{
  "checked_on": "2026-10-01",
  "scope": "Selected public research outcomes and contributions. Not a count of all collaborations or all formalized results.",
  "cases": [
    {
      "id": "cloitre-padovan",
      "result": "A Padovan-automatic description of a nested recurrence",
      "status": "Public three-author arXiv preprint and versioned reproducibility archive",
      "sources": [
        "https://arxiv.org/abs/2609.33421",
        "https://doi.org/10.5281/zenodo.22979217"
      ],
      "verification_scope": "Existing Lean proof archive; 69 modules and 18 final theorem audits refer to September 23. No new Lean execution for this presentation.",
      "image_source": "https://github.com/the-omega-institute/a076502-padovan/blob/v1.0.1/manuscript/output/pdf/paper.pdf"
    },
    {
      "id": "sahbi-subquorum",
      "result": "Hypercube sub-quorum coloring number is 2^(n-1) for every n >= 2",
      "status": "Public arXiv v3 lists Haobo Ma, Rafik Sahbi and Zhang Wenlin",
      "sources": [
        "https://arxiv.org/abs/2609.25128v3",
        "https://github.com/the-omega-institute/subquorum-colorings/blob/d8f15435f8ea2b4b97ac94008f1611204cbd1c0b/docs/GRID_RESULTS.md"
      ],
      "verification_scope": "Hypercube proof has existing Lean formalization. Subsequent grid work contains written proofs and exact strip certificates. General grid conjecture remains open."
    },
    {
      "id": "nikandish-orthogonality",
      "result": "Exact dimension-six chromatic numbers: chi(Gamma_6) = 12 for lines and chi(O_6*) = 15 for all nonzero subspaces; a separation of three colors",
      "status": "Joint manuscript and collaborator code in an open PR",
      "source_commit": "6696170d84c5c4cb31be31920e9cb273a9384a02",
      "sources": [
        "https://github.com/the-omega-institute/binary-subspace-orthogonality/pull/1",
        "https://github.com/the-omega-institute/binary-subspace-orthogonality/blob/6696170d84c5c4cb31be31920e9cb273a9384a02/README.md",
        "https://github.com/the-omega-institute/binary-subspace-orthogonality/blob/6696170d84c5c4cb31be31920e9cb273a9384a02/notes/formal-verification-audit.md",
        "https://github.com/the-omega-institute/binary-subspace-orthogonality/blob/6696170d84c5c4cb31be31920e9cb273a9384a02/notes/line-coloring.md"
      ],
      "collaborator_commits": [
        "ae4844d520",
        "ad269bf5cf",
        "144ce045ec",
        "076b518af8",
        "0a239c8989"
      ],
      "verification_scope": "Independent finite certificates establish both exact chromatic numbers, with a weighted independent-set lower bound for the line graph. The full-subspace checker tests 2824 vertices and 3986076 pairs. Historical Lean build has native-evaluation axioms; do not describe as pure kernel validation. The all-dimensional clique formula has a separate existing Lean proof. Dimension seven and a structural full-subspace coloring rule remain open."
    },
    {
      "id": "campbell-cloitre-recurrences",
      "result": "Explicit Campbell recurrence formula with ratio liminf 2/5 and limsup 3/4; global golden structure and Fibonacci collars for Cloitre's recurrence",
      "status": "Public PR #8 merged: exact cap-defect-two/three sublevels, five-row recursive closure and stronger local dispersion bounds",
      "source_commit": "77ad361d14bfb91152ba8862ca61030a0398f359",
      "sources": [
        "https://github.com/the-omega-institute/nested-recurrences/blob/77ad361d14bfb91152ba8862ca61030a0398f359/README.md",
        "https://github.com/the-omega-institute/nested-recurrences/blob/77ad361d14bfb91152ba8862ca61030a0398f359/STATUS.md"
      ],
      "verification_scope": "Written proofs and exact finite checks. New interval and closure results hold on their stated Fibonacci-block domains with supplied context; global convergence and uniform dispersion remain open. No Lean formalization or agreed joint paper authorship."
    },
    {
      "id": "kotesovec-hankel",
      "result": "Analytic next-order constants for Catalan-square Hankel determinants, with exact c_(r+1) = 4 c_r for fixed integer shifts",
      "status": "Substantive exchange and OEIS publication interest; joint note proposed",
      "sources": [
        "https://oeis.org/A277829",
        "https://oeis.org/A278770"
      ],
      "verification_scope": "Initial logarithmic-limit results have existing Lean formalization. Subsequent analytic constants and shift extensions are written proofs; quoted numerical quadrature is not interval-certified. No joint paper or new Lean formalization claimed."
    }
  ],
  "headline_metrics": {
    "recorded_problem_resolutions": 439,
    "proved": 336,
    "refuted": 103,
    "problem_manifest": "problem-resolutions.json",
    "joint_arxiv_preprints": 4,
    "researchers_display": "30+",
    "researcher_count_scope": "38 unique named researchers across research exchanges, proof review, requests for further verification, positive acknowledgement and interest in reviewing the work. Includes copied coauthors already recorded in substantive two-way threads; aliases are deduplicated. This is a research-contact count, not a count of agreed collaborations, papers or institutional endorsements. Automated replies, delivery failures and unanswered outbound messages are excluded.",
    "counted_researchers": [
      "John M. Campbell",
      "Benoit Cloitre",
      "Reza Nikandish",
      "Dan Rust",
      "Gandhar Joshi",
      "Gennadiy Averkov",
      "Katherina von Dichter",
      "Ivan Soprunov",
      "Louxin Zhang",
      "Cedric Chauve",
      "Maximilian F. Hasler",
      "Neetu Chanania",
      "Mohan",
      "R. Venkatesh",
      "Chaithra P",
      "Shushma Rani",
      "Sanjaye Ramgoolam",
      "Garreth Kemp",
      "Sen-Peng Eu",
      "Thomas Ordowski",
      "Václav Kotěšovec",
      "Robert Israel",
      "Wieb Bosma",
      "Jonathan Grube",
      "Robbert Fokkink",
      "Thian Tromp",
      "Rafik Sahbi",
      "Charles E. Hughes",
      "Max A. Alekseyev",
      "Peter Cameron",
      "Peter Kagey",
      "Yusaku Nishimura",
      "Pantelimon Stănică",
      "Aslı Pekcan",
      "Cristina Ballantine",
      "Madeline Locus Dawsey",
      "Chiara Paletta",
      "Zhi-Hong Sun"
    ],
    "published_problem_resolutions": 427,
    "frozen_theorem_statements": 35010,
    "theorem_census_manifest": "theorem-growth.json",
    "researcher_count": 38
  },
  "selected_researcher_affiliations": [
    {
      "person": "John M. Campbell",
      "affiliation_or_community_role": "OpenAI",
      "research_topic": "Nested recurrences",
      "source": "https://github.com/the-omega-institute/nested-recurrences/blob/77ad361d14bfb91152ba8862ca61030a0398f359/STATUS.md",
      "exchange_scope": "Current affiliation directly confirmed by the researcher; public research repository linked.",
      "affiliation_basis": "Directly confirmed by the researcher in current correspondence; no institutional partnership is claimed.",
      "current_research_display": "Explicit formulas and recursive information",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Rafik Sahbi",
      "affiliation_or_community_role": "University of Carthage",
      "research_topic": "Sub-quorum colorings of hypercubes and grids",
      "source": "https://github.com/the-omega-institute/subquorum-colorings/blob/d8f15435f8ea2b4b97ac94008f1611204cbd1c0b/docs/GRID_RESULTS.md",
      "exchange_scope": "Joint public paper and ongoing work on the general grid conjecture. Affiliation is given on the paper first page.",
      "current_research_display": "Hypercube proof and uniform grid bounds",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Benoit Cloitre",
      "affiliation_or_community_role": "OEIS contributor",
      "research_topic": "Padovan and nested recurrences",
      "source": "https://github.com/the-omega-institute/nested-recurrences/blob/77ad361d14bfb91152ba8862ca61030a0398f359/STATUS.md",
      "exchange_scope": "Joint public paper and continuing recurrence research. OEIS contributor is a community role, not an employment affiliation.",
      "current_research_display": "Padovan recurrences and Conway-type dynamics",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Louxin Zhang",
      "affiliation_or_community_role": "National University of Singapore",
      "research_topic": "Word neighborhoods",
      "source": "https://arxiv.org/html/2505.13796v2",
      "exchange_scope": "Research discussion invited; follow-up direction under discussion.",
      "current_research_display": "Minimal word neighborhoods and classification",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Reza Nikandish",
      "affiliation_or_community_role": "Jundi-Shapur University of Technology",
      "research_topic": "Orthogonality graphs",
      "source": "https://github.com/the-omega-institute/binary-subspace-orthogonality/blob/6696170d84c5c4cb31be31920e9cb273a9384a02/notes/line-coloring.md",
      "exchange_scope": "Active shared manuscript and code contributions in public PR #1. Exact dimension-six chromatic numbers now established: 12 for the induced graph on lines and 15 for all nonzero subspaces, using independent finite certificates.",
      "current_research_display": "Dimension six: 12 line colors, 15 subspace colors",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "R. Venkatesh",
      "affiliation_or_community_role": "Indian Institute of Science",
      "research_topic": "Hypergraph positivity",
      "source": "https://arxiv.org/html/2507.20847v1",
      "exchange_scope": "Written positivity classification and sufficient-condition proofs sent in response to the revised problem.",
      "current_research_display": "Hypergraph positivity and sunflower conditions",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Sanjaye Ramgoolam",
      "affiliation_or_community_role": "Queen Mary, London",
      "research_topic": "Determinants and geometry",
      "source": "https://arxiv.org/html/2603.05259",
      "exchange_scope": "Comparing approaches and exploring geometric connections; no joint paper agreed.",
      "current_research_display": "Graph determinants and geometric multiplicities",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Wieb Bosma",
      "affiliation_or_community_role": "Radboud University",
      "research_topic": "Sumfree periodicity",
      "source": "https://www.math.ru.nl/~bosma/",
      "exchange_scope": "Explicit interest in further seed families; general certificate argument shared.",
      "current_research_display": "Sumfree periodic families and finite certificates",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Dan Rust",
      "affiliation_or_community_role": "The Open University, UK",
      "research_topic": "Fibonacci-word bounds",
      "source": "https://arxiv.org/html/2501.05830v2",
      "exchange_scope": "Readable proof and sharper bounds sent; reserved orbit-analysis scope respected.",
      "current_research_display": "Sharper Fibonacci-word progression bounds",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Charles E. Hughes",
      "affiliation_or_community_role": "University of Central Florida",
      "research_topic": "Insertion languages",
      "source": "https://www.cs.ucf.edu/~ceh/",
      "exchange_scope": "Pegasus Professor Emeritus. Reviewed padding proof and proposed further directions.",
      "current_research_display": "Insertion-degree spectra and No-Gap questions",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Sen-Peng Eu",
      "affiliation_or_community_role": "National Taiwan Normal University",
      "research_topic": "Raney spectra",
      "source": "https://arxiv.org/html/2609.25742",
      "exchange_scope": "Interested in exact parameter classification and retrieving existing data.",
      "current_research_display": "Raney zero-run spectra and parameter classification",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Maximilian F. Hasler",
      "affiliation_or_community_role": "OEIS contributor",
      "research_topic": "Digit-substring sequences",
      "source": "https://oeis.org/A359482",
      "exchange_scope": "Proof review and new predecessor data/questions exchanged. OEIS contributor is a community role, not an employment affiliation.",
      "current_research_display": "Digit-substring sequences and predecessor patterns",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Max A. Alekseyev",
      "affiliation_or_community_role": "George Washington University · OEIS",
      "research_topic": "Translation-action proof and formalization feedback",
      "current_research_display": "Translation-action proof and formalization feedback",
      "source": "https://oeis.org/A146557",
      "affiliation_sources": [
        "https://blogs.gwu.edu/maxal/cv/",
        "https://oeis.org/wiki/User:Max_Alekseyev"
      ],
      "affiliation_basis": "University and OEIS profile verified; no unverified editorial office is asserted.",
      "exchange_scope": "Recognized the translation-action argument as an interesting alternative viewpoint and the value of formalization for the project; suggested less immediate test problems.",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Peter Cameron",
      "affiliation_or_community_role": "University of St Andrews",
      "research_topic": "Review of the five-element semigroup counterexample",
      "current_research_display": "Reviewed our semigroup counterexample",
      "source": "https://arxiv.org/abs/2608.25092",
      "affiliation_sources": [
        "https://www.st-andrews.ac.uk/mathematics-statistics/people/pjc20"
      ],
      "affiliation_basis": "University profile identifies Emeritus Professor of Mathematics.",
      "exchange_scope": "Read the five-element semigroup counterexample note and confirmed that the argument looked sound.",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Robert Israel",
      "affiliation_or_community_role": "University of British Columbia · OEIS",
      "research_topic": "Laguerre parity and cyclotomic sequence proofs",
      "current_research_display": "Proof feedback and OEIS update support",
      "source": "https://oeis.org/A160627",
      "affiliation_sources": [
        "https://oeis.org/wiki/User:Robert_Israel"
      ],
      "affiliation_basis": "OEIS profile identifies emeritus Associate Professor of Mathematics at UBC.",
      "exchange_scope": "Found the results interesting and offered to update OEIS with proof links and author attribution.",
      "topic_checked_on": "2026-10-01"
    },
    {
      "person": "Václav Kotěšovec",
      "affiliation_or_community_role": "OEIS · Asymptotics",
      "research_topic": "Catalan-square Hankel determinant asymptotics",
      "current_research_display": "Hankel constants, precision checks and OEIS links",
      "source": "https://oeis.org/A277829",
      "affiliation_sources": [
        "https://oeis.org/wiki/User:Vaclav_Kotesovec"
      ],
      "affiliation_basis": "OEIS contributor and asymptotics researcher, verified through his OEIS profile. No executive or editorial title is asserted.",
      "exchange_scope": "Responded positively to the analytic next-order constants, supplied a high-precision numerical calculation and asked for a publication link to include in OEIS. His follow-up question led to written fixed-shift and growing-shift results. A joint note has been proposed; authorship has not been agreed.",
      "topic_checked_on": "2026-10-01"
    }
  ],
  "eqt2": {
    "official_results": {
      "source": "Official My Submission page text supplied by the team owner in the conversation on 2026-10-01",
      "verification_method": "User-provided official page transcription; not independently fetched",
      "participations": [
        {
          "track": "Solo",
          "model": "Google: Gemma 4 31B",
          "official_rank": 1,
          "score": 200,
          "maximum_score": 200
        },
        {
          "track": "Solo",
          "model": "OpenAI: gpt-oss-120b",
          "official_rank": 1,
          "score": 200,
          "maximum_score": 200
        },
        {
          "track": "Marathon",
          "model": "Google: Gemma 4 31B",
          "official_rank": 1,
          "score": 200,
          "maximum_score": 200
        },
        {
          "track": "Marathon",
          "model": "OpenAI: gpt-oss-120b",
          "official_rank": 1,
          "score": 200,
          "maximum_score": 200
        }
      ]
    },
    "internal_testing": {
      "certified": 1889,
      "total": 1889,
      "scope": "Internal test suite, not the official competition score. Scope explicitly clarified by the team owner on 2026-10-01."
    },
    "solver_llm_calls": 0,
    "model_label_scope": "Competition participation settings; these labels do not imply the solver invoked either model.",
    "public_artifacts": [
      "https://github.com/the-omega-institute/sair-eqt2-stage2-solver",
      "https://arxiv.org/abs/2609.00706"
    ]
  },
  "update_sources": {
    "checked_on": "2026-10-01",
    "published_catalog_source_commit": "b6b6592977e7132b4d568b164ec8f20f914dcf7d",
    "growth_endpoint_source_commit": "b6b6592977e7132b4d568b164ec8f20f914dcf7d",
    "scope": "Counts retain their original definitions. New collaboration results link to public proofs and finite certificates; correspondence is not included.",
    "research_contact_review": "2026-10-01; verified reply records and public profiles, with the user-requested scope extended to recognition and research feedback."
  },
  "research_contact_updates": [
    {
      "person": "Max A. Alekseyev",
      "relationship": "proof_feedback",
      "scope": "Recognized the translation-action argument as an interesting alternative viewpoint and the value of formalization for the project; suggested less immediate test problems."
    },
    {
      "person": "Peter Cameron",
      "relationship": "proof_review",
      "scope": "Read the five-element semigroup counterexample note and confirmed that the argument looked sound."
    },
    {
      "person": "Peter Kagey",
      "relationship": "research_feedback",
      "scope": "Encouraged wider-grid recurrence research and requested OEIS updates."
    },
    {
      "person": "Yusaku Nishimura",
      "relationship": "new_problem_invitation",
      "scope": "Invited investigation of the exact zero set in the seven-point harmonic-strength problem."
    },
    {
      "person": "Pantelimon Stănică",
      "relationship": "acknowledgement_and_review_interest",
      "scope": "Acknowledged the results and expressed an intention to examine the research repository."
    },
    {
      "person": "Aslı Pekcan",
      "relationship": "review_interest",
      "scope": "Responded with a plan to examine the four-soliton argument and provide feedback."
    },
    {
      "person": "Cristina Ballantine",
      "relationship": "verification_request",
      "scope": "Requested a specific n=14 verification while declining ongoing AI collaboration."
    },
    {
      "person": "Madeline Locus Dawsey",
      "relationship": "recognition_and_research_direction",
      "scope": "Recognized the counterexample contribution, identified the fixed-size/fixed-length direction and offered future acknowledgement; currently unavailable for collaboration."
    },
    {
      "person": "Chiara Paletta",
      "relationship": "recognition_and_acknowledgement",
      "scope": "Confirmed the conjecture issue and offered acknowledgement in the authors’ forthcoming revision."
    },
    {
      "person": "Zhi-Hong Sun",
      "relationship": "proof_review",
      "scope": "Expressed confidence in the proof and supplied an earlier paper solving the conjectures; the result is an independent proof, not a first resolution."
    },
    {
      "person": "Václav Kotěšovec",
      "relationship": "research_exchange_and_oeis_publication_interest",
      "scope": "Recognized the analytic Hankel-constant results, supplied a high-precision calculation and requested a public note link for OEIS. Already included in the 38-person count; no additional contact counted."
    }
  ],
  "displayed_researchers": [
    "John M. Campbell",
    "Rafik Sahbi",
    "Benoit Cloitre",
    "Louxin Zhang",
    "Reza Nikandish",
    "Max A. Alekseyev",
    "Sanjaye Ramgoolam",
    "Peter Cameron",
    "Robert Israel",
    "Wieb Bosma",
    "R. Venkatesh",
    "Václav Kotěšovec"
  ],
  "workshop_papers": [
    {
      "title": "Cross-Engine Admission Contracts for Autonomous Formalization",
      "authors": [
        "Haobo Ma",
        "Wenlin Zhang"
      ],
      "venue": "NeurIPS 2026, 6th MATH-AI Workshop",
      "status": "Accepted for poster presentation",
      "acceptance_basis": "Direct author confirmation in the conversation on 2026-10-01.",
      "manuscript_sha256": "3c8dd439bf9b6e0078bb5f3b015ba68f4c3969a3f646a44985f1e51c7b1bab5a",
      "scope": "Earlier harness engineering: distinct checks for kernel validity, allowed dependencies and statement faithfulness, with differential testing of admission-policy adapters. Not a complete account of current Trureturing.",
      "public_url": null
    }
  ],
  "joint_arxiv_papers": {
    "count": 4,
    "checked_on": "2026-10-01",
    "scope": "Includes the two earlier community collaborations with Manuel Israel Cázares, as confirmed by the project owner. Title and authorship for both earlier papers verified against public arXiv records. The separate Math-AI workshop poster is not counted here.",
    "papers": [
      {
        "id": "cloitre-padovan",
        "title": "A Padovan-automatic description of a nested recurrence",
        "url": "https://arxiv.org/abs/2609.33421",
        "collaborator": "Benoit Cloitre"
      },
      {
        "id": "sahbi-subquorum",
        "title": "Sub-quorum colorings of graphs",
        "url": "https://arxiv.org/abs/2609.25128v3",
        "collaborator": "Rafik Sahbi"
      },
      {
        "id": "eqt2-cascade",
        "title": "A Certificate-Producing Cascade for Equational Implication: The SAIR EQT2 Stage 2 Solver",
        "url": "https://arxiv.org/abs/2609.00706",
        "authors": [
          "Haobo Ma",
          "Wenlin Zhang",
          "Manuel Israel Cázares"
        ],
        "collaborator": "Manuel Israel Cázares",
        "published": "2026-09-01"
      },
      {
        "id": "mechanism-routing",
        "title": "Mechanism-level routing failure in LLMs over Lean-verified algebraic structures",
        "url": "https://arxiv.org/abs/2607.04534",
        "authors": [
          "Manuel Israel Cázares",
          "Wenlin Zhang",
          "Haobo Ma"
        ],
        "collaborator": "Manuel Israel Cázares",
        "published": "2026-07-05"
      }
    ]
  }
}