"locator_id":"native-5","kind":"artifact","url":"https://en.wikipedia.org/wiki/Borsuk%27s_conjecture"}],"metadata":{"name":"Borsuk Conjecture lowest-ever counterexample (N=63)","slug":"borsuk-conjecture-lowest-ever-counterexample-n-63","field":"Discrete geometry","model":"GPT-5.5 Pro","ai_role":"For the first solve, Grinsztajn's README states: \"The construction and proof were obtained with assistance from GPT-5.5 Pro\", with a dedicated \"Disclose GPT assistance\" commit; no finer division of labour is given, so the tier is the floor for an unspecific disclosure. The independent August 2026 rediscovery by Nicholas Konz with Claude (Fable 5 and Opus 5) carries a much fuller disclosure - Claude produced the counterexample and an exact certificate over $\\mathbb{Q}(\\sqrt{222})$ - and would rate ai-discovered on its own, but the entry's tier follows the solve it records, which is the first one.","posed_by":"Karol Borsuk","citations":null,"statement":"Borsuk's conjecture asked whether every bounded set in $\\mathbb{R}^n$ can be partitioned into $n+1$ subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in $\\mathbb{R}^{63}$ whose smaller-diameter subsets have at most 5 points, so at least $\\lceil 321/5\\rceil = 65 > 64$ parts are required. The previous record dimension was 64 (Jenrich-Brouwer, 2014), and the first failing dimension remains open for $4 \\le n \\le 62$. The construction modifies Bondarenko's $G_2(4)$ two-distance set: a 320-point rank-63 subconfiguration plus one added scaled projected point, which makes the set three-distance - precisely why it was not reachable inside the two-distance framework in which all previous work took place.","resolution":"partial","short_name":"Borsuk Counterexample N=63","solve_date":"2026-05-26","solve_type":"disproved","source_url":"https://github.com/maaxgrin/borsuk-63-counterexample","year_posed":1933,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":"Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitter of this entry surfaced the earlier work themselves after publication. Dimension 63 is the current record; whether Borsuk's conjecture fails for any dimension in 4..62 remains open.","source_name":"Max Grinsztajn's proof note and certificates","significance":30,"verification":"site-confirmed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Borsuk's conjecture is a famous named problem with a Wikipedia article and a 90-year history; the conjecture itself was already refuted by Kahn-Kalai in 1993, so what this entry records is the current record for the smallest failing dimension, 64 to 63, a serious but incremental step on a well-known question (the record has moved six times since 1993). Scored level with the Hadamard order-668 construction: both are the current record instance of a famous conjecture rather than the conjecture itself.","verification_note":"Both derivations reproduced by this site on 12 August 2026, independently of each other. For the first solve (Grinsztajn, May 2026): the repository's exact verifier - pure Python integer arithmetic over F16, read before running - was executed locally and passes all checks: it rebuilds the G2(4) strongly regular graph with parameters (416,100,36,20), the B1/B2/B3/C partition and degree data behind the dimension drop, and the clique obstructions forcing every smaller-diameter subset to size at most 5. The repo's GitHub creation date of 2026-05-26 is not forgeable after the fact, and Terence Tao's optimization-problems ledger (constant 28a) independently credits the 63 bound to Grinsztajn, citing this repository. For the August rediscovery (Konz + Claude): we ran the author's stand-alone verifier against the published 321x63 coordinate file and confirmed affine dimension exactly 63, the squared-distance spectrum (53-sqrt(222))/156, 1/4 and 1/3, and independence number 5 for the diameter graph by Bron-Kerbosch, forcing ceil(321/5) = 65 parts where Borsuk allows 64; the distance-class gap is far wider than any float tolerance. Neither write-up is peer-reviewed; neither is on arXiv.","human_collaborators":"[\"Max Grinsztajn\"]"},"metadata_root":"sha256:e9cee6aa9ecaaa22b6b252e36abf0a1579b5d901928aa5e33541d101f9144239","content_root":"sha256:d825109ef2b4daacfce97bc441e6a0c0713d5eb7ba4021cdc7e6541afa78b1f0","availability":"reference_only","row_root":"sha256:0bad7aec3e603b7016e2f82d649b1ed23e0037fabb7ccb216e5fcaac55b36d1d"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bosonic-quantum-capacity-non-gaussian","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Bosonic Quantum Communication Beyond the Thermal Threshold","summary":"Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bosonic-quantum-capacity-non-gaussian"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.27449"}],"metadata":{"name":"Bosonic Quantum Communication Beyond the Thermal Threshold","slug":"bosonic-quantum-capacity-non-gaussian","field":"Quantum Information, Bosonic Channels","model":"ChatGPT 5.6 Sol","ai_role":"From the paper's own AI-assisted research statement: some of the authors had tried hard to find such a non-Gaussian counterexample several years ago and failed; they tried again with ChatGPT 5.5 and failed again; ChatGPT 5.6 Sol then gave them the families of non-Gaussian counterexamples. They call the AI crucial to the work while taking full responsibility for its content.","posed_by":"Alexander Holevo and Reinhard Werner","citations":null,"statement":"Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.","resolution":"resolved","short_name":"Non-Gaussian bosonic capacity","solve_date":"2026-07-29","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.27449","year_posed":1999,"field_group":"Quantum information & computing","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Breaks a lower bound standing since Holevo and Werner in 1999 and shows a complete theory of noisy bosonic communication cannot stay within single-mode Gaussian states.","verification_note":"No independent review, but stronger than a bare claim: Appendix A turns the numerical evaluation into a rigorous bound by enclosing every quantity in certified real intervals, and a second certified witness is given in Appendix B. Preprint, not refereed.","human_collaborators":"[\"Francesco Anna Mele\",\"Giuseppe Catalano\",\"Marco Fanizza\",\"Vittorio Giovannetti\",\"Ludovico Lami\"]"},"metadata_root":"sha256:ced5e9a0d012ad29b2d8bc6cf1ef1e022871469df6f0f7496990644140982b33","content_root":"sha256:b4cdc7067c4a9cfaf59c1bd514b035e1612049bf1e3bfb0359156a47f7179dfd","availability":"reference_only","row_root":"sha256:786e2bb425d41f72f0056bb35d2b363ae7f8fda9c5a5d988a764fcaf71aaf1fb"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:boucksom-local-analytic-bertini","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Boucksom's Local Analytic Bertini Conjecture","summary":"Does the analytic Bertini restriction theorem for multiplier ideals hold locally, outside a pluripolar exceptional set of fibers? Proved in full generality.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/boucksom-local-analytic-bertini"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.25230"}],"metadata":{"name":"Boucksom's Local Analytic Bertini Conjecture","slug":"boucksom-local-analytic-bertini","field":"Pluripotential theory","model":"Rethlas (GPT-5.6 Sol)","ai_role":"The author had the proof idea before the AI era; Rethlas running GPT-5.6 Sol first carried out the details, and Xia then simplified and largely rewrote the proof.","posed_by":"Sébastien Boucksom","citations":null,"statement":"Does the analytic Bertini restriction theorem for multiplier ideals hold locally, outside a pluripolar exceptional set of fibers? Proved in full generality.","resolution":"resolved","short_name":"Local analytic Bertini","solve_date":"2026-07-28","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.25230","year_posed":null,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.25230 - Analytic Bertini theorem II: the local case","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A named Boucksom conjecture in complex/non-archimedean geometry.","verification_note":"Author-rewritten arXiv preprint. Not yet peer-reviewed.","human_collaborators":"[\"Mingchen Xia\"]"},"metadata_root":"sha256:11bc4ed57142024fff708b47132c78cf6dbb0ba27df4b871b7c48c3af51da139","content_root":"sha256:e2dea7aff82d6f79f5ee3fb5d168059a454d42d067d31340c59a89f3a21bbe93","availability":"reference_only","row_root":"sha256:0f838fb361bf75e712506d1751af00d8d9102b6cd06d69dcb3ed74c58717ec93"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bounded-circumference-cycle-counting","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Counting Fixed Cycles in Graphs with Bounded Circumference","summary":"Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an $n$-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed $s \\ge 3$ and $L \\ge 2s+2$ and all large $n$, $\\mathrm{ex}(n, C_{2s+1}, \\mathcal{C}_{\\ge L+1}) = N(C_{2s+1}, H(n,L))$. Together with the companion even-cycle result this settles the conjecture.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bounded-circumference-cycle-counting"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.10779"}],"metadata":{"name":"Counting Fixed Cycles in Graphs with Bounded Circumference","slug":"bounded-circumference-cycle-counting","field":"Extremal graph theory","model":"GPT-5.6","ai_role":"The declaration credits the model with solving one case of Theorem 1.2, in particular the calculations in that proof, and with rewriting the Section 2.4 argument in the language of directed graphs; the rest is readability and exposition. The authors reviewed and verified the proofs and take sole responsibility.","posed_by":"Zhu, Gyori, He, Lv, Salia, Xiao","citations":null,"statement":"Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an $n$-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed $s \\ge 3$ and $L \\ge 2s+2$ and all large $n$, $\\mathrm{ex}(n, C_{2s+1}, \\mathcal{C}_{\\ge L+1}) = N(C_{2s+1}, H(n,L))$. Together with the companion even-cycle result this settles the conjecture.","resolution":"resolved","short_name":"Bounded circumference cycles","solve_date":"2026-07-12","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.10779","year_posed":2023,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":"the odd-cycle half; the even-cycle half is a companion paper by the same authors","source_name":"arXiv:2607.10779 - Counting Odd Cycles in Graphs with Bounded Circumference","significance":9,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A 2023 Bulletin of the LMS conjecture in generalized Turan theory, real and cited but recent and specialized.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Xiamiao Zhao\",\"Yuanpei Wang\"]"},"metadata_root":"sha256:7efbc9536a32c9bc94530d006ab812472c31cb14e3b2b070050bc569f7bd9e87","content_root":"sha256:bd8f136e7ddee5fb52f92e611e7b351155e0c5e73d9e084a0eca7df5afc1fed3","availability":"reference_only","row_root":"sha256:96cab8b884ba5fa32fa88f651dfada4cea66b8192ccd4faf61cde99c851a1388"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bounded-oracle-noise-nonconvex-lower-bound","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Bounded Oracle Error in Nonconvex Stochastic Optimization","summary":"Arjevani et al. asked whether almost-surely bounded oracle error permits a better rate than bounded variance for smooth nonconvex stochastic optimization. It does not: every randomized adaptive algorithm still needs Omega(dL/eps^2 + dL sigma^2/eps^4) queries, matching the standard upper bound.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bounded-oracle-noise-nonconvex-lower-bound"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.09004"}],"metadata":{"name":"Bounded Oracle Error in Nonconvex Stochastic Optimization","slug":"bounded-oracle-noise-nonconvex-lower-bound","field":"Stochastic optimization","model":"GPT-5.6 Sol","ai_role":"Stated in the abstract itself, not buried in an acknowledgment: \"The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only for checking the proof and revising and polishing the manuscript.\"","posed_by":"Yossi Arjevani, Yair Carmon, John C. Duchi, Dylan J. Foster, Nathan Srebro, Blake Woodworth","citations":null,"statement":"Arjevani et al. asked whether almost-surely bounded oracle error permits a better rate than bounded variance for smooth nonconvex stochastic optimization. It does not: every randomized adaptive algorithm still needs Omega(dL/eps^2 + dL sigma^2/eps^4) queries, matching the standard upper bound.","resolution":"resolved","short_name":"Bounded oracle error question","solve_date":"2026-08-10","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.09004","year_posed":2023,"field_group":"Algorithms & optimization","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":12,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A question posed explicitly in a well-cited lower-bounds paper, familiar to the optimization-theory community but recent and specialist.","verification_note":"A preprint days old, with no independent review.","human_collaborators":"[\"Jikai Jin\"]"},"metadata_root":"sha256:99928a88e9195883bb6f4cb231f48eb299c1a0ab07c591b6e92b2a345deb82c4","content_root":"sha256:026a74ed972754d0b3b01730c335ba2602d29051bc3dbae0ff365b692d84fef4","availability":"reference_only","row_root":"sha256:d80ebb7f6ba0c89824344b38f3767668c5cbf5ec2992cd590446c869c36eeebd"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:brezis-degree-inequality-constants","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Brezis's Problems on Degenerate Constants in Degree Inequalities","summary":"Two degree inequalities for circle-valued Sobolev maps have constants that degenerate as $p \\to 1^+$ or $\\delta \\to 0^+$. Brezis posed the problem of sharpening them; both are now sharpened, by the same power trick with elementary estimates.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/brezis-degree-inequality-constants"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.24626"},{"locator_id":"native-3","kind":"artifact","url":"https://arxiv.org/abs/2607.23598"}],"metadata":{"name":"Brezis's Problems on Degenerate Constants in Degree Inequalities","slug":"brezis-degree-inequality-constants","field":"Calculus of variations","model":"Rethlas","ai_role":"The abstract says the proofs were obtained by generative AI and verified by the authors, and the body names Rethlas as producing the proofs of Theorems 1.3 and 1.4. The raw system output is published alongside, so the provenance is inspectable rather than asserted.","posed_by":"Haim Brezis","citations":null,"statement":"Two degree inequalities for circle-valued Sobolev maps have constants that degenerate as $p \\to 1^+$ or $\\delta \\to 0^+$. Brezis posed the problem of sharpening them; both are now sharpened, by the same power trick with elementary estimates.","resolution":"resolved","short_name":"Brezis degree constants","solve_date":"2026-05-23","solve_type":"proved","source_url":"https://arxiv.org/abs/2605.24626","year_posed":null,"field_group":"Analysis","model_maker":"Frenzy Math","publication":"preprint","result_note":null,"source_name":"arXiv:2605.24626 - Degenerate constants in degree inequalities for Sobolev circle maps: on some problems posed by Brezis","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Two problems from Brezis's own list, the third of his problems this catalog now records as settled with AI in two months.","verification_note":"The raw Rethlas output is publicly linked, which is unusually good provenance. Author-verified, arXiv preprint, not peer-reviewed.","human_collaborators":"[\"Xu'an Dou\",\"Zeyu Jin\"]"},"metadata_root":"sha256:0b856f8a8076a8b8aeb813a84e57940de19c0574482922532d4d291900982e91","content_root":"sha256:413b46b2d6acc617db283868efedb35a4d6e955c97dfb74daecb071d0e7b96ab","availability":"reference_only","row_root":"sha256:23adf7eefd1581d681c6fb7afa3dd298a0a95a3ada5638a9f89d9ed1f1ec3022"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:brezis-mironescu-circle-minimizers","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Brezis-Mironescu Open Problems 23 and 24 on Minimizing Maps to the Circle","summary":"For $s \\in (1/4,1)$ and any degree, the only $W^{s,1/s}$-minimizers among maps $\\mathbb{S}^1 \\to \\mathbb{S}^1$ are Blaschke products. This resolves Open Problems 23 and 24 of Brezis and Mironescu's book on mappings to the circle, and Brezis's Favorite Open Problem 5.4 in the same range of $s$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/brezis-mironescu-circle-minimizers"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2606.15713"},{"locator_id":"native-3","kind":"artifact","url":"https://arxiv.org/abs/2607.23598"}],"metadata":{"name":"Brezis-Mironescu Open Problems 23 and 24 on Minimizing Maps to the Circle","slug":"brezis-mironescu-circle-minimizers","field":"Calculus of variations","model":"ChatGPT","ai_role":"The LLM usage note says the authors used ChatGPT to assist with conceptualization and computations, with all mathematical validation their own.","posed_by":"Haim Brezis, Petru Mironescu","citations":null,"statement":"For $s \\in (1/4,1)$ and any degree, the only $W^{s,1/s}$-minimizers among maps $\\mathbb{S}^1 \\to \\mathbb{S}^1$ are Blaschke products. This resolves Open Problems 23 and 24 of Brezis and Mironescu's book on mappings to the circle, and Brezis's Favorite Open Problem 5.4 in the same range of $s$.","resolution":"partial","short_name":"Circle minimizers","solve_date":"2026-06-14","solve_type":"proved","source_url":"https://arxiv.org/abs/2606.15713","year_posed":2021,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":"in the range s in (1/4,1); Brezis's Problem 5.4 outside that range is untouched","source_name":"arXiv:2606.15713 - On minimizing W^{s,1/s}-maps between circles","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Numbered open problems from the Brezis-Mironescu monograph plus an entry on Brezis's own favourite-problems list, both recognized reference lists in analysis.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Dorian Martino\",\"Katarzyna Mazowiecka\",\"Armin Schikorra\"]"},"metadata_root":"sha256:5c410c1fa5618467aa6a58b11c2bfce011a0cdd6fa82ed4123b57d9281a89ecb","content_root":"sha256:b5082653ffe07ca12e41e60aa4753dcafefb91ba7ced0f5a4f4c7c8e7cc2bbf4","availability":"reference_only","row_root":"sha256:d408327e524b7b17b8f1c892c1ee9b20bd8dc0476f9c9dd10ecd8f5ed7215bdd"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:brezis-open-problem-5-6-fourier-summation","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Brezis's Open Problem 5.6 on Universal Fourier Summation","summary":"Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does $\\sum_n \\sigma_{n,\\varepsilon} n |\\hat f(n)|^2 \\to \\deg f$ hold for Holder maps below the threshold? No. For every $0 < \\alpha < 1/3$ there is an $f \\in C^{0,\\alpha}(S^1;S^1)$ for which the sum fails to converge to $\\deg f$, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all $p > 3$. The endpoint $C^{0,1/3}$ is left unresolved.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/brezis-open-problem-5-6-fourier-summation"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.23598"}],"metadata":{"name":"Brezis's Open Problem 5.6 on Universal Fourier Summation","slug":"brezis-open-problem-5-6-fourier-summation","field":"Harmonic analysis","model":"ChatGPT","ai_role":"The disclosure states that ChatGPT generated preliminary drafts of the proofs in the manuscript. The author then verified each argument in detail, revised the proofs where necessary, checked the cited sources, determined the final formulation of all results, and takes sole responsibility for the content.","posed_by":"Haim Brezis","citations":null,"statement":"Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does $\\sum_n \\sigma_{n,\\varepsilon} n |\\hat f(n)|^2 \\to \\deg f$ hold for Holder maps below the threshold? No. For every $0 < \\alpha < 1/3$ there is an $f \\in C^{0,\\alpha}(S^1;S^1)$ for which the sum fails to converge to $\\deg f$, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all $p > 3$. The endpoint $C^{0,1/3}$ is left unresolved.","resolution":"partial","short_name":"Brezis OP 5.6","solve_date":"2026-07-26","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.23598","year_posed":null,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":"negative below the 1/3 threshold; the endpoint case is still open","source_name":"arXiv:2607.23598 - Nonexistence of universal Fourier summation formulas for the degree below the Holder threshold","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A numbered entry on Brezis's published list of favourite open problems, a recognized problem list in analysis, with the degree-and-Fourier-moduli circle of questions attached to it.","verification_note":"Single-author arXiv preprint; the construction combines degree-zero quotients of Blaschke factors with a Baire category argument. Not yet peer-reviewed.","human_collaborators":"[\"Michal Cieszynski\"]"},"metadata_root":"sha256:32b8721a5dba107b59cd4ab396baedf9eb68094ff965f16a3d297a101d0c3692","content_root":"sha256:a1822172b9924ca9319c669fe6c93727c38016922c8d1941cebc27b5d39d75a2","availability":"reference_only","row_root":"sha256:30a35cd6e1596797d9761acae415d9c9bfa99256082d8aebcacd22ac1db2290c"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:brualdi-interchange-graph-hamiltonicity","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Brualdi's Question on Hamiltonicity of Interchange Graphs","summary":"The interchange graph $G(R,S)$ has the $(0,1)$-matrices with row sums $R$ and column sums $S$ as vertices, adjacent when they differ by a single $2\\times 2$ interchange. Brualdi asked whether $G(R,S)$ is always Hamiltonian. It satisfies more: it is maximally Hamiltonian, Hamilton-laceable when bipartite and Hamilton-connected when not.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/brualdi-interchange-graph-hamiltonicity"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.13165"}],"metadata":{"name":"Brualdi's Question on Hamiltonicity of Interchange Graphs","slug":"brualdi-interchange-graph-hamiltonicity","field":"Combinatorial matrix theory","model":"Claude, GPT/Codex","ai_role":"The declaration says the computational search and verification programs and the Lean 4 formalization were developed with AI-assisted tools under author direction, and that no AI system is an author. The structural induction carrying the proof is the authors'.","posed_by":"Richard A. Brualdi","citations":null,"statement":"The interchange graph $G(R,S)$ has the $(0,1)$-matrices with row sums $R$ and column sums $S$ as vertices, adjacent when they differ by a single $2\\times 2$ interchange. Brualdi asked whether $G(R,S)$ is always Hamiltonian. It satisfies more: it is maximally Hamiltonian, Hamilton-laceable when bipartite and Hamilton-connected when not.","resolution":"resolved","short_name":"Interchange graphs","solve_date":"2026-07-14","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.13165","year_posed":1980,"field_group":"Combinatorics","model_maker":"Anthropic / OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.13165 - Interchange graphs of (0,1)-matrices are maximally Hamiltonian","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A long-standing question of Brualdi in combinatorial matrix theory, standard background for anyone working with interchange classes of 0-1 matrices.","verification_note":"The paper reports a Lean 4 formalization alongside computational search and verification programs. We have not compiled it. arXiv preprint, not yet peer-reviewed.","human_collaborators":"[\"Jeffrey S. Baggett\",\"Huiya Yan\"]"},"metadata_root":"sha256:79d5ae403c45866837bca959d4dd1ca294ff6364b7baedd91f022d70997ad370","content_root":"sha256:c4df738c2c93d5f3b4947f22a1ae4f2d86797a1ff4943ddae1900bdca7078ea3","availability":"reference_only","row_root":"sha256:1443fb6da5b006fe1cd41671bc3bb0ef8a48d7b0b0a88c1052ba0a50683ad8b1"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:bruhat-hypercube-intervals","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Large Hypercube Intervals in Bruhat Order","summary":"How large can a Bruhat interval in $S_n$ that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension $O(n \\log n)$ for $n$ a power of 2, matching the largest possible dimension up to a constant - in the problem circle around the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/bruhat-hypercube-intervals"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2601.01235"}],"metadata":{"name":"Large Hypercube Intervals in Bruhat Order","slug":"bruhat-hypercube-intervals","field":"Coxeter combinatorics","model":"AlphaEvolve","ai_role":"AlphaEvolve, searching evolutionarily rather than exhaustively, \"produced a pattern which performed well for the n tested, and which we show works well for general n\" - the agent found the construction, the humans proved it works in general.","posed_by":null,"citations":null,"statement":"How large can a Bruhat interval in $S_n$ that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension $O(n \\log n)$ for $n$ a power of 2, matching the largest possible dimension up to a constant - in the problem circle around the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.","resolution":"partial","short_name":"Bruhat hypercubes","solve_date":"2026-01-03","solve_type":"proved","source_url":"https://arxiv.org/abs/2601.01235","year_posed":null,"field_group":"Combinatorics","model_maker":"Google DeepMind","publication":"preprint","result_note":"Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.","source_name":"arXiv","significance":10,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A question in the active problem circle around combinatorial invariance of Kazhdan-Lusztig polynomials, pursued by leading figures of the field.","verification_note":null,"human_collaborators":"[\"Jordan Ellenberg\",\"Nicolas Libedinsky\",\"David Plaza\",\"José Simental\",\"Geordie Williamson\"]"},"metadata_root":"sha256:5b22cdff6e1dbe287dfa7c9f5f5c6dbe32248cb3da4aaed60a933ebbc7670dd0","content_root":"sha256:a763cd390df29bf98126f22c2011dcdc12ca61653d9da4de990646d3be03c02c","availability":"reference_only","row_root":"sha256:27b0db5d7b923cf1c2f0d0bc6e1157a002e78005c50f00e17ad966eefc3992ef"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:canonical-closure-completeness-conp","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Completeness of Canonical Closure Representations Is coNP-Complete","summary":"A finite closure system can be given by implications or by a list of subsets closed under intersection. Deciding whether one specification of each kind defines the same family had remained open in several settings; the paper proves the problem coNP-complete.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/canonical-closure-completeness-conp"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.18534"}],"metadata":{"name":"Completeness of Canonical Closure Representations Is coNP-Complete","slug":"canonical-closure-completeness-conp","field":"Computational Complexity, Closure Systems","model":"GPT-5.6 Pro","ai_role":"The disclosure states the main proof was obtained by GPT-5.6 Pro through ChatGPT and checked by the author.","posed_by":"Open in the closure-systems literature","citations":null,"statement":"A finite closure system can be given by implications or by a list of subsets closed under intersection. Deciding whether one specification of each kind defines the same family had remained open in several settings; the paper proves the problem coNP-complete.","resolution":"resolved","short_name":"Closure completeness","solve_date":"2026-07-20","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.18534","year_posed":null,"field_group":"Theoretical computer science","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Settles the complexity of a decision problem left open across several formulations in the closure-systems literature.","verification_note":"No independent review; the author reports checking the model's proof himself. Preprint, not refereed.","human_collaborators":"[\"Mikhail Babin\"]"},"metadata_root":"sha256:3e255f525213aac1b45d885477240c0b63fcef448a1c197057c67822350f96a7","content_root":"sha256:9b34de9df2ba15c17b03e87b38174ce338c10d23f40aabf4ad89d1d4356b0d73","availability":"reference_only","row_root":"sha256:cf56ac1c1b40f5882defd624d6ef4e6cc0f89c1852f79deaa7b6de6ef2ae9a19"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:carbery-almost-orthogonality","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Carbery's Almost-Orthogonality Inequality in Lp","summary":"For $p \\ge 2$, does Carbery's proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power $2$ - and if not, what is the largest possible exponent?","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/carbery-almost-orthogonality"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.05192"}],"metadata":{"name":"Carbery's Almost-Orthogonality Inequality in Lp","slug":"carbery-almost-orthogonality","field":"Functional analysis","model":"Grok Heavy, Grok 4.20 Heavy","ai_role":"The authors knew a counterexample should exist from unstructured brute-force search; Grok produced a construction with a clear structural pattern, which revealed the optimal exponent p'.","posed_by":"Anthony Carbery","citations":null,"statement":"For $p \\ge 2$, does Carbery's proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power $2$ - and if not, what is the largest possible exponent?","resolution":"resolved","short_name":"Carbery inequality","solve_date":"2026-05","solve_type":"disproved","source_url":"https://arxiv.org/abs/2605.05192","year_posed":2009,"field_group":"Analysis","model_maker":"xAI","publication":"preprint","result_note":"exponent 2 fails for every p > 2; the sharp exponent p' form is proved for integer p ≥ 2","source_name":"arXiv:2605.05192 - Almost-orthogonality in Lp spaces: a case study with Grok","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A named Carbery question in harmonic analysis.","verification_note":"Author-checked public arXiv preprint. Not yet peer-reviewed.","human_collaborators":"[\"Ziang Chen\",\"Jaume de Dios Pont\",\"Paata Ivanisvili\",\"Jose Madrid\",\"Haozhu Wang\"]"},"metadata_root":"sha256:37b467d4afd6b57a3e33779c385fc81d9cde1c877fc97587e8ea807b317716e6","content_root":"sha256:6d9ca97a82dcdb8593a04864fa5a1f028fd50ddfe0a53aead15663d88f1c66c9","availability":"reference_only","row_root":"sha256:1beefb16957b5158ec237ae2a7e141a22aaf344d6999465aafc0fbcb45b95335"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:carlson-depth-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Carlson's Associated-Prime Depth Conjecture","summary":"Is the depth of the mod-$p$ cohomology ring of every finite group realized as the dimension of one of its associated primes? For $G = \\operatorname{SmallGroup}(128, 859)$ over $\\overline{\\mathbb{F}}_2$ the ring has depth $2$ while every associated-prime quotient has dimension at least $3$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/carlson-depth-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.23732"}],"metadata":{"name":"Carlson's Associated-Prime Depth Conjecture","slug":"carlson-depth-conjecture","field":"Group cohomology","model":"TARS agent system","ai_role":"The candidate group was found by the TARS agent system (foundation model not disclosed); the counterexample is certified by exact GAP/Singular computations audited by the human authors.","posed_by":"Jon F. Carlson","citations":null,"statement":"Is the depth of the mod-$p$ cohomology ring of every finite group realized as the dimension of one of its associated primes? For $G = \\operatorname{SmallGroup}(128, 859)$ over $\\overline{\\mathbb{F}}_2$ the ring has depth $2$ while every associated-prime quotient has dimension at least $3$.","resolution":"resolved","short_name":"Carlson depth","solve_date":"2026-07-26","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.23732","year_posed":1995,"field_group":"Algebra","model_maker":null,"publication":"preprint","result_note":null,"source_name":"arXiv:2607.23732 - An exact counterexample to Carlson's associated-prime depth conjecture","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Carlson's 1995 conjecture in modular representation theory.","verification_note":"Exact computational certificate (verifier and certificates ship with the paper, checkable with the Python standard library alone) plus a human proof audit. Not yet peer-reviewed.","human_collaborators":"[\"Xinan Dai\",\"Wenhao Deng\",\"Yingdong Shi\",\"Tailin Wu\",\"Yuchen Yang\"]"},"metadata_root":"sha256:5e711cdba10aedf5fb1b9cd0bcbbc1e1d17e2a05e409f10f3664dd1794e6da20","content_root":"sha256:a5ff4f749c443f795346c3cc56270a16dbe3d413df3012c4cef23929567d5d8e","availability":"reference_only","row_root":"sha256:419a898f2fe2dff594a9feaf819808e47ae8c1061c963c730fafd66025e485a9"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:carrasco-conjecture-ozf","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Carrasco's Conjecture on the O'Shea-Zames-Falb Test","summary":"Is the O'Shea-Zames-Falb multiplier test necessary for robust stability of Lur'e systems with slope-restricted nonlinearities, as conjectured by Carrasco? No: there is a stable Lur'e interconnection, certified by a full-block multiplier, that admits no OZF multiplier.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/carrasco-conjecture-ozf"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.23599"}],"metadata":{"name":"Carrasco's Conjecture on the O'Shea-Zames-Falb Test","slug":"carrasco-conjecture-ozf","field":"Control theory","model":"ChatGPT 5.5","ai_role":"A finite-horizon counterexample provided by ChatGPT 5.5 motivated the construction; the author built and certified the full interconnection.","posed_by":"Joaquin Carrasco","citations":null,"statement":"Is the O'Shea-Zames-Falb multiplier test necessary for robust stability of Lur'e systems with slope-restricted nonlinearities, as conjectured by Carrasco? No: there is a stable Lur'e interconnection, certified by a full-block multiplier, that admits no OZF multiplier.","resolution":"resolved","short_name":"Carrasco/OZF conjecture","solve_date":"2026-07-26","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.23599","year_posed":null,"field_group":"Differential equations","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.23599 - Existence of stable Lur'e systems for which the O'Shea-Zames-Falb stability test fails","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A known conjecture on the standard multiplier test in absolute stability theory.","verification_note":"Single-author arXiv note; not yet peer-reviewed.","human_collaborators":"[\"Andrey Kharitenko\"]"},"metadata_root":"sha256:0742ce8ad7a4d331547c348a121358503545686f1267bd60f8c142c6d6d8e510","content_root":"sha256:6c486127c21ecef5d7d7200e2fe52f0a2c3cf12fe905d175e7d25cde24197ca6","availability":"reference_only","row_root":"sha256:edb5f3f512e3ae5f86dbde7c76868155bacf108cbebbfd31f00e65ce8651411f"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:cerny-one-cluster","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Černý Conjecture for One-Cluster Automata","summary":"Does every synchronizing one-cluster automaton on $n$ states admit a reset word of length at most $(n-1)^2$? The new bound $(m-1)(n-1) + m\\ell \\le (n-1)^2$ settles the one-cluster case of the Černý conjecture.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/cerny-one-cluster"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.19675"}],"metadata":{"name":"Černý Conjecture for One-Cluster Automata","slug":"cerny-one-cluster","field":"Automata theory","model":"OpenAI Codex (GPT-5.6 Sol Ultra)","ai_role":"The annular spectral descent argument was obtained in interaction with OpenAI Codex running GPT-5.6 Sol Ultra and verified by the author; the paper also proves the positive-level relative-extending-word conjecture of Kisielewicz, Kowalski and Szykuła.","posed_by":null,"citations":null,"statement":"Does every synchronizing one-cluster automaton on $n$ states admit a reset word of length at most $(n-1)^2$? The new bound $(m-1)(n-1) + m\\ell \\le (n-1)^2$ settles the one-cluster case of the Černý conjecture.","resolution":"resolved","short_name":"Černý, one-cluster","solve_date":"2026-07-25","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.19675","year_posed":2016,"field_group":"Theoretical computer science","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.19675 - The Černý conjecture for one-cluster automata via annular spectral descent","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A recognized major case of the Černý conjecture, automata theory's oldest open problem.","verification_note":"Author-verified arXiv preprint. Not yet peer-reviewed.","human_collaborators":"[\"Yinfeng Zhu\"]"},"metadata_root":"sha256:dd3629dc7f519ac3ab5960e897773d9e3a0bdd7671004da57e137200db3498f4","content_root":"sha256:167bd4e76c4ae3c7d64bd73e7cc49889c5abba2f2cb7a792c0442cb44b67eaa3","availability":"reference_only","row_root":"sha256:d2575430151c4d6aea729911a45d3d0b7e0ae13dad244cd9076b4fa126b66bfb"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:cesaro-means-firmly-nonexpansive","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Strong Convergence of Cesaro Means of Firmly Nonexpansive Iterates","summary":"Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/cesaro-means-firmly-nonexpansive"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.25491"}],"metadata":{"name":"Strong Convergence of Cesaro Means of Firmly Nonexpansive Iterates","slug":"cesaro-means-firmly-nonexpansive","field":"Fixed point theory","model":"ChatGPT 5.5","ai_role":"The acknowledgement credits use of ChatGPT 5.5 as leading eventually to the realization of the explicit curve, which is the object the counterexample is built from.","posed_by":null,"citations":null,"statement":"Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.","resolution":"resolved","short_name":"Cesaro means","solve_date":"2026-05-25","solve_type":"disproved","source_url":"https://arxiv.org/abs/2605.25491","year_posed":1975,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2605.25491 - Cesaro means of firmly nonexpansive iterates need not converge strongly","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A natural follow-up to the Genel-Lindenstrauss counterexample that had stood since 1975, in the convex optimization and monotone operator literature.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Heinz H. Bauschke\",\"Tran Thanh Tung\"]"},"metadata_root":"sha256:fdc9b1daa0d5cf8e7cecf64a0fd3bb261b5117d22a2561bc2a4d89f4500b1573","content_root":"sha256:5c34d36416fc063ddb35c546594751b07c0a59425e51335d8da36a89572092d6","availability":"reference_only","row_root":"sha256:a15f9be24b4e06551bbeae559b100c2b653e1b25c5f0359e9d3639ca37e85b01"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:chafai-dadoun-youssef-log-energy-monotonicity","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Chafai-Dadoun-Youssef Questions on Logarithmic Energy Monotonicity","summary":"Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a finite-energy Wigner counterexample and a one-parameter family of Gaussian-regularised Bernoulli entry laws answer both questions negatively.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/chafai-dadoun-youssef-log-energy-monotonicity"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.24170"}],"metadata":{"name":"Chafai-Dadoun-Youssef Questions on Logarithmic Energy Monotonicity","slug":"chafai-dadoun-youssef-log-energy-monotonicity","field":"Random matrix theory","model":"ChatGPT Pro 5.5, ChatGPT 5.6 Sol Ultra","ai_role":"The disclosure says the tools were used for ideation, technical help, editing and checking the proofs, and puts the contribution at the level of a co-author, while the author retains responsibility for mistakes. It does not separate which of the two counterexamples came from where.","posed_by":"Djalil Chafai, Benjamin Dadoun, Pierre Youssef","citations":null,"statement":"Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a finite-energy Wigner counterexample and a one-parameter family of Gaussian-regularised Bernoulli entry laws answer both questions negatively.","resolution":"resolved","short_name":"Log-energy monotonicity","solve_date":"2026-07-27","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.24170","year_posed":null,"field_group":"Probability & statistics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.24170 - On non-monotonicity of logarithmic energy for random matrices","significance":10,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Two numbered questions from a recent random-matrix paper, documented but read within one community.","verification_note":"Single-author arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Theodoros Assiotis\"]"},"metadata_root":"sha256:f85642d93ffcfaf2cd2d43902c38885b8855929929fa4e40cef0feb567d7a019","content_root":"sha256:751f476284f8d98d8c9b3822590ad0e5a2adca3b3d56587327bce0edcbe47735","availability":"reference_only","row_root":"sha256:6004556a19f525a3ef3a42326b27ff1901618503047534cb2fd6780682f8a394"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:chen-gendron-spin-parity","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Chen-Gendron Spin-Parity Identity for k-Differentials","summary":"For odd $k$ with $\\gcd(n,k) = \\gcd(n+1,k) = 1$, is $N_k(n) \\equiv \\lfloor (k+1)/4 \\rfloor \\pmod 2$, where $N_k(n)$ counts pairs $1 \\le b_i \\le (k-1)/2$ with $b_1 + b_2 \\ge (k+1)/2$ and $b_2 \\equiv n b_1 \\pmod k$? Conjectured by Chen and Gendron; its proof removes a conditional step in the genus-zero and genus-one spin-parity classification.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/chen-gendron-spin-parity"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2602.03722"}],"metadata":{"name":"Chen-Gendron Spin-Parity Identity for k-Differentials","slug":"chen-gendron-spin-parity","field":"Flat surfaces & moduli","model":"AxiomProver","ai_role":"Proved by the AxiomProver system with a Lean-checked core.","posed_by":"Dawei Chen & Quentin Gendron","citations":null,"statement":"For odd $k$ with $\\gcd(n,k) = \\gcd(n+1,k) = 1$, is $N_k(n) \\equiv \\lfloor (k+1)/4 \\rfloor \\pmod 2$, where $N_k(n)$ counts pairs $1 \\le b_i \\le (k-1)/2$ with $b_1 + b_2 \\ge (k+1)/2$ and $b_2 \\equiv n b_1 \\pmod k$? Conjectured by Chen and Gendron; its proof removes a conditional step in the genus-zero and genus-one spin-parity classification.","resolution":"resolved","short_name":"Spin parity identity","solve_date":"2026-02-03","solve_type":"proved","source_url":"https://arxiv.org/abs/2602.03722","year_posed":2022,"field_group":"Geometry & topology","model_maker":null,"publication":"preprint","result_note":null,"source_name":"arXiv:2602.03722 - Parity of k-differentials in genus zero and one","significance":10,"verification":"lean-checked","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A concrete identity conjectured in the strata-of-differentials literature.","verification_note":"Core argument Lean-checked, with an expert-written exposition. Tier: the Lean-checked core comes from the proving system itself; the statement correspondence is not independently audited.","human_collaborators":"[]"},"metadata_root":"sha256:317a473fbf0a2ef043495f99fafc3395de461744d5a52737ff16c8934e3960cb","content_root":"sha256:badda2e41a0eaa2b225f4e35143050f78cd8ee983230f45d38e5946add890a30","availability":"reference_only","row_root":"sha256:037e160d0f3d394625b69f668d7310e39e5294b382b4df9dc7a7cc163f981153"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:chen-lawrencenko-conjectures","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Chen-Lawrencenko Conjectures on Cyclic Colorations","summary":"A cyclic coloration of a triangulation of a closed 2-manifold gives the faces around every vertex distinct colors. Chen and Lawrencenko made two conjectures about the cyclic chromatic number of minimal triangulations in 1999. Their second is proved here and their first disproved.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/chen-lawrencenko-conjectures"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.06863"}],"metadata":{"name":"The Chen-Lawrencenko Conjectures on Cyclic Colorations","slug":"chen-lawrencenko-conjectures","field":"Topological graph theory","model":"GPT-5.6 Pro","ai_role":"\"We succeed, based on our extensive interactions with GPT-5.6 Pro, in proving and disproving, respectively, Chen and Lawrencenko's second and first conjectures.\" The acknowledgement places the model in the exploratory and proof-development stages, with all suggestions substantially revised, corrected and independently verified by the author.","posed_by":"Beifang Chen, Serge Lawrencenko","citations":null,"statement":"A cyclic coloration of a triangulation of a closed 2-manifold gives the faces around every vertex distinct colors. Chen and Lawrencenko made two conjectures about the cyclic chromatic number of minimal triangulations in 1999. Their second is proved here and their first disproved.","resolution":"resolved","short_name":"Chen-Lawrencenko conjectures","solve_date":"2026-08-07","solve_type":"disproved","source_url":"https://arxiv.org/abs/2608.06863","year_posed":1999,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":"One conjecture each way: the second proved, the first disproved. Two further Chen-Lawrencenko conjectures remain open and are flagged as such in the paper.","source_name":"arXiv","significance":12,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Named conjectures standing since 1999 in topological graph theory, with a documented line through Enomoto and Hornak on cyclic colorings; specialist reach.","verification_note":"A preprint days old, with no independent review.","human_collaborators":"[\"John M. Campbell\"]"},"metadata_root":"sha256:ec97c5aabca641ef7b76d69affd772325d5cf647bad67fad158039ec0f90c59c","content_root":"sha256:01bdf90da3fb6a635c5dcd30dd94b896efc0704f5d1045fd15ddf137ea268838","availability":"reference_only","row_root":"sha256:1f896ff331ca1320813c5ee746d29368dc1be643dc1f9f8b51067af373797b1a"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:chern-class-positivity-symmetric-powers","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Positivity of Chern Classes of Symmetric Powers","summary":"The total Chern class of $\\mathrm{Sym}^d(\\mathbb{C}^n)$ as a torus representation is a symmetric polynomial whose coefficients were conjectured positive, with a binomial log-concavity refinement. Both are established.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/chern-class-positivity-symmetric-powers"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.25271"}],"metadata":{"name":"Positivity of Chern Classes of Symmetric Powers","slug":"chern-class-positivity-symmetric-powers","field":"Enumerative geometry","model":"AlphaEvolve, ChatGPT 5.5 Pro","ai_role":"The paper calls itself a case study in synchronized AI-assisted mathematics and separates the two roles: AlphaEvolve served first as a searching engine and later as what the authors call a detective for structural recurrences, including a section-long account of one conjecture; ChatGPT 5.5 Pro acted as an interactive assistant for symbolic derivations.","posed_by":"classical enumerative geometry","citations":null,"statement":"The total Chern class of $\\mathrm{Sym}^d(\\mathbb{C}^n)$ as a torus representation is a symmetric polynomial whose coefficients were conjectured positive, with a binomial log-concavity refinement. Both are established.","resolution":"resolved","short_name":"Chern class positivity","solve_date":"2026-05-24","solve_type":"proved","source_url":"https://arxiv.org/abs/2605.25271","year_posed":null,"field_group":"Geometry & topology","model_maker":"Google DeepMind / OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2605.25271 - Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Positivity of Chern classes of symmetric powers, a classical question in enumerative geometry with a concrete polynomial formulation.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Gergely Berczi\",\"Laszlo M. Feher\"]"},"metadata_root":"sha256:b5ab200bf54244770402b11c1c620c5ba092b3316273fefd72233b16bf351bdb","content_root":"sha256:e1c1b0a37b4a9d0f441051bebc7b154b92e8600fe9bca2b24f10e74d29b36811","availability":"reference_only","row_root":"sha256:2f0fba31e6a10c92a6e28fa9d6dd33a4cb78b096d338e191b2392b9745152af8"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:chernoff-density-strong-log-concavity","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Strong Log-Concavity of Chernoff's Density","summary":"Is the density of Chernoff's distribution - the law of $\\operatorname{argmax}_t \\{W(t) - t^2\\}$ for two-sided Brownian motion $W$ - strongly log-concave, as conjectured by Balabdaoui and Wellner in 2014?","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/chernoff-density-strong-log-concavity"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.18619"}],"metadata":{"name":"Strong Log-Concavity of Chernoff's Density","slug":"chernoff-density-strong-log-concavity","field":"Probability & statistics","model":"GPT-5.6 Sol","ai_role":"The proof was generated in its entirety by GPT-5.6 Sol; the authors checked it and prepared the manuscript.","posed_by":"Fadoua Balabdaoui & Jon A. Wellner","citations":null,"statement":"Is the density of Chernoff's distribution - the law of $\\operatorname{argmax}_t \\{W(t) - t^2\\}$ for two-sided Brownian motion $W$ - strongly log-concave, as conjectured by Balabdaoui and Wellner in 2014?","resolution":"resolved","short_name":"Chernoff log-concavity","solve_date":"2026-07-21","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.18619","year_posed":2014,"field_group":"Probability & statistics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.18619 - Chernoff's density is strongly log-concave","significance":10,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A specialist question in shape-constrained statistics.","verification_note":"Author-checked public arXiv preprint. Not yet peer-reviewed.","human_collaborators":"[\"Xianyang Zhang\",\"Quan Zhou\"]"},"metadata_root":"sha256:81cf893c122e02ba7d0a15d9171be38492f04ffa014256931532bd5346120385","content_root":"sha256:5ad78c04219a3b652171d147edb94c8a0e11539902b17b4dd1892424b46370cd","availability":"reference_only","row_root":"sha256:249474c1a9df689eccb4e4b90d46002648d77a12253a724391b0c03fe77e166f"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:chip-firing-middle-stair","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Middle Stair of Parallel Chip-Firing","summary":"Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/chip-firing-middle-stair"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.04153"}],"metadata":{"name":"The Middle Stair of Parallel Chip-Firing","slug":"chip-firing-middle-stair","field":"Combinatorial dynamics","model":"GPT-5.6 Sol","ai_role":"The paper presents the proof plainly as found by the model: \"The proof presented in Section 3 was found by the large language model GPT-5.6-Sol. The authors verified the resulting argument and take full responsibility.\"","posed_by":"David Ji, Michael Li, Daniel Wang","citations":null,"statement":"Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.","resolution":"resolved","short_name":"Chip-firing middle stair","solve_date":"2026-08-04","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.04153","year_posed":2024,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":12,"verification":"expert-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A documented 2024 conjecture unifying thirty years of partial results on the devil's staircase, known within the chip-firing community.","verification_note":"Beyond the authors' own verification, the acknowledgments thank David Ji and Michael Li, two of the conjecture's posers, for assisting with reviewing the proof.","human_collaborators":"[\"Daniel Wang\",\"Nathan Lannan\"]"},"metadata_root":"sha256:f831c2572942e366e1cf4e953cb981088d37d28462730a8d50d3fce762295671","content_root":"sha256:8acc10eb5f8eb89be18b9977cc178ef50d4399db4453145e39d7e7ba1327acb0","availability":"reference_only","row_root":"sha256:7be96d4ed75bae822965182c08d9d68bc0be2260ba87f24e4c97e7488636da0b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:ciliberto-non-movable-divisor","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Ciliberto et al. Question on Effective Divisors of Positive Self-Intersection","summary":"Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa and Testa asked a question about effective divisors of positive self-intersection on smooth projective surfaces. The answer is negative, witnessed by a very non-movable effective divisor.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/ciliberto-non-movable-divisor"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2605.20594"}],"metadata":{"name":"The Ciliberto et al. Question on Effective Divisors of Positive Self-Intersection","slug":"ciliberto-non-movable-divisor","field":"Algebraic geometry","model":"ChatGPT 5.5 Pro and Rethlas","ai_role":"The author states the main result is obtained by generative AI, particularly ChatGPT 5.5 Pro and the Rethlas system. This is the fourth note in a short series where the same author publishes machine-obtained answers to named questions in birational geometry, each with the same one-sentence disclosure.","posed_by":"Ciro Ciliberto, Andreas Leopold Knutsen, John Lesieutre, Victor Lozovanu, Rick Miranda, Yusuf Mustopa, Damiano Testa","citations":null,"statement":"Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa and Testa asked a question about effective divisors of positive self-intersection on smooth projective surfaces. The answer is negative, witnessed by a very non-movable effective divisor.","resolution":"resolved","short_name":"Very non-movable divisor","solve_date":"2026-05-20","solve_type":"disproved","source_url":"https://arxiv.org/abs/2605.20594","year_posed":null,"field_group":"Geometry & topology","model_maker":null,"publication":"preprint","result_note":null,"source_name":"arXiv:2605.20594 - An example of a very non-movable effective divisor","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A question posed by seven authors on positivity of effective divisors on surfaces, answered negatively.","verification_note":"The counterexample is an explicit divisor, so it is checkable directly. arXiv preprint, not peer-reviewed.","human_collaborators":"[\"Jihao Liu\"]"},"metadata_root":"sha256:c6d171e3b168719eb6d3d015fcc5516f1f152c2e06c2b0e8bcf381367d0e3311","content_root":"sha256:50fae2578b8a61874bec70d78122efe84c41908648a6fbb3d4eb36bde2e7a68d","availability":"reference_only","row_root":"sha256:cb57fe67a42b949a23c4da9715f40b09a3c876b0339fb09c4cb78128896fc195"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:classical-smith-ward-problem","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Classical Smith-Ward Problem","summary":"The Smith-Ward theorem realizes the first $k$ essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of $k$, equivalently whether the identity map on a three-dimensional operator system $\\mathrm{span}\\{1,q(D),q(K)\\}$ in the Calkin algebra always lifts. It need not: an explicit three-dimensional hyperrigid operator system has no unital completely positive lift, and its dual is the first three-dimensional operator system that fails to be exact.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/classical-smith-ward-problem"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.04274"}],"metadata":{"name":"The Classical Smith-Ward Problem","slug":"classical-smith-ward-problem","field":"Operator algebras","model":"ChatGPT","ai_role":"The one-line disclosure says the model was used to perform literature search and to accelerate the search for the operator system. Since that operator system is the counterexample, the contribution touches the mathematics, but the wording does not say the model found it.","posed_by":"R. R. Smith, J. D. Ward","citations":null,"statement":"The Smith-Ward theorem realizes the first $k$ essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of $k$, equivalently whether the identity map on a three-dimensional operator system $\\mathrm{span}\\{1,q(D),q(K)\\}$ in the Calkin algebra always lifts. It need not: an explicit three-dimensional hyperrigid operator system has no unital completely positive lift, and its dual is the first three-dimensional operator system that fails to be exact.","resolution":"resolved","short_name":"Smith-Ward problem","solve_date":"2026-07-13","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.04274","year_posed":null,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":"Harris had settled the generalized problem in dimension four; this reaches dimension three","source_name":"arXiv:2607.04274 - A Three-Dimensional Operator System without the Smith-Ward Property","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A named lifting problem in operator algebras with a documented line of attack through Paulsen and more recently Harris.","verification_note":"Single-author arXiv preprint (v2), isolating and strengthening an argument of Harris; not yet peer-reviewed.","human_collaborators":"[\"Marcel Scherer\"]"},"metadata_root":"sha256:90916e89aebf10e1e21080411c90ec475ed1881be47dc2559bc49583a5d6375c","content_root":"sha256:b2d416af1ec1382e93e310709037bc8130bfbe0682b9c1ca7b9d2a5812d1473d","availability":"reference_only","row_root":"sha256:24a80c3b4c3851f3e6761d76b625b7ae25860548c76049641b99d5701f19584b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:cohen-cyclic-number-conjectures","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Cohen's 22 Conjectures on Cyclic Numbers","summary":"22 conjectures of Cohen about cyclic numbers (integers with $\\gcd(n, \\varphi(n)) = 1$) settled at once - 16 proved, 6 disproved - together with a complete resolution of a related OEIS problem on sequences whose running averages are Fibonacci numbers (Fried's Conjecture 2).","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/cohen-cyclic-number-conjectures"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2509.26138"}],"metadata":{"name":"Cohen's 22 Conjectures on Cyclic Numbers","slug":"cohen-cyclic-number-conjectures","field":"Elementary number theory","model":"GPT-5","ai_role":"\"All proofs in the paper were assisted by GPT-5\" - a blanket credit across 22 resolutions, with no individual attribution, so the lower tier applies.","posed_by":"Cohen (OEIS conjectures); Fried","citations":null,"statement":"22 conjectures of Cohen about cyclic numbers (integers with $\\gcd(n, \\varphi(n)) = 1$) settled at once - 16 proved, 6 disproved - together with a complete resolution of a related OEIS problem on sequences whose running averages are Fibonacci numbers (Fried's Conjecture 2).","resolution":"retracted","short_name":"Cyclic number conjectures","solve_date":"2025-09-30","solve_type":"proved","source_url":"https://arxiv.org/abs/2509.26138","year_posed":null,"field_group":"Number theory","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":6,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"OEIS-attached conjectures with documented statements but a one-community audience; the bundle of 22 sits just above the single machine-generated-conjecture baseline.","verification_note":"Paper has been retracted from arXiv","human_collaborators":"[\"Duc Hieu Le\"]"},"metadata_root":"sha256:d957687b5b187929b023e40fd8fd34df10c4a9506be08f73bc0e8622fc6d77fb","content_root":"sha256:556a2cafcf3455dc9bb68cd9c14eb2d0c7e58e6b743fec87db08ca4f8f644f52","availability":"reference_only","row_root":"sha256:c5a66da08e55d46481852e108816a9708fb280535402e61af6a090161ea00cdd"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:cohn-elkies-bound-dimension-36","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Tightness of the Cohn-Elkies Bound in Dimension 36","summary":"Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension $36$ as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-$18$ modular forms for $\\Gamma_0(24)$, shows the two-point linear programming bound in dimension $36$ exceeds the density of the Kschischang-Pasupathy packing by a factor of at least $32.91$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/cohn-elkies-bound-dimension-36"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.11319"}],"metadata":{"name":"Tightness of the Cohn-Elkies Bound in Dimension 36","slug":"cohn-elkies-bound-dimension-36","field":"Sphere packing","model":"Claude Fable 5, Claude Opus 4.8, Codex (GPT-5.6)","ai_role":"The disclosure reports substantial assistance: the Claude models were used for the construction of the certificate itself, for the verification tooling and for drafting, and Codex was used as an independent cross-check of the certificate computations.","posed_by":"Henry Cohn, Noam Elkies","citations":null,"statement":"Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension $36$ as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-$18$ modular forms for $\\Gamma_0(24)$, shows the two-point linear programming bound in dimension $36$ exceeds the density of the Kschischang-Pasupathy packing by a factor of at least $32.91$.","resolution":"resolved","short_name":"Sphere packing in dim 36","solve_date":"2026-07-13","solve_type":"disproved","source_url":"https://arxiv.org/abs/2607.11319","year_posed":2003,"field_group":"Geometry & topology","model_maker":"Anthropic / OpenAI","publication":"preprint","result_note":"rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown","source_name":"arXiv:2607.11319 - A dual linear programming bound for sphere packing in dimension 36","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"The Cohn-Elkies linear program is the method behind the dimension 8 and 24 solutions, so where it provably cannot work is a live question in the sphere packing programme.","verification_note":"The result is an explicit dual-feasible certificate, so it is checkable in principle by evaluating the constructed function; the paper reports an independent cross-check of the computations by a second model. We have not reproduced it. arXiv preprint, not peer-reviewed.","human_collaborators":"[\"Rifat Jumagulov\"]"},"metadata_root":"sha256:5fb2d1edad9cfc7585e4dc8af60c1f7a4b0e99abb65298b7cc5c805056810aa5","content_root":"sha256:3926ce1608f4888d4f1f620212ea948062187bbc992ca49887c92ebe16e40180","availability":"reference_only","row_root":"sha256:ac8b4f1903b03154872431ec2313d967872de537781818a57dce4b9f39faeaa5"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:complementary-bell-fibers","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Finitude of the Fibers of Complementary Bell Numbers","summary":"Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers $f(n) = B_n(-1)$ take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result whose techniques connect to Wilf's conjecture on the vanishing of $f(n)$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/complementary-bell-fibers"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.00575"}],"metadata":{"name":"Finitude of the Fibers of Complementary Bell Numbers","slug":"complementary-bell-fibers","field":"Enumerative combinatorics","model":"GPT-5.6 Pro","ai_role":"Solved \"through our extensive interactions with GPT-5.6 Pro\" during the exploratory and proof-development stages; all AI-generated suggestions were substantially revised, corrected and independently verified by the author.","posed_by":"M. V. Subbarao, A. Verma","citations":null,"statement":"Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers $f(n) = B_n(-1)$ take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result whose techniques connect to Wilf's conjecture on the vanishing of $f(n)$.","resolution":"resolved","short_name":"Complementary Bell fibers","solve_date":"2026-08-01","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.00575","year_posed":1999,"field_group":"Combinatorics","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":12,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A named 1999 problem adjacent to Wilf's conjecture, with a real literature (Yang solved the first two problems of the same set in 2001).","verification_note":null,"human_collaborators":"[\"John M. Campbell\"]"},"metadata_root":"sha256:987f8ef5c0ac755b1395472b5cb4d30f2b7ec9e1bce523517329c8906526a856","content_root":"sha256:ee413ced3a454381f52df3de09d5c77546e949f382696ecd6ef77f55aa187b72","availability":"reference_only","row_root":"sha256:2aa90da0e84c28d08ead2b36ec3b961028637785cc6c736da5e740f88bd7bc50"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$","summary":"For rational-valued signals $f,g:C_{30}\\to\\mathbb Q$ with exact Fourier support $U_{30}$, equality of autocorrelations through order five is completely classified. After translating $g$, there are $\\alpha\\in\\mathbb Q(\\zeta_{30})^\\times$ and $z\\in\\mathbb Q(\\zeta_6)^\\times$, with $z\\bar z=1$, such that\n$$\n\\widehat f(u)=\\sigma_u(\\alpha),\\qquad\n\\widehat g(u)=\\sigma_u(z\\alpha)\n$$\nfor every $u\\in U_{30}$. Conversely, every such pair, extended by zero off $U_{30}$, is rational-valued and agrees through order five. Normalized parameters are translation-equivalent exactly modulo $\\mu_6$, and the sixth-order data agree exactly when $z^6=1$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/blob/b4bbfccdd508caa4a5a6e145abd78ac082195c16/paper/Autocorrelation_Phase_Lattices_on_Cyclic_Groups.pdf"},{"locator_id":"native-3","kind":"artifact","url":"https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups"},{"locator_id":"native-4","kind":"artifact","url":"https://arxiv.org/abs/2604.13310"},{"locator_id":"native-5","kind":"artifact","url":"https://github.com/aconsciousfractal/Gate-Disciplined-Computational-Mathematics"}],"metadata":{"name":"Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$","slug":"complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u","field":"Finite harmonic analysis / phase retrieval on cyclic groups","model":"GPT-5.6 Sol (Codex)","ai_role":"Following the self created protocol PAPP, that you can find here:https://github.com/aconsciousfractal/Gate-Disciplined-Computational-Mathematics\n GPT-5.6 Sol, operating through Codex under my direction, developed the central relation-lattice reduction, the phase-ratio normal form, and the Galois-descent argument that confines the relative phase to the norm-one torus in $\\mathbb Q(\\zeta_6)$. It also generated exact symbolic replays and the verification package. I selected the problem and scope, controlled the literature and claim boundary, directed repeated adversarial reviews, checked the mathematical outputs, and revised the manuscript after each finding. An earlier AI-agent package supplied preliminary computational observations; the workflow independently rederived and checked them before use.","posed_by":"Aaron Agulnick, Toby Busick-Warner","citations":null,"statement":"For rational-valued signals $f,g:C_{30}\\to\\mathbb Q$ with exact Fourier support $U_{30}$, equality of autocorrelations through order five is completely classified. After translating $g$, there are $\\alpha\\in\\mathbb Q(\\zeta_{30})^\\times$ and $z\\in\\mathbb Q(\\zeta_6)^\\times$, with $z\\bar z=1$, such that\n$$\n\\widehat f(u)=\\sigma_u(\\alpha),\\qquad\n\\widehat g(u)=\\sigma_u(z\\alpha)\n$$\nfor every $u\\in U_{30}$. Conversely, every such pair, extended by zero off $U_{30}$, is rational-valued and agrees through order five. Normalized parameters are translation-equivalent exactly modulo $\\mu_6$, and the sixth-order data agree exactly when $z^6=1$.","resolution":"candidate","short_name":"The $U_{30}$ phase-ambiguity question","solve_date":"2026-08-07","solve_type":"proved","source_url":"https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/blob/b4bbfccdd508caa4a5a6e145abd78ac082195c16/paper/Autocorrelation_Phase_Lattices_on_Cyclic_Groups.pdf","year_posed":2026,"field_group":"Analysis","model_maker":"OpenAI","publication":"announcement","result_note":"Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support $U_{30}$ and conjectured that it was not a complete classification because it did not use the full field $\\mathbb Q(\\zeta_{30})$. This work proves the complete classification. The larger field enlarges the common amplitude $\\alpha$, while every relative ambiguity remains a norm-one parameter in $\\mathbb Q(\\zeta_6)$. The result is a specialization of a theorem for every exact unit support $U_{6m}$.\n\nThe entry does not claim a complete parametrization for arbitrary supports: on the 255 automorphism-stable supports treated elsewhere in the paper, the broader result is a closing-degree classification. It does not treat noisy data or noncyclic groups, and it makes no novelty, priority, or firstness claim.","source_name":"Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports and the 2pq Orbit-Stable Classification","significance":4,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A support-specific conjecture stated in one April 2026 paper and answered four months later. Precisely posed and genuinely open - Agulnick and Busick-Warner write that they \"conjecture that it is not even a complete classification on this particular support\" - but with no literature behind it and a narrow specialist audience. Level with the other 2026 conjecture-in-a-recent-paper entries at 4.","verification_note":"No named independent domain expert has endorsed the theorem. The repository carries exact symbolic checks, fail-closed verification scripts, frozen manifests and an adversarial review report, but those are author-side and agent-side assurance, so this stays Unreviewed and Candidate.\n\nThis site ran its own checks, written from the statement rather than from the repository's scripts. Building a pair from a chosen $\\alpha$ and $z$: both inverse transforms are rational at all 30 points, the support is exactly $U_{30}$, the pair agrees through order five and differs at order six, and replacing $z$ by a sixth root of unity restores order-six agreement - the claimed $z^6=1$ boundary, exactly. The phase lattice was recomputed independently by Smith normal form: $\\mathbb Z^8/L$ has free rank 1 through order five and rank 0 at order six, so a one-parameter ambiguity survives order five and dies at six. And the Agulnick-Busick-Warner pair itself fits the classification - its Fourier ratio is Galois-equivariant with $z+\\bar z=13/7$ and $z\\bar z=1$, so $z=(13\\pm3\\sqrt{-3})/14$ lies in $\\mathbb Q(\\zeta_6)$.\n\nNot checked: completeness itself, which is the novelty. The converse direction, the lattice skeleton and the known example are all consistent with it without establishing it.","human_collaborators":"[]"},"metadata_root":"sha256:2646433004e7d29ccff24b3b40cdb4ae704991db19539438d3eb3630d21e41aa","content_root":"sha256:167609128f4cbad58f3a14295b9826ecd32c042490c40ebc0f18c1d16356d237","availability":"reference_only","row_root":"sha256:6d9826f43b4ca3ec91f9b7ab7c686e066ac0ca0dfe42e6322a14cb70c93548ee"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:composites-among-xi-7-n","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Composites Among $[\\xi 7^n]$ and Right-Truncatable Primes in Base 7","summary":"For every real $\\xi>0$ the sequence of integer parts $[\\xi 7^{n}]$, $n=0,1,2,\\dots$, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~$7$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/composites-among-xi-7-n"},{"locator_id":"native-2","kind":"artifact","url":"https://www.researchgate.net/publication/412294027_On_the_composites_among_x7"},{"locator_id":"native-3","kind":"artifact","url":"https://github.com/rwst/On-Composites"}],"metadata":{"name":"Composites Among $[\\xi 7^n]$ and Right-Truncatable Primes in Base 7","slug":"composites-among-xi-7-n","field":"Number theory - integer sequences","model":"Fable 5, Opus 4.8","ai_role":"After showing that a finite computation can provably resolve the problem, it wrote programs to do the computation, resulting in a checkable certificate. It then was directed to formalize all proofs and the certificate check in Lean. Finally it was directed to write a draft of the paper from the Lean.","posed_by":"Forman and Shapiro (1967), Dubickas and Novikas (2005)","citations":null,"statement":"For every real $\\xi>0$ the sequence of integer parts $[\\xi 7^{n}]$, $n=0,1,2,\\dots$, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~$7$.","resolution":"resolved","short_name":"Composites among $[\\xi 7^n]$","solve_date":"2026-08-15","solve_type":"proved","source_url":"https://www.researchgate.net/publication/412294027_On_the_composites_among_x7","year_posed":2005,"field_group":"Number theory","model_maker":"Anthropic","publication":"preprint","result_note":null,"source_name":"On the composites among [ξ7ⁿ]","significance":10,"verification":"lean-checked","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"computation","significance_note":"A concrete question in the Forman-Shapiro tradition, posed for base 7 by Dubickas and Novikas in 2005 and open twenty-one years. Specialist but genuinely standing - level with the numbered-Erdos band at 10.","verification_note":"Reviewed and corrected 17 August 2026. The first review said no public Lean repository was linked; that was wrong - it is linked in appendix A of the paper. The error is recorded here rather than quietly dropped.\n\nSource audit of github.com/rwst/On-Composites at commit 2e49c4d, 17 files, comments stripped before counting: no sorry, admit or axiom anywhere on the proof path. The 18 sorry occurrences are all in Challenge.lean, which nothing imports - the leanprover/comparator \"statement of record\", which re-declares the definitions against Mathlib alone so the solution's constants, axiom profile and fresh-export kernel re-acceptance can be checked.\n\nThe tier stops at Lean-checked for a precise reason: all five comparator configs permit exactly propext, Quot.sound and Classical.choice, and the two theorems this entry claims - infinite_composites_seven and no_infiniteTruncatablePrime_seven - are in none of them, because they rest on three native_decide calls, which decide via the compiled evaluator rather than the kernel. The repository documents that quarantine itself.\n\nTwo things bound the risk: floorPow, CompositeInt and InfiniteTruncatablePrime are verbatim identical to the Mathlib-only re-declarations comparator certifies at std3 for bases 3-6, so definitional drift is ruled out; and cond.c, cycles.c and compress.py recompute the hypotheses outside Lean. Not built here - no toolchain, and the repo has no CI - so this is a source audit, not a compile.","human_collaborators":"[]"},"metadata_root":"sha256:d369a11d372d0b462bd355331f43fd429be47f1879184086c05c2108a3f13e36","content_root":"sha256:3e11354a5f593b34d0c1112dd09f9bd7b0d4bbff71daad61ffe606222b1b4cb2","availability":"reference_only","row_root":"sha256:c9580f4bbaca92f0c2fb32800fbf5a51703f58d26a96d9d0cb350ccedc8deb35"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:cone-theorem-effective-fourfold-pairs-char-p","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Cone Theorem for Effective Fourfold Pairs in Characteristic $p > 5$","summary":"Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to $X$, the cone theorem holds for projective log canonical, $\\mathbb{Q}$-factorial fourfold pairs $(X, \\Delta)$ with $K_X + \\Delta \\equiv M \\ge 0$, over bases of positive and mixed characteristic $p > 5$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/cone-theorem-effective-fourfold-pairs-char-p"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.14236"}],"metadata":{"name":"The Cone Theorem for Effective Fourfold Pairs in Characteristic $p > 5$","slug":"cone-theorem-effective-fourfold-pairs-char-p","field":"Birational geometry (positive characteristic)","model":"ChatGPT 5.6 Sol, Codex","ai_role":"The paper's AI statement, in the author's words: he worked out the outline with all essential ingredients in Spring 2025, but one gap remained that he could not repair; he gave the unfinished proof to ChatGPT 5.6 Sol in Summer 2026 asking it to fill the gap, and it modified the approach to avoid the issue, which after further editing by hand and with Codex became the current version.","posed_by":"The char-p minimal model program (Birkar, Hacon, Xu, Waldron and others)","citations":null,"statement":"Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to $X$, the cone theorem holds for projective log canonical, $\\mathbb{Q}$-factorial fourfold pairs $(X, \\Delta)$ with $K_X + \\Delta \\equiv M \\ge 0$, over bases of positive and mixed characteristic $p > 5$.","resolution":"partial","short_name":"Cone theorem, fourfolds, char p>5","solve_date":"2026-08-14","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.14236","year_posed":null,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"The cone theorem for effective fourfold pairs in characteristic p>5","significance":25,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"The minimal model program in positive characteristic is a central program of modern algebraic geometry, and the cone theorem for fourfolds is a real step it has been waiting for - conditional on log resolution, which keeps it below the unconditional band.","verification_note":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14236, Waldron, Michigan State): the AI statement is verbatim as quoted - among the frankest in the catalog, a professional birational geometer crediting the model with repairing a gap he could not. The result is conditional on the log resolution conjecture and is recorded as Partial for that reason. The proof was not checked here; days-old preprint, no independent review.","human_collaborators":"[\"Joe Waldron\"]"},"metadata_root":"sha256:c22eda373cb50dee0d530da2a1ec5e9cf6cd07b9855e33e1ef0c58ad8f1fe441","content_root":"sha256:a890d48bf61b004646d000435bed57393db5b9eb04db227059ae732ee02c7bb5","availability":"reference_only","row_root":"sha256:ecdba931ac13f7b3499ad1e61ba4374447947cec382a2c6e911946fb9d0e7a6b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:connes-rigidity-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Connes' Rigidity Conjecture","summary":"Are ICC property (T) groups remembered by their von Neumann algebras - if $L(\\Gamma) \\cong L(\\Lambda)$ for such groups, must $\\Gamma \\cong \\Lambda$? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/connes-rigidity-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://openai.com/index/ten-advances-in-mathematics/"},{"locator_id":"native-3","kind":"artifact","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf"},{"locator_id":"native-4","kind":"artifact","url":"https://github.com/openai/ten-proofs/blob/main/ConnesRigidity.lean"},{"locator_id":"native-5","kind":"artifact","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"metadata":{"name":"Connes' Rigidity Conjecture","slug":"connes-rigidity-conjecture","field":"Operator algebras","model":"Astra (internal preview)","ai_role":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","posed_by":"Alain Connes","citations":null,"statement":"Are ICC property (T) groups remembered by their von Neumann algebras - if $L(\\Gamma) \\cong L(\\Lambda)$ for such groups, must $\\Gamma \\cong \\Lambda$? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.","resolution":"candidate","short_name":"Connes rigidity","solve_date":"2026-08-01","solve_type":"disproved","source_url":"https://openai.com/index/ten-advances-in-mathematics/","year_posed":1980,"field_group":"Analysis","model_maker":"OpenAI","publication":"announcement","result_note":null,"source_name":"OpenAI: Ten advances in mathematics and theoretical computer science","significance":45,"verification":"lean-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Posed by Connes around 1980; the organizing conjecture of W*-rigidity theory for four decades.","verification_note":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","human_collaborators":"[]"},"metadata_root":"sha256:a3fd8ed773a7e6a8b9817e706d54dc849080cdc15b9907399c0084749bbaae63","content_root":"sha256:9b3ab6ca5d2f54e61632549168bfaf50b2686a9ff433425c2bdf752d7067c885","availability":"reference_only","row_root":"sha256:3d831bc2bcbf46385453739545e95b6e62d99a4a11208c90744f5e052999d1fb"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:connes-rigidity-icc-property-t","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Connes' Rigidity Conjecture for ICC Property (T) Groups","summary":"Connes' rigidity conjecture asks whether an ICC group with Kazhdan's property (T) is determined by its group von Neumann algebra. Disproved for this class: two explicit countable discrete groups $\\Gamma_1$ and $\\Gamma_2$, both ICC and property (T), are non-isomorphic as groups while $L(\\Gamma_1) \\cong L(\\Gamma_2)$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/connes-rigidity-icc-property-t"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.02327"}],"metadata":{"name":"Connes' Rigidity Conjecture for ICC Property (T) Groups","slug":"connes-rigidity-icc-property-t","field":"Operator algebras","model":"GPT-5.6 Sol; Codex; Danus","ai_role":"The AI use statement says the construction underlying the main result was found mainly by GPT-5.6 Sol, Codex and the Danus multi-agent research system, under the author's mathematical guidance, with Lean 4.32.1 used alongside.","posed_by":"Alain Connes","citations":null,"statement":"Connes' rigidity conjecture asks whether an ICC group with Kazhdan's property (T) is determined by its group von Neumann algebra. Disproved for this class: two explicit countable discrete groups $\\Gamma_1$ and $\\Gamma_2$, both ICC and property (T), are non-isomorphic as groups while $L(\\Gamma_1) \\cong L(\\Gamma_2)$.","resolution":"resolved","short_name":"Connes rigidity (ICC, T)","solve_date":"2026-08-03","solve_type":"disproved","source_url":"https://arxiv.org/abs/2608.02327","year_posed":null,"field_group":"Algebra","model_maker":"OpenAI","publication":"preprint","result_note":"Settles the ICC property (T) case. The paper records that the result was obtained independently of and concurrently with work by OpenAI.","source_name":"arXiv:2608.02327 - ICC property(T) groups without W*-superrigidity","significance":35,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Connes' rigidity conjecture is among the best-known open problems in von Neumann algebras, and W*-superrigidity for property (T) groups has organised a substantial part of that literature since the 1980s.","verification_note":"arXiv preprint, not yet peer-reviewed. The paper reports Lean 4.32.1 used during the work; the site has not checked that development.","human_collaborators":"[\"Shuoxing Zhou\"]"},"metadata_root":"sha256:066c059cf43ed8343cb1ce837266c22f59b1088658de6c3ccdcefd11948ad461","content_root":"sha256:a565723768b74e845e78b8b87c3fc65f540bdb29cc6f05f205204325139c24fc","availability":"reference_only","row_root":"sha256:92d26db314c7ea6058e5264672a2a8d7fc13ada5c39dac692618dd40bc1fbe77"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:courtade-kumar-coordinatewise","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Courtade and Kumar's Coordinate-wise Mutual Information Question","summary":"The Courtade-Kumar conjecture (2014) posits that dictatorship functions maximize mutual information between a Boolean function's output and a noisy input. The paper resolves an open question posed by Courtade and Kumar themselves - a sharp bound of $1-H(\\alpha)$ on the sum of coordinate-wise mutual informations for arbitrary bias - and extends the proven high-noise range of the main conjecture via optimal entropy bounds.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/courtade-kumar-coordinatewise"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2601.09679"}],"metadata":{"name":"Courtade and Kumar's Coordinate-wise Mutual Information Question","slug":"courtade-kumar-coordinatewise","field":"Boolean functions; information theory","model":"Gemini Deep Think (larger internal version)","ai_role":"\"The results in this paper were obtained with significant interaction with a larger version of Google's Deep Think Gemini-based model. The authors verified the entire paper and take full responsibility.\" The acknowledgments thank the Deep Think team by name.","posed_by":"Thomas Courtade, Gowtham Kumar","citations":null,"statement":"The Courtade-Kumar conjecture (2014) posits that dictatorship functions maximize mutual information between a Boolean function's output and a noisy input. The paper resolves an open question posed by Courtade and Kumar themselves - a sharp bound of $1-H(\\alpha)$ on the sum of coordinate-wise mutual informations for arbitrary bias - and extends the proven high-noise range of the main conjecture via optimal entropy bounds.","resolution":"partial","short_name":"Courtade-Kumar progress","solve_date":"2026-01-14","solve_type":"proved","source_url":"https://arxiv.org/abs/2601.09679","year_posed":2014,"field_group":"Probability & statistics","model_maker":"Google DeepMind","publication":"preprint","result_note":"Fully resolves the posed coordinate-wise question; the main Courtade-Kumar conjecture itself remains open outside the extended high-noise range.","source_name":"arXiv","significance":22,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"The Courtade-Kumar conjecture is one of the best-known open problems in the analysis of Boolean functions, attacked steadily since 2014 across information theory and TCS.","verification_note":null,"human_collaborators":"[\"Adel Javanmard\",\"David P. Woodruff\"]"},"metadata_root":"sha256:a61457a0775e882f6c55831183bdcfc4071e805aa9e3c11359bb40ce92fc22d6","content_root":"sha256:b10cf7707cbd85c6882945d1f2214bf63560f95623d8bf7138980938a3b26cbb","availability":"reference_only","row_root":"sha256:3f1d57d6035534f00f26e371ad4340d3247044706f44bca6e6dae983d1201049"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:covering-number-c-12-6-4","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Covering Number $C(12,6,4)$","summary":"A $t$-$(v,k,\\lambda)$ covering is a family of $k$-subsets of a $v$-set meeting every $t$-subset at least $\\lambda$ times, and $C(v,k,t)$ is the least number of blocks. The recorded bounds for $C(12,6,4)$ were $40 \\le C(12,6,4) \\le 41$. No $4$-$(12,6,1)$ covering with $40$ blocks exists, so $C(12,6,4) = 41$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/covering-number-c-12-6-4"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.23766"}],"metadata":{"name":"The Covering Number $C(12,6,4)$","slug":"covering-number-c-12-6-4","field":"Design theory","model":"GPT-5.6 Sol, GPT-5.6 Terra, Claude Fable 5","ai_role":"The disclosure lists exploratory analysis, computational search, supporting code, manuscript drafting and revision, and proofreading, without separating which step came from where, so the lowest tier applies.","posed_by":"covering design tables","citations":null,"statement":"A $t$-$(v,k,\\lambda)$ covering is a family of $k$-subsets of a $v$-set meeting every $t$-subset at least $\\lambda$ times, and $C(v,k,t)$ is the least number of blocks. The recorded bounds for $C(12,6,4)$ were $40 \\le C(12,6,4) \\le 41$. No $4$-$(12,6,1)$ covering with $40$ blocks exists, so $C(12,6,4) = 41$.","resolution":"resolved","short_name":"Covering number C(12,6,4)","solve_date":"2026-07-26","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.23766","year_posed":null,"field_group":"Combinatorics","model_maker":"OpenAI / Anthropic","publication":"preprint","result_note":"closes a one-block gap in the covering tables; the analogous next case is not reachable by this method","source_name":"arXiv:2607.23766 - The covering number C(12, 6, 4) is 41","significance":10,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"computation","significance_note":"An open value in the covering design tables, the kind of documented specific gap that design theorists track but that has no audience beyond them.","verification_note":"The non-existence argument is a forced-structure search: a counting argument pins every point to degree exactly 20 and forces each point link to be an optimal 3-(11,5,1) covering, collapsing the search space. Single-author arXiv preprint, not peer-reviewed.","human_collaborators":"[\"Charlie Krug\"]"},"metadata_root":"sha256:f554681a8660fdc1c6d63c4cdc89750360920d199b73d94fd23e22c56a34f10b","content_root":"sha256:335e0b0cbc57f0952e82d333b755f3f5e4daf5a64fd3e82eb052f764d9a5627e","availability":"reference_only","row_root":"sha256:a2ef0cfa79c3c2a82f07da2e10b2e6229f1544080be7f7cc46204de6b6185d98"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:coxeter-code-minimum-distance","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Coxeter Code Minimum Distance Conjecture","summary":"Coble and Barg introduced binary Coxeter codes, the span of indicators of standard cosets of fixed rank in a finite Coxeter system, generalizing Reed-Muller codes, and proposed a conjectural value for the minimum distance of a general Coxeter code. The conjecture is true, and it yields a decoding consequence.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/coxeter-code-minimum-distance"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.10774"}],"metadata":{"name":"The Coxeter Code Minimum Distance Conjecture","slug":"coxeter-code-minimum-distance","field":"Coding theory","model":"ChatGPT-5.4, Claude Sonnet 4.6","ai_role":"The authors say the models were used as exploratory tools in connection with the problem, and bound that use explicitly: brainstorming, discussion of possible approaches, and preliminary checking of ideas.","posed_by":"Nolan Coble, Alexander Barg","citations":null,"statement":"Coble and Barg introduced binary Coxeter codes, the span of indicators of standard cosets of fixed rank in a finite Coxeter system, generalizing Reed-Muller codes, and proposed a conjectural value for the minimum distance of a general Coxeter code. The conjecture is true, and it yields a decoding consequence.","resolution":"resolved","short_name":"Coxeter codes","solve_date":"2026-07-12","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.10774","year_posed":2025,"field_group":"Combinatorics","model_maker":"OpenAI / Anthropic","publication":"preprint","result_note":null,"source_name":"arXiv:2607.10774 - Minimum distance and decoding of Coxeter codes","significance":7,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A conjectural minimum distance proposed in the paper that introduced the codes; recent and specialized, though Reed-Muller codes give it wider context.","verification_note":"arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Alexander Barg\",\"Qendrim R. Gashi\",\"Tianyuan Xu\"]"},"metadata_root":"sha256:d9a5cd6292e137c3874a69cbe172d468e7f2d94f2b7c76c22f4cfa06287fd0bd","content_root":"sha256:98493a03f6669636c2b4a518a846e4f6854a7de1bb160e77b666d858e481e4bb","availability":"reference_only","row_root":"sha256:0ef598923caadd4e0fdf93e66c4a2a764a593e7abfeaa6e604a38eeeafc94437"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:crouzeix-s-conjecture","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Crouzeix's Conjecture","summary":"Crouzeix conjectured in 2004 that for every square complex matrix $A$ and every polynomial $p$, $\\lVert p(A)\\rVert \\leq 2 \\max_{z \\in W(A)} |p(z)|$, where $W(A)$ is the numerical range of $A$ - that is, the numerical range is a 2-spectral set. Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to $1+\\sqrt{2}$ in 2017; the conjectured constant 2 is attained by $2\\times 2$ matrices. Jin proves the sharp bound by a function-theoretic route whose key theorem reduces the problem, via a sampling strategy, to a positivity condition; Lorist and Schwenninger independently prove it days later by combining double-layer potential machinery with a perturbation lemma for 2-dilations.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/crouzeix-s-conjecture"},{"locator_id":"native-2","kind":"artifact","url":"https://www.preprints.org/manuscript/202607.1919"},{"locator_id":"native-3","kind":"artifact","url":"https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf"},{"locator_id":"native-4","kind":"artifact","url":"https://github.com/jinshanmu/CrouzeixConjecture"},{"locator_id":"native-5","kind":"artifact","url":"https://arxiv.org/abs/2608.03841"},{"locator_id":"native-6","kind":"artifact","url":"https://aimath.org/pastworkshops/crouzeix.html"},{"locator_id":"native-7","kind":"artifact","url":"https://en.wikipedia.org/wiki/Crouzeix%27s_conjecture"}],"metadata":{"name":"Crouzeix's Conjecture","slug":"crouzeix-s-conjecture","field":"Matrix analysis","model":"GPT-5.6 Sol; ChatGPT 5.6 Pro","ai_role":"For the first proof: Jin, a neurosurgery resident with no specialized mathematical training, reports that the key result (Theorem 2) emerged during an approximately sixteen-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode - a public prompt adapted from the Cycle Double Cover run, web access denied, a branching portfolio of subagent strategies under adversarial audit, and no human intervention once started. Jin then simplified and verified the output; the repository publishes the prompt, successive manuscripts, a Lean formalization and an axiom audit. For the independent second proof, Lorist and Schwenninger disclose that ChatGPT 5.6 Pro was used to review previous approaches to the weaker spectral constant $1 + \\sqrt{2}$ and to identify a possible source of improvement in estimates involving iterates $f^n$ of extremal or approximately extremal functions.","posed_by":"Michel Crouzeix","citations":null,"statement":"Crouzeix conjectured in 2004 that for every square complex matrix $A$ and every polynomial $p$, $\\lVert p(A)\\rVert \\leq 2 \\max_{z \\in W(A)} |p(z)|$, where $W(A)$ is the numerical range of $A$ - that is, the numerical range is a 2-spectral set. Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to $1+\\sqrt{2}$ in 2017; the conjectured constant 2 is attained by $2\\times 2$ matrices. Jin proves the sharp bound by a function-theoretic route whose key theorem reduces the problem, via a sampling strategy, to a positivity condition; Lorist and Schwenninger independently prove it days later by combining double-layer potential machinery with a perturbation lemma for 2-dilations.","resolution":"resolved","short_name":"Crouzeix's conjecture","solve_date":"2026-07-27","solve_type":"proved","source_url":"https://www.preprints.org/manuscript/202607.1919","year_posed":2004,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":"Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma for 2-dilations - produced with ChatGPT 5.6 Pro exploring proof strategies. The entry's headline axes record Jin's proof; the earlier version of this entry recorded Lorist-Schwenninger's as primary while Jin's AI provenance was still unknown.","source_name":"Jin, The Numerical Range Is a 2-Spectral Set","significance":35,"verification":"expert-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A named 2004 conjecture at the centre of matrix analysis and operator theory: two decades of partial results, its own AIM workshop (2017), its own survey, and Wikipedia articles in two languages. Field-famous rather than household - level with Feige and Krauth-Mezard at 35, above the strong specialist band at 30 where it previously sat; the AIM workshop and the constant-lowering literature are the concrete differentiators.","verification_note":"Independently expert-verified, publicly on record: Townsend and Greenbaum's essay of 14 August 2026 states that both authors and Michel Crouzeix himself \"have checked the proof thoroughly and believe that Dr. Jin's manuscript is correct\" - the conjecture's own poser among the verifiers, and Greenbaum co-organized the 2017 AIM workshop on the problem. This site read that essay in full and audited Jin's repository (commit 9df0783): 82 Lean files with zero sorry, zero axiom declarations and zero native_decide with comments stripped, on toolchain v4.28.0, alongside an Annals-formatted manuscript and the complete autonomous-run prompt - though the Lean was not compiled here and its statement-to-conjecture correspondence not audited, so the tier rests on the expert endorsement, not the formalization. The independent second proof by Lorist and Schwenninger (arXiv:2608.03841) has no comparable public endorsement yet and the essay stops short of vouching for it. Neither manuscript is refereed.","human_collaborators":"[\"Shanmu Jin\",\"Emiel Lorist\",\"Felix L. 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Disproved: an explicit competitive 3-neuron TLN has a stable fixed point whose support strictly contains another's, and 3 neurons is proven smallest possible.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/curto-minimal-fixed-points"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2511.05517"}],"metadata":{"name":"Curto et al.'s Minimality Conjecture for Threshold-Linear Networks","slug":"curto-minimal-fixed-points","field":"Mathematical neuroscience","model":"GPT-5","ai_role":"\"We used GPT-5 to find the 3-neuron counterexample and to draft some of the expository text.\" The minimality of the construction and the expansions to larger networks are the author's.","posed_by":"Carina Curto et al.","citations":null,"statement":"Curto et al. (Advances in Applied Mathematics, 2024) conjectured that every stable fixed point of a threshold-linear network is minimal. 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The proofs are described as short, which makes them checkable by a reader who knows the areas, but no independent verification is recorded and the autonomy claim covers every proof in the paper.","human_collaborators":"[\"Colin Defant\"]"},"metadata_root":"sha256:98a22aaa3fa267a03863a086d48963e0f32fb99d252100419476b11f4b373bb4","content_root":"sha256:c54f99fc217e51d1933d6f186fc84a7313d42e936aa74390b408c25c6bc56772","availability":"reference_only","row_root":"sha256:67761e3a6d741f28da1842ae2c6298b6cc9b80b8bf06044518fcb71467d76cf5"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:denjoy-theorem-sharpness","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Sharpness of Denjoy's Theorem","summary":"Denjoy's 1932 theorem says a $C^{1+\\mathrm{bv}}$ circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity $\\omega$ weaker than Lipschitz, there is a $C^{1+\\omega}$ circle diffeomorphism with irrational rotation number and a wandering interval. The case $\\omega(t) = t\\log(1/t)$ settles an open problem going back to Herman's 1979 work, which had constructions only for $\\omega(t) = t\\log(1/t)^{1+\\varepsilon}$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/denjoy-theorem-sharpness"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2608.02380"}],"metadata":{"name":"Sharpness of Denjoy's Theorem","slug":"denjoy-theorem-sharpness","field":"Dynamical systems","model":"GPT-5.6 Sol Ultra; Claude Fable 5","ai_role":"The AI use section says the author prompted GPT-5.6 Sol Ultra to construct a Denjoy example for the modulus $t\\log(1/t)$, which is the corollary settling Herman's case, and used Claude Fable 5 to search for errors.","posed_by":"Michael Herman","citations":null,"statement":"Denjoy's 1932 theorem says a $C^{1+\\mathrm{bv}}$ circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity $\\omega$ weaker than Lipschitz, there is a $C^{1+\\omega}$ circle diffeomorphism with irrational rotation number and a wandering interval. The case $\\omega(t) = t\\log(1/t)$ settles an open problem going back to Herman's 1979 work, which had constructions only for $\\omega(t) = t\\log(1/t)^{1+\\varepsilon}$.","resolution":"resolved","short_name":"Denjoy sharpness","solve_date":"2026-08-03","solve_type":"proved","source_url":"https://arxiv.org/abs/2608.02380","year_posed":1979,"field_group":"Analysis","model_maker":null,"publication":"preprint","result_note":null,"source_name":"arXiv:2608.02380 - On the sharpness of Denjoy's theorem","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"Closes the regularity gap left by Herman's 1979 constructions, which had stood as the boundary of Denjoy's theorem for over forty years.","verification_note":"arXiv preprint, not yet peer-reviewed.","human_collaborators":"[\"Rohil Prasad\"]"},"metadata_root":"sha256:4ba720ea26b5889322d6677d899c47585364c7327aafa3f8853516e8d1ac94d8","content_root":"sha256:315bac1872b027ba7546427d5bc6557622d343130e2990f9d3a65cfc6fc3fe38","availability":"reference_only","row_root":"sha256:9e517a3cbdef01ae35eb77ebaac18ec210850e106707e6d5131adf98aec933c1"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:density-large-dilates","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Density Thresholds for Large Dilates of Point Configurations","summary":"Near-optimal density thresholds forcing a measurable set in $\\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/density-large-dilates"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2604.18544"}],"metadata":{"name":"Density Thresholds for Large Dilates of Point Configurations","slug":"density-large-dilates","field":"Euclidean density theorems","model":"ChatGPT 5.4 Pro","ai_role":"ChatGPT 5.4 Pro \"was used to suggest and draft approaches to Proposition 5\"; in particular the random pattern thinning argument, \"somewhat novel in this context,\" was suggested by the model. Main ideas and final proofs are the authors'.","posed_by":null,"citations":null,"statement":"Near-optimal density thresholds forcing a measurable set in $\\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.","resolution":"partial","short_name":"Density of large dilates","solve_date":"2026-04-20","solve_type":"proved","source_url":"https://arxiv.org/abs/2604.18544","year_posed":null,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":"Near-optimal rather than optimal: the bounds match up to logarithmic-type factors.","source_name":"arXiv","significance":8,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-assisted","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A stated question in the Euclidean density-theorem line (Bourgain, Graham tradition), specialist but documented.","verification_note":null,"human_collaborators":"[\"Vjekoslav Kovač\",\"Adian Anibal Santos Sepčić\"]"},"metadata_root":"sha256:be5be4cdb91b0314e6fedf5aaccd4593823cf4b67d971f5d73a4ab070068d769","content_root":"sha256:777dd0a09c441670e5a78f7d0c34e86be780f4bfe23dd3727e53666aaf7d0d63","availability":"reference_only","row_root":"sha256:a9f8a0f073688513d5ceb8bdfd301b139d95f53cf35af946f4da45aed4d2d315"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:depth-of-the-in-tree-of-s-in-the-graph-of-q-qsq-1-in-the-symmetric-group","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Depth of the in-tree of $s$ under $q \\mapsto q s q^{-1}$ on $n$-cycles","summary":"Fix an $n$-cycle $s$ and map every $n$-cycle $q$ to its conjugate $D(q) = q s q^{-1}$, which is the same as reading the one-line word $(q(0), \\ldots, q(n-1))$ back as a cycle. Iterating $D$ turns the $(n-1)!$ $n$-cycles into a functional graph. Its only fixed point is $s$, and the cycles that eventually reach $s$ form a tree feeding into it. How deep is that tree?\n\nExactly $\\varphi(n)$ cycles map directly onto $s$, and the tree stays shallow - depth 1 - unless $8 \\mid n$ or $p^2 \\mid n$ for an odd prime $p$, which is the Hull-Dobell threshold for the existence of a full-period non-translation affine map on $\\mathbb{Z}/n$. Past it the depth is $p^{e-1}$ for $n = p^e$ with $p$ odd, $2^{e-1} - 1$ for $n = 2^e$, and for general $n$ the largest of these over the prime powers dividing $n$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/depth-of-the-in-tree-of-s-in-the-graph-of-q-qsq-1-in-the-symmetric-group"},{"locator_id":"native-2","kind":"artifact","url":"https://www.datastrategies.fr/fr/blog/vibemathing-45-ans-plus-tard"},{"locator_id":"native-3","kind":"artifact","url":"https://www.datastrategies.fr/sites/default/files/documents/2026-07/depth_companion_1.pdf"},{"locator_id":"native-4","kind":"artifact","url":"https://www.datastrategies.fr/sites/default/files/documents/2026-07/derivation_2.pdf"},{"locator_id":"native-5","kind":"artifact","url":"https://web.archive.org/web/2012/http://permutationc.free.fr/B1703659376/"}],"metadata":{"name":"Depth of the in-tree of $s$ under $q \\mapsto q s q^{-1}$ on $n$-cycles","slug":"depth-of-the-in-tree-of-s-in-the-graph-of-q-qsq-1-in-the-symmetric-group","field":"Permutation combinatorics / functional graphs","model":"Claude FABLE 5","ai_role":"Claude Opus 4.8 in a first phase, then Claude FABLE 5, which obtained the result, did 99.9% of the research, in manual mode (that is, with more than 50 human prompts by an amateur, and step-by-step approvals). OpenAI's GPT-5.5 and others were used to proofread; all corrections after human and AI proofreading were made by Claude. Diagrams were made by Claude under human instructions.","posed_by":"Frédéric Lefebvre-Naré","citations":null,"statement":"Fix an $n$-cycle $s$ and map every $n$-cycle $q$ to its conjugate $D(q) = q s q^{-1}$, which is the same as reading the one-line word $(q(0), \\ldots, q(n-1))$ back as a cycle. Iterating $D$ turns the $(n-1)!$ $n$-cycles into a functional graph. Its only fixed point is $s$, and the cycles that eventually reach $s$ form a tree feeding into it. How deep is that tree?\n\nExactly $\\varphi(n)$ cycles map directly onto $s$, and the tree stays shallow - depth 1 - unless $8 \\mid n$ or $p^2 \\mid n$ for an odd prime $p$, which is the Hull-Dobell threshold for the existence of a full-period non-translation affine map on $\\mathbb{Z}/n$. Past it the depth is $p^{e-1}$ for $n = p^e$ with $p$ odd, $2^{e-1} - 1$ for $n = 2^e$, and for general $n$ the largest of these over the prime powers dividing $n$.","resolution":"candidate","short_name":"In-tree depth under cyclic conjugation","solve_date":"2026-07-24","solve_type":"proved","source_url":"https://www.datastrategies.fr/fr/blog/vibemathing-45-ans-plus-tard","year_posed":2006,"field_group":"Combinatorics","model_maker":"Anthropic","publication":"announcement","result_note":"Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: \"The proof turns conjugation, near $s$, into base-$p$ arithmetic.\" A cycle near the fixed point splits into a coarse base permutation and a vector of carries in $\\mathbb{Z}/p$, $D$ acts on the carries by the carrying of ordinary base-$p$ addition, and the depth comes out as the nilpotency length of a shift difference - exactly that for odd $p$, one less for $p = 2$. The single missing carry that odd primes absorb and $2$ cannot is what produces the two-branch answer.\n\nA companion survey paper covers the rest of the graph: the other periodic orbits, congruences on basin sizes, and a cyclic-sieving count. The depth theorem is the substantive part.","source_name":"Data Stratégies (the author's blog)","significance":3,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Below the anchor at 5, which covers machine-generated conjectures and recent one-paper questions. This one is a real, precisely posed question about a natural object, but it was asked by a high-school pupil in about 1980, put on a personal blog in 2006, and engaged with by nobody in the twenty years since - no citations, no literature, no community. A Graffiti conjecture at least lands on a list a research community reads. Above the floor because the question is genuine rather than generated, and the submitter says the same thing: this is not about an important issue in mathematics.","verification_note":"Not refereed, not formalised, and the paper says so itself on page one: \"An end-to-end verification of the assembled whole, and human peer review, remain to be done; the argument should be examined critically before being relied upon.\" The author, an amateur, states plainly that he cannot check the proof. So this stays Unreviewed and Candidate.\n\nThe claim is unusually checkable, though, and this site checked it independently, from the statement rather than from the author's code. Enumerating every $n$-cycle and building the whole functional graph for $n \\le 10$: $|D^{-1}(s)| = \\varphi(n)$ at every $n$, and the depth matches the formula at every $n$, including both nontrivial cases in range, depth 3 at $n = 8$ and at $n = 9$. Walking the tree backwards, which costs $n$ checks per node instead of $(n-1)!$, reaches depth 7 at $n = 16$, 5 at $n = 25$, 3 at $n = 18$, and 9 at $n = 27$ across 472,392 nodes - every one the predicted value. The Hull-Dobell mechanism was checked directly: preimages of a translation are exactly the affine maps with that multiplier, Hull-Dobell decides which are $n$-cycles, and the threshold predicts a nonempty second level exactly, no exceptions at $n = 6, 8, 9, 12, 16, 18, 25, 27$. The survey's basin-count sequence recounts to 1, 2, 2, 6, 7, 18, 17, 29 for $n = 3 \\ldots 10$, matching, and OEIS returns nothing for it.\n\nWhat none of that touches is the proof, which is the novelty: a formula confirmed at every $n$ reachable is not a theorem for all $n$.","human_collaborators":"[]"},"metadata_root":"sha256:712250cac8df6ee71c3a1805ead0f12ece815ee51a694f5af75b3cab42386bb1","content_root":"sha256:edeadd389fbc610e76e8a3e6a9ddbe29577a09dde1a21fdea2137175cfecbffa","availability":"reference_only","row_root":"sha256:501fecf8997e0c8e109515b93dcd3670cce3124b71c77019b63a4aa25dba0138"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:derivative-free-convex-oracle-gap","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Oracle-Complexity Gap in Derivative-Free Convex Optimization","summary":"For deterministically minimizing a convex 1-Lipschitz function on the $d$-dimensional ball using only exact function values, the query complexity sat between $\\Omega(d)$ and $O(d^2 \\log^2 d)$ since 1996. The paper proves a near-quadratic lower bound $\\Omega(d^2 / \\log(d+1))$, closing the gap: $Q(d, \\sim d^{-1/2}) = \\Theta(d^2)$, a polynomial separation from full first-order information.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/derivative-free-convex-oracle-gap"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.13335"}],"metadata":{"name":"Oracle-Complexity Gap in Derivative-Free Convex Optimization","slug":"derivative-free-convex-oracle-gap","field":"Optimization (Oracle Complexity)","model":"GPT-5.6 Sol Pro","ai_role":"Kerger reports that GPT-5.6 Sol Pro solved the problem rather than the author, following a workflow like OpenAI's Cycle Double Cover effort. It first proved a $\\tilde{\\Omega}(d^2)$ lower bound at accuracy of order $d^{-3}$ (after ~148 minutes), which was then refined to the order-$d^{-1/2}$ result via a further ~230-minute run. The author verified the arguments by hand and takes full responsibility.","posed_by":"Vladimir Protasov (gap since 1996)","citations":null,"statement":"For deterministically minimizing a convex 1-Lipschitz function on the $d$-dimensional ball using only exact function values, the query complexity sat between $\\Omega(d)$ and $O(d^2 \\log^2 d)$ since 1996. The paper proves a near-quadratic lower bound $\\Omega(d^2 / \\log(d+1))$, closing the gap: $Q(d, \\sim d^{-1/2}) = \\Theta(d^2)$, a polynomial separation from full first-order information.","resolution":"resolved","short_name":"Zeroth-order oracle gap","solve_date":"2026-07-14","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.13335","year_posed":1996,"field_group":"Algorithms & optimization","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2607.13335","significance":15,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A 30-year oracle-complexity gap in zeroth-order optimization.","verification_note":"arXiv preprint 2607.13335 (14 Jul 2026) by Phillip Kerger (UC Berkeley), not yet peer-reviewed. The weaker-accuracy $\\tilde{\\Omega}(d^2)$-at-$d^{-3}$ lower bound was formally verified in Lean (github.com/PhillipKerger/zero-order-bounds-lean-verification); the headline improvement to accuracy $d^{-1/2}$ is not yet Lean-formalized (it needs convex-geometry results like Urysohn's inequality absent from current Lean libraries) and rests on the author's hand verification.","human_collaborators":"[\"Phillip Kerger\"]"},"metadata_root":"sha256:e07569b20e94ef1429e3a1035f67b4044fbca8c2b02d712ec231c1d8553d3002","content_root":"sha256:06c1b33128d0ba0171cd8fa208f318f64e9b5d075bb2cc67c38a9ec77add177c","availability":"reference_only","row_root":"sha256:48280ccd7bad494b8b855bd93c9cf5c6b23b43696aed72292d81b68e03e2ff7b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:dfjp-interpolation-schauder-basis","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Weakly Compact Factorization Through a Space With a Basis","summary":"Davis, Figiel, Johnson and Pełczyński showed their interpolation space admits a Schauder basis when the range space has a shrinking one. Can the DFJP space always be chosen with a basis whenever the range space has a basis? The paper proves it can.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/dfjp-interpolation-schauder-basis"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.17388"},{"locator_id":"native-3","kind":"artifact","url":"https://mathoverflow.net/questions/240472"}],"metadata":{"name":"Weakly Compact Factorization Through a Space With a Basis","slug":"dfjp-interpolation-schauder-basis","field":"Banach Space Factorization","model":"ChatGPT 5.5 Pro","ai_role":"ChatGPT 5.5 Pro, driven directly and through Codex agents over a filesystem, generated the key ideas and the proof; the authors verified and refined it. The paper reports that the model sometimes misattributed a theorem, cited a result imprecisely, or presented steps as immediate when they still needed checking, so the human verification was load-bearing. The authors state none of these results came from the newer generation of models.","posed_by":"Davis, Figiel, Johnson and Pełczyński; raised again by Kevin Beanland","citations":null,"statement":"Davis, Figiel, Johnson and Pełczyński showed their interpolation space admits a Schauder basis when the range space has a shrinking one. Can the DFJP space always be chosen with a basis whenever the range space has a basis? The paper proves it can.","resolution":"resolved","short_name":"DFJP basis factorization","solve_date":"2026-07-19","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.17388","year_posed":1974,"field_group":"Analysis","model_maker":"OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"The oldest of the batch: a question about the DFJP interpolation construction that the paper states remained open until this proof.","verification_note":"No independent check. The proof stands on verification by the paper's authors, who describe correcting misattributions and filling in steps the model presented as immediate. Preprint, not refereed.","human_collaborators":"[\"Antonio Acuaviva\",\"Pablo Acuaviva\"]"},"metadata_root":"sha256:abc15b666332d8d3683411942709742d711d4aa5fcc2e7c5add50e7ac29c8cda","content_root":"sha256:bce71244990faf0aa246948f91e11e81b18d210e3a0b70d3f3eec0e16defb073","availability":"reference_only","row_root":"sha256:06997b2d1b748a7e81262a38e014504eba5ac57b1612544003d3b672096dad28"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Dihedral and cyclic Ramsey numbers of the alternating 3-path","summary":"$R_{\\mathrm{dih}}(P_3^{\\mathrm{alt}}, K_b) = R_{\\mathrm{cyc}}(P_3^{\\mathrm{alt}}, K_b) = 2b - 1$ for all $b \\in \\mathbb{N}$ — the $a = 3$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/ZestyWombat854/dihedral-ramsey"},{"locator_id":"native-3","kind":"artifact","url":"https://arxiv.org/abs/2607.06817"},{"locator_id":"native-4","kind":"artifact","url":"https://arxiv.org/abs/2604.16188"},{"locator_id":"native-5","kind":"artifact","url":"https://github.com/ZestyWombat854/dihedral-ramsey/tree/01a50c7/sat"}],"metadata":{"name":"Dihedral and cyclic Ramsey numbers of the alternating 3-path","slug":"dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path","field":"Permutational Ramsey theory","model":"Claude Fable 5","ai_role":"The model produced the proof (the $\\mathrm{Dih}(3) = \\mathrm{Sym}(3)$ collapse, the Chvátal reduction, the cyclic corollary), the Lean 4 formalization, and the Python verification script autonomously. Human direction was limited to initiation and operational supervision.","posed_by":"Damnjanović–Đorđević (Conj 4.9); Bašić–Damnjanović–Stevanović–Stošić (Conj 4.23)","citations":null,"statement":"$R_{\\mathrm{dih}}(P_3^{\\mathrm{alt}}, K_b) = R_{\\mathrm{cyc}}(P_3^{\\mathrm{alt}}, K_b) = 2b - 1$ for all $b \\in \\mathbb{N}$ — the $a = 3$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).","resolution":"resolved","short_name":"$R_{dih}(P_3^{alt},K_b)=2b-1$","solve_date":"2026-08-12","solve_type":"proved","source_url":"https://github.com/ZestyWombat854/dihedral-ramsey","year_posed":2026,"field_group":"Combinatorics","model_maker":"Anthropic","publication":"preprint","result_note":"The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.","source_name":"GitHub repo (preprint + Lean proof + Python checker)","significance":5,"verification":"site-confirmed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Small, and the preprint says so itself. This closes one slice (a = 3) of a conjecture stated about five weeks earlier, and it closes it by a group coincidence plus a citation: Dih(3) happens to equal Sym(3), so the permutational condition collapses to ordinary subgraph containment and Chvatal's 1977 theorem finishes it. The note is candid that the cyclic values for b = 3..8 were already tabulated by Basic et al., and that what is new is the closed form, not the numbers. Scored near the bottom of the spine, below the Erdos entries at 10, which are decades-old rather than weeks-old.","verification_note":"Reproduced here on 13 August 2026. The Lean development builds clean (exit 0) on the pinned toolchain (v4.12.0, core only, no Mathlib), and #print axioms shows all five main theorems depending on exactly propext, Classical.choice and Quot.sound. No Lean.ofReduceBool; with comments stripped the source has zero sorry, admit, axiom declarations and native_decide, and its 23 decide calls are kernel-reduced. A naive grep says otherwise only because those words appear in the file's own docs. The Python checker runs as described: Dih(3) has order 6 and equals Sym(3), and the lower-bound witnesses hold for b = 2..8. The general upper bound is not formalized; it cites Chvatal 1977, whose arithmetic holds. The SAT claim, unconfirmed at review, was substantiated the same day at commit 01a50c7. The DRAT files were not replayed, since replaying a shipped proof is the weaker check; instead all twelve CNFs were re-solved here with CaDiCaL, every verdict matching their kissat logs - satisfiable at $n=2b-2$, unsatisfiable at $n=2b-1$, for b = 2..7. The six satisfiable instances had their witnesses re-substituted clause by clause and all satisfy, and the b = 3 legs agree with this site's own exhaustive enumeration, anchoring their encoder against an independent computation. The certificates are regenerated rather than the originals, disclosed unprompted, which costs nothing here. Still unconfirmed: no human peer review, this being a self-submission reviewed by AI agents in-pipeline.","human_collaborators":"[]"},"metadata_root":"sha256:be4d728ffdb3464dd190826c3547527d62ef5e46549d7b7aff9b0a73e3701f6e","content_root":"sha256:9c65a509dcf4052c6cc9f5e5a117f726fa727cc0829fc0b2a7ea6160653cd8c6","availability":"reference_only","row_root":"sha256:9964cfa7c67056935f498029dc90d67a49b514177f4f4c41f239bfbc50ae9c2b"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)","summary":"$R_{\\mathrm{dih}}(P_a^{\\mathrm{alt}}, K_b) = 1 + (a-1)(b-1)$ for all $a \\geq 4$, $b \\geq 1$ — the $a \\geq 4$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the $a = 3$ case (see sibling entry), this resolves Conjecture 4.9 in full for $a \\geq 3$.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/ZestyWombat854/alternating-path-ramsey"},{"locator_id":"native-3","kind":"artifact","url":"https://arxiv.org/abs/2607.06817"}],"metadata":{"name":"Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)","slug":"dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1","field":"Permutational Ramsey theory","model":"Claude Fable 5","ai_role":"The proof was produced by a sealed, multi-agent research process: independently-launched Claude agents across three rounds, convergent results cross-validated. Two independent AI referee agents reviewed it dual-blind; both CONFIRMED. Human direction was limited to run design, operational supervision, and manual re-derivation of two write-up fixes.","posed_by":"Damnjanović–Đorđević (Conj 4.9)","citations":null,"statement":"$R_{\\mathrm{dih}}(P_a^{\\mathrm{alt}}, K_b) = 1 + (a-1)(b-1)$ for all $a \\geq 4$, $b \\geq 1$ — the $a \\geq 4$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the $a = 3$ case (see sibling entry), this resolves Conjecture 4.9 in full for $a \\geq 3$.","resolution":"resolved","short_name":"$R_{dih}(P_a^{alt},K_b)=1+(a-1)(b-1)$, $age4$","solve_date":"2026-08-13","solve_type":"proved","source_url":"https://github.com/ZestyWombat854/alternating-path-ramsey","year_posed":2026,"field_group":"Combinatorics","model_maker":"Anthropic","publication":"preprint","result_note":"The dihedral case only, for every $a \\ge 4$ and $b \\ge 1$; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim $1+(a-1)(b-1)$ for all $a \\ge 3$; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue $R_{cyc}(P_a^{alt}, K_b)$ for $a \\ge 4$ remains open. The engine is a self-contained inequality of independent interest: for any graph on a linearly ordered vertex set, the alternating-path reach statistics satisfy $\\sum_m [P(m)+Q(m)] \\ge 2|E(G)|$, from which the theorem falls out by averaging and a pivot decomposition.","source_name":"GitHub repo (proof + referee reports + verification code + partial Lean formalization)","significance":8,"verification":"site-confirmed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"Resolves the dihedral side of a conjecture posed five weeks earlier in a single paper with no independent citations yet - a young question in a niche new area, permutational Ramsey theory. Above the a = 3 slice (5), which fell to a group coincidence plus a citation, because this is the general theorem with a genuinely new combinatorial inequality behind it; below the Erdos entries at 10, which are decades-old problems with real literatures.","verification_note":"Reproduced by this site on 13 August 2026, working from the pinned statement alone - the proof's machinery, both referee reports and the shipped CNFs were not consulted by the checker. Confirmed independently: the orbit anchor ($|Dih(a)$-orbit of $P_a^{alt}| = a$ for a = 3..14); the Ramsey value at nine (a,b) cells in both directions - a good coloring exists at $n = (a-1)(b-1)$ and none at $n+1$ - exhaustively over every 2-coloring at (4,2), (5,2), (6,2), (7,2) and (4,3), and via an independently written CNF encoding solved with CaDiCaL at (8,2), (5,3), (6,3) and (4,4); and the proof's load-bearing inequality, the Aggregate Sum Theorem, by a third implementation built from the P/Q definitions rather than the recursion, over all 33,868 labeled graphs on up to six vertices - zero violations, minimum slack 0, so the bound is tight. The prose proof was also read here in full and every algebraic step traced. Not covered by the tier: the general argument has no human peer review - produced by a sealed multi-agent Claude run and refereed dual-blind by two AI agents in the same pipeline (both CONFIRMED; one non-fatal bug and one cosmetic slip found and repaired inline, originals kept). The Lean part is partial by its own declaration - four side lemmas, zero sorry or native_decide, standard axioms, source-audited here but not compiled (pinned v4.30.0 + Mathlib, no CI runs). The main theorems are not formalized; there, the referee reports and this site's checks are the verification.","human_collaborators":"[]"},"metadata_root":"sha256:af44af734287e792498b439fc375ea0d8569177fec6e2dc5274aea7a237514a1","content_root":"sha256:2a1ad345b4464d4e26d6aec1c2856078b8497d22e5adc27ef108fe3c1e5734d3","availability":"reference_only","row_root":"sha256:8ed7e18292d4fdf61e59ca17f888c8b0440b508173f8dfa8b73327dab49398d0"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:dimer-constant-cubic-lattice","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"The Dimer Constant of the Cubic Lattice","summary":"The dimer constant of $\\mathbb{Z}^3$, the exponential growth rate of perfect matchings of the cubic lattice, has no closed form and is pinned only by bounds. The upper bound improves from Lundow's $0.457547$, standing since 2001, to $0.452130$, via diagonal transfer layers and an inequality of Csikvari relating the spectral radius of the transfer matrix to the constant.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/dimer-constant-cubic-lattice"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2607.28810"}],"metadata":{"name":"The Dimer Constant of the Cubic Lattice","slug":"dimer-constant-cubic-lattice","field":"Statistical mechanics","model":"GPT-5.6 Sol Ultra","ai_role":"The acknowledgement attributes the paper's two key ingredients to the model: the diagonal transfer layers, which replace the symmetry argument special to the rectangular torus, and the connection with Csikvari's inequality.","posed_by":"classical lattice statistics","citations":null,"statement":"The dimer constant of $\\mathbb{Z}^3$, the exponential growth rate of perfect matchings of the cubic lattice, has no closed form and is pinned only by bounds. The upper bound improves from Lundow's $0.457547$, standing since 2001, to $0.452130$, via diagonal transfer layers and an inequality of Csikvari relating the spectral radius of the transfer matrix to the constant.","resolution":"partial","short_name":"Dimer constant of Z^3","solve_date":"2026-07-30","solve_type":"proved","source_url":"https://arxiv.org/abs/2607.28810","year_posed":null,"field_group":"Mathematical physics","model_maker":"OpenAI","publication":"preprint","result_note":"a record upper bound; the exact constant remains unknown","source_name":"arXiv:2607.28810 - A new upper bound on the dimer constant of Z^3","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"computation","significance_note":"The three-dimensional dimer constant is a classical unsolved quantity in lattice statistical mechanics, with the planar case exactly solved and the cubic case reduced to a bound ladder.","verification_note":"Single-author arXiv preprint; not yet peer-reviewed.","human_collaborators":"[\"Qidong He\"]"},"metadata_root":"sha256:bb2fcca659b93edd513fb4b4e9390654cdfede732240f1917bd31d39efb7de87","content_root":"sha256:fd9c010fc107d471708ad86474a23a39554165c0b4e684c2128b471ecc0fe3ef","availability":"reference_only","row_root":"sha256:35d23baf970a615d525a77d137c113e02f99a6c74381e557189f926a0a6d60b7"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:dinitz-garg-goemans-unsplittable-flow","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Dinitz-Garg-Goemans Conjecture","summary":"For single-source unsplittable flow, every fractional flow can be rounded to an unsplittable flow whose cost is no higher than the fractional cost, while each arc's load is exceeded by at most the maximum demand. (The cost version of Goemans' unsplittable-flow conjecture.)","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/dinitz-garg-goemans-unsplittable-flow"},{"locator_id":"native-2","kind":"artifact","url":"https://x.com/DmitryRybin1/status/2079904005652893709"},{"locator_id":"native-3","kind":"artifact","url":"https://vibemathed.com/problem/planar-four-terminal-dgg"}],"metadata":{"name":"Dinitz-Garg-Goemans Conjecture","slug":"dinitz-garg-goemans-unsplittable-flow","field":"Combinatorial Optimization","model":"GPT-5.6 Pro","ai_role":"Rybin used GPT-5.6 Pro to search for and construct an explicit counterexample: a graph whose fractional flow cost is 58, while every unsplittable flow with capacity violation at most 15 costs at least 60 - so no cost-preserving rounding exists.","posed_by":"Yefim Dinitz, Naveen Garg, Michel Goemans","citations":null,"statement":"For single-source unsplittable flow, every fractional flow can be rounded to an unsplittable flow whose cost is no higher than the fractional cost, while each arc's load is exceeded by at most the maximum demand. (The cost version of Goemans' unsplittable-flow conjecture.)","resolution":"resolved","short_name":"Dinitz-Garg-Goemans","solve_date":"2026-07-22","solve_type":"disproved","source_url":"https://x.com/DmitryRybin1/status/2079904005652893709","year_posed":1999,"field_group":"Algorithms & optimization","model_maker":"OpenAI","publication":"announcement","result_note":null,"source_name":"Dmitry Rybin (X)","significance":20,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-discovered","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A well-known 1999 conjecture in flow approximation algorithms.","verification_note":"Announced on X by Dmitry Rybin (2026-07-22) with a shared GPT-5.6 Pro chat. The counterexample is a concrete finite graph checkable by direct computation (fractional cost 58 vs. minimum unsplittable cost 60 under capacity violation $\\le 15$), but it is not yet peer-reviewed or formally verified. Not to be confused with the separate 'Dinitz conjecture' on Latin-square colourings.","human_collaborators":"[\"Dmitry Rybin\"]"},"metadata_root":"sha256:92a7a51d75ca7181da2f0a918e13f9185d18abab7a0f648a83a1abdab1d16a0f","content_root":"sha256:b06f2bb33360d9e88a2be6f80dd3d1d62fb522066826ebfe2b94b22cd4bdfb40","availability":"reference_only","row_root":"sha256:1606328027d0aeb0b3414c63af85913b7ef56f6ea18a2cd51baec25c772e2a8d"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:directed-3-torus-hamilton","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Hamilton Decompositions of the Directed 3-Torus","summary":"For $D_3(m) = \\vec{C}_m \\square \\vec{C}_m \\square \\vec{C}_m$, can the full arc set be partitioned into three directed Hamilton cycles for every integer $m \\ge 3$?","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/directed-3-torus-hamilton"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2603.24708"}],"metadata":{"name":"Hamilton Decompositions of the Directed 3-Torus","slug":"directed-3-torus-hamilton","field":"Graph decompositions","model":"Claude Opus 4.6, GPT-5.3 Codex, GPT-5.4 Pro","ai_role":"The return-map and odometer reduction, the Kempe-swap constructions for odd $m$, and the clock-and-carry analysis for even $m$ were developed across three frontier models.","posed_by":null,"citations":null,"statement":"For $D_3(m) = \\vec{C}_m \\square \\vec{C}_m \\square \\vec{C}_m$, can the full arc set be partitioned into three directed Hamilton cycles for every integer $m \\ge 3$?","resolution":"resolved","short_name":"3-torus decomposition","solve_date":"2026-03-25","solve_type":"proved","source_url":"https://arxiv.org/abs/2603.24708","year_posed":2026,"field_group":"Combinatorics","model_maker":"Anthropic / OpenAI","publication":"preprint","result_note":null,"source_name":"arXiv:2603.24708 - Hamilton decompositions of the directed 3-torus","significance":5,"verification":"lean-verified","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"argument","significance_note":"A recent concrete decomposition question with a one-paper audience.","verification_note":"A Lean 4 formalization accompanies the construction; arXiv preprint.","human_collaborators":"[]"},"metadata_root":"sha256:1170c30c2862b9a9dfd09b5d6ef5b93fbac890684c0e64f78bd44a01ff44ef39","content_root":"sha256:93aafd2a4acea4aad6e4dc4ad96633a818927f1614e43f0f0e8365853ab1cc6a","availability":"reference_only","row_root":"sha256:6d6d832963aa6a222d6a4abd1ef0db2177556ef848a7aee04b412cde8c358a46"},{"schema":"vela.math-native-record.v1","source_id":"source:vibemathed","observation_root":"sha256:1586c920e7bb919f66bbae81e6563de4c0e57cbe573d2d181df0ceb5248ae264","native_id":"vibemathed:directed-5-torus-hamilton","native_kind":"attributed_activity","native_revision":"sha256:1b075ecdd98f2aad25613e84d61661f7fe173b9e146f9f32a94c69bf9c03f977","title":"Hamilton Decompositions of the Directed 5-Torus, Odd Modulus","summary":"The directed five-dimensional torus $D_5(m)$ has a Hamilton decomposition for every odd $m \\geq 3$, extending the decomposition program for directed tori beyond the three-dimensional case.","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://vibemathed.com/problem/directed-5-torus-hamilton"},{"locator_id":"native-2","kind":"artifact","url":"https://arxiv.org/abs/2604.27140"}],"metadata":{"name":"Hamilton Decompositions of the Directed 5-Torus, Odd Modulus","slug":"directed-5-torus-hamilton","field":"Graph decompositions","model":"GPT-5.5 Pro, GPT-5.5 Codex, Claude Opus 4.7","ai_role":"GPT-5.5 Pro \"contributed to proof exploration, including selector design and block-recurrence case analysis\"; GPT-5.5 Codex drafted the Lean 4 formalization; Claude Opus 4.7 contributed exposition. All mathematical content author-verified.","posed_by":null,"citations":null,"statement":"The directed five-dimensional torus $D_5(m)$ has a Hamilton decomposition for every odd $m \\geq 3$, extending the decomposition program for directed tori beyond the three-dimensional case.","resolution":"resolved","short_name":"Directed 5-torus","solve_date":"2026-04-29","solve_type":"proved","source_url":"https://arxiv.org/abs/2604.27140","year_posed":null,"field_group":"Combinatorics","model_maker":"OpenAI, Anthropic","publication":"preprint","result_note":null,"source_name":"arXiv","significance":5,"verification":"unreviewed","citations_url":null,"problem_number":null,"ai_contribution":"ai-co-developed","citations_paper":null,"citations_source":null,"claim_issue_note":null,"resolution_method":"construction","significance_note":"A concrete decomposition question with a one-paper audience, sibling to the directed 3-torus entry already in the catalog.","verification_note":"A Lean 4 formalization draft exists (cited in the paper) but its completeness is not stated.","human_collaborators":"[\"SangHyun Park\"]"},"metadata_root":"sha256:efcd3774f27601ba75ae3e02e8b31a57bfdf67f3cad0c0dcea735dc5939213d8","content_root":"sha256:15cd3bf2cce9e0b6463e1afee7a47959e4c0b2d5352749af4281e9a28b876e1b","availability":"reference_only","row_root":"sha256:cf52e1071c7026a159b927b69d7a2cfe13a27b218d37751c5405e5a5e55ebdd6"}],"repository_bindings":[],"next_cursor":"WyJuYXRpdmUiLCJzb3VyY2U6dmliZW1hdGhlZCIsInNoYTI1NjoxNTg2YzkyMGU3YmI5MTlmNjZiYmFlODFlNjU2M2RlNGMwZTU3Y2JlNTczZDJkMTgxZGYwY2ViNTI0OGFlMjY0IiwiYXR0cmlidXRlZF9hY3Rpdml0eSIsInZpYmVtYXRoZWQ6ZGlyZWN0ZWQtNS10b3J1cy1oYW1pbHRvbiJd","next_binding_cursor":null}