{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:ame-states-five-open-cases", "title": "Absolutely Maximally Entangled States in Five Open Cases", "canonical_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 $\\mathrm{AME}(17,13)$.", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:ame-states-five-open-cases:event", "problem_id": "vibemathed:ame-states-five-open-cases", "type": "proved", "status": "resolved", "occurred_at": "2026-08-06T00:00:00.000Z", "title": "Symmetry-guided constructions of AME states in five open cases", "summary": "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.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:ame-states-five-open-cases:attempt" ], "method_family_ids": [ "vibemathed:ame-states-five-open-cases:method" ], "source_assertion_ids": [ "vibemathed:ame-states-five-open-cases:assertion" ] } ], "attempts": [ { "id": "vibemathed:ame-states-five-open-cases:attempt", "problem_id": "vibemathed:ame-states-five-open-cases", "model": "Claude Fable 5, ChatGPT 5.6 Sol", "provider": "Anthropic, OpenAI", "attempted_at": "2026-08-06T00:00:00.000Z", "human_collaborators": [ "Samuel Bevins", "Yunus Bidav" ], "exposure": "unknown", "prompt_public": null, "outcome": "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.", "failed_routes": [], "artifacts": [ { "label": "Huber and Wyderka, Table of AME states - the existence table these cells come from", "url": "https://tp.nt.uni-siegen.de/ame/ame.html", "kind": "problem-record", "license": null }, { "label": "Grassl, Tables of linear codes and quantum codes", "url": "https://www.codetables.de/", "kind": "problem-record", "license": null } ], "sources": [ { "label": "Symmetry-guided constructions of AME states in five open cases", "url": "https://arxiv.org/abs/2608.05781", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Absolutely Maximally Entangled States in Five Open Cases", "url": "https://vibemathed.com/problem/ame-states-five-open-cases", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:ame-states-five-open-cases:verification", "problem_id": "vibemathed:ame-states-five-open-cases", "solution_event_id": "vibemathed:ame-states-five-open-cases:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "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.", "sources": [ { "label": "VibeMathed: Absolutely Maximally Entangled States in Five Open Cases", "url": "https://vibemathed.com/problem/ame-states-five-open-cases", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Symmetry-guided constructions of AME states in five open cases", "url": "https://arxiv.org/abs/2608.05781", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Symmetry-guided constructions of AME states in five open cases", "url": "https://arxiv.org/abs/2608.05781", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Absolutely Maximally Entangled States in Five Open Cases", "url": "https://vibemathed.com/problem/ame-states-five-open-cases", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:ame-states-five-open-cases:signal:watch_only", "problem_id": "vibemathed:ame-states-five-open-cases", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }