{
  "title": "The frontiers of mathematics",
  "reviewed": "2026-10-20",
  "scope": "Selected historical milestones; not an exhaustive current-record database.",
  "problems": [
    {
      "id": "primes",
      "name": "Twin prime gaps",
      "kind": "bound",
      "category": "01 / Closing in on a target",
      "metric": "Selected upper bound on recurring prime gaps",
      "question": "Are there infinitely many prime pairs just two apart?",
      "explanation": "Prime pairs such as 11 and 13, or 17 and 19, differ by 2. The twin-prime conjecture says there are infinitely many such pairs.",
      "unit": "Prime gap H₁",
      "start": 2013,
      "end": 2027,
      "yearTicks": [
        2013,
        2016,
        2020,
        2023,
        2026
      ],
      "domain": [
        1,
        100000000
      ],
      "ticks": [
        1,
        100,
        10000,
        1000000,
        100000000
      ],
      "target": 2,
      "targetLabel": "Twin-prime target: 2",
      "chartDescription": "Selected upper bounds on the limit inferior of consecutive prime gaps, on a logarithmic scale.",
      "chartNote": "H₁ is the limit inferior of consecutive prime gaps. A bound means infinitely many pairs are at least this close, not that all gaps are this small. Dates use public reports.",
      "interpretation": "The published bound fell from 70 million to 246 in just over a year. The conjecture asks for a gap of 2. The reported 186 result has a conditional Lean formalization, not a complete formal proof.",
      "note": "An independent August 31, 2026 <a href=\"https://arxiv.org/abs/2608.31126\" target=\"_blank\" rel=\"noopener\">preprint by Julia Stadlmann</a> gives 240. It is not another step here: the 186 manuscript is dated August 30. These are manuscript dates, not a priority judgment.",
      "milestones": [
        {
          "date": "2013-05",
          "label": "May 2013",
          "value": 70000000,
          "who": "Yitang Zhang",
          "title": "The first finite bound",
          "detail": "Zhang proved that infinitely many consecutive prime pairs have a gap below 70 million. Accepted in May 2013; published in Annals of Mathematics in 2014.",
          "url": "https://doi.org/10.4007/annals.2014.179.3.7",
          "status": "published",
          "display": "< 70,000,000",
          "short": "Zhang"
        },
        {
          "date": "2013-09-02",
          "label": "Sep 2013",
          "value": 4680,
          "who": "D. H. J. Polymath / Polymath8a",
          "title": "Polymath reduces the bound to 4,680",
          "detail": "A worldwide collaboration rapidly reduced the bound. Tao’s September 2013 account records 4,680; the paper followed in 2014.",
          "url": "https://terrytao.wordpress.com/2013/09/02/polymath8-writing-the-paper-ii/comment-page-1/",
          "status": "published",
          "display": "≤ 4,680",
          "short": "Polymath8a"
        },
        {
          "date": "2013-11-19",
          "label": "Nov 2013",
          "value": 600,
          "who": "James Maynard; method independently developed by Terence Tao",
          "title": "Maynard’s sieve gives 600",
          "detail": "Maynard’s preprint gives a bound of 600 with a different sieve method. Published in Annals of Mathematics in 2015.",
          "url": "https://arxiv.org/abs/1311.4600",
          "status": "published",
          "display": "≤ 600",
          "short": "Maynard"
        },
        {
          "date": "2014-07-18",
          "label": "Jul 2014",
          "value": 246,
          "who": "D. H. J. Polymath / Polymath8b",
          "title": "Polymath reduces the bound to 246",
          "detail": "The collaboration refined the new method and proved the unconditional bound 246. Published in Research in the Mathematical Sciences in 2014.",
          "url": "https://arxiv.org/abs/1407.4897",
          "status": "published",
          "display": "≤ 246",
          "short": "Polymath8b"
        },
        {
          "date": "2026-08-30",
          "label": "Aug 2026",
          "value": 186,
          "who": "OpenAI / GPT-6 Astra",
          "title": "A reported bound of 186",
          "detail": "The manuscript claims an unconditional mathematical bound of 186. Its Lean formalization assumes three input axioms, covering exponential-sum estimates and numerical bounds. The repository makes these dependencies explicit.",
          "url": "https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf",
          "status": "preprint",
          "display": "≤ 186",
          "short": "Preprint · partial formalization",
          "ai": "The paper credits GPT-6 Astra with the proof. The Lean development remains conditional on input axioms; numerical checks use Python/FLINT.",
          "supportUrl": "https://github.com/openai/PrimeGaps186/blob/main/README.md"
        }
      ]
    },
    {
      "id": "riemann",
      "name": "Riemann hypothesis",
      "kind": "proportion",
      "category": "02 / Critical-line proportions & zero-free regions",
      "metric": "Selected lower bound on the critical-line proportion",
      "question": "Do all nontrivial zeta zeros lie on one line?",
      "explanation": "The Riemann hypothesis places every nontrivial zero on the line with real part ½. The proportion chart tracks how many zeros must lie there; the zero-free view bounds where any zero can lie.",
      "unit": "Asymptotic proportion on the critical line",
      "start": 1970,
      "end": 2028,
      "yearTicks": [
        1974,
        1989,
        2000,
        2010,
        2026
      ],
      "domain": [
        0,
        100
      ],
      "ticks": [
        0,
        25,
        50,
        75,
        100
      ],
      "target": 100,
      "targetLabel": "RH would imply 100%",
      "chartDescription": "Selected proven lower bounds on the asymptotic proportion of zeta zeros on the critical line.",
      "chartNote": "This is an asymptotic lower density, not a count of zeros checked by a computer. Earlier published results are plotted at their public or publication year.",
      "interpretation": "Even a proven proportion of 100% would not by itself prove the Riemann hypothesis: a zero-density set of exceptions could remain. This chart is not a percentage solved.",
      "note": "An independent <a href=\"https://arxiv.org/abs/2609.02882\" target=\"_blank\" rel=\"noopener\">September 2026 preprint by Youness Lamzouri</a> gives a simpler proof of the same 67.25% bound. It adds evidence, not another upward step. This is a selected history, not an exhaustive ledger of every recent claim.",
      "milestones": [
        {
          "date": "1974",
          "label": "1974",
          "value": 34.74,
          "who": "Norman Levinson",
          "title": "Levinson’s 34.74% bound",
          "detail": "Levinson proved that more than one third of the zeros lie on the critical line. The numerical 34.74% version is documented in the Bui–Conrey–Young paper’s historical introduction.",
          "url": "https://doi.org/10.1016/0001-8708(74)90074-7",
          "status": "published",
          "display": "≥ 34.74%",
          "short": "Levinson",
          "supportUrl": "https://arxiv.org/html/1002.4127v2"
        },
        {
          "date": "1989",
          "label": "1989",
          "value": 40.88,
          "who": "J. Brian Conrey",
          "title": "Conrey’s 40.88% bound",
          "detail": "Conrey’s published result increased the lower bound to more than 40.88%. The numerical value is documented in the Pratt–Robles–Zaharescu–Zeindler introduction.",
          "url": "https://doi.org/10.1515/crll.1989.399.1",
          "status": "published",
          "display": "> 40.88%",
          "short": "Conrey",
          "supportUrl": "https://arxiv.org/html/1802.10521v3"
        },
        {
          "date": "2010-02-22",
          "label": "2010",
          "value": 41.05,
          "who": "Hung Bui, Brian Conrey, Matthew Young",
          "title": "A lower bound of 41.05%",
          "detail": "Theorem 1.1 establishes a lower bound of 41.05%. First posted in 2010 and published in Acta Arithmetica in 2011.",
          "url": "https://arxiv.org/abs/1002.4127",
          "status": "published",
          "display": "≥ 41.05%",
          "short": "Bui · Conrey · Young"
        },
        {
          "date": "2018-02-28",
          "label": "2018",
          "value": 41.7293,
          "who": "Kyle Pratt, Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler",
          "title": "A lower bound of 41.7293%",
          "detail": "The paper’s refined numerical bound is 41.7293%, slightly better than five twelfths. The preprint appeared in 2018; publication followed in 2020.",
          "url": "https://arxiv.org/abs/1802.10521",
          "status": "published",
          "display": "> 41.7293%",
          "short": "Pratt et al."
        },
        {
          "date": "2026-08-13",
          "label": "2026",
          "value": 67.25,
          "who": "Levent Alpöge and Ralph Furman",
          "title": "A reported lower bound of 67.25%",
          "detail": "The preprint proves a lower bound of 67.25% for zeros that are both simple and on the critical line. The authors say they verified the argument and take responsibility for it.",
          "url": "https://arxiv.org/abs/2608.13637",
          "status": "preprint",
          "display": "≥ 67.25%",
          "short": "Alpöge · Furman",
          "ai": "The authors credit Claude with discovering and writing the argument. They checked it themselves; journal peer review is not established here."
        }
      ],
      "zeroFreeRegions": [
        {
          "id": "eleven-twelfths",
          "label": "11/12",
          "numerator": 11,
          "denominator": 12,
          "date": "2026-10-05",
          "dateLabel": "October 5, 2026",
          "status": "preprint",
          "who": "OpenAI",
          "title": "A reported zero-free half-plane beyond 11/12",
          "detail": "The preprint states that ζ(s) has no zeros with real part greater than 11/12. Symmetry then confines all nontrivial zeros to 1/12 ≤ β ≤ 11/12. This paper isolates a simpler argument for a weaker bound than 7/8.",
          "url": "https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf",
          "ai": "OpenAI is the listed author of the preprint."
        },
        {
          "id": "seven-eighths",
          "label": "7/8",
          "numerator": 7,
          "denominator": 8,
          "date": "2026-09-30",
          "dateLabel": "September 30, 2026",
          "status": "preprint",
          "who": "OpenAI",
          "title": "A reported zero-free half-plane beyond 7/8",
          "detail": "The preprint states that ζ(s) has no zeros with real part greater than 7/8. Symmetry then confines all nontrivial zeros to 1/8 ≤ β ≤ 7/8. This narrower strip still permits zeros away from the critical line.",
          "url": "https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf",
          "ai": "OpenAI is the listed author of the preprint."
        },
        {
          "id": "rh",
          "label": "Full RH",
          "numerator": 1,
          "denominator": 2,
          "date": "1859",
          "dateLabel": "Conjectured in 1859",
          "status": "conjecture",
          "who": "Bernhard Riemann · overview by the Clay Mathematics Institute",
          "title": "The conjectured limit: every zero on the line",
          "detail": "Full RH requires every nontrivial zero to have real part exactly ½, without exceptions. The permitted strip would collapse to the critical line. This is a conjectured reference, not an established bound.",
          "url": "https://www.claymath.org/millennium/riemann-hypothesis/"
        }
      ]
    },
    {
      "id": "matrix",
      "name": "Matrix multiplication",
      "kind": "bound",
      "category": "03 / Lowering an exponent",
      "metric": "Selected upper bound on the exponent ω",
      "question": "How fast can we multiply two square matrices?",
      "explanation": "The standard algorithm takes roughly n³ arithmetic operations. Faster algorithms reduce the exponent. The conjectured optimum is 2.",
      "unit": "Arithmetic-complexity exponent ω · linear scale",
      "start": 1969,
      "end": 2028,
      "yearTicks": [
        1969,
        1990,
        2010,
        2026
      ],
      "domain": [
        2,
        3
      ],
      "ticks": [
        2,
        2.25,
        2.5,
        2.75,
        3
      ],
      "target": 2,
      "targetLabel": "Conjectured target: 2",
      "chartDescription": "Selected upper bounds on the matrix multiplication exponent, with conjectured target 2.",
      "chartNote": "ω describes asymptotic field-operation complexity, not benchmarked runtime. The zoom view reveals recent improvements without moving the target.",
      "interpretation": "The exponent has fallen by less than 0.002 since 1990. These asymptotic bounds describe performance as matrix sizes grow; they do not necessarily improve practical runtimes.",
      "milestones": [
        {
          "date": "1969",
          "label": "1969",
          "value": 2.807354922057604,
          "who": "Volker Strassen",
          "title": "Strassen’s recursive algorithm",
          "detail": "Strassen broke the cubic barrier with a recursive algorithm, giving an exponent of log₂(7), approximately 2.807355.",
          "url": "https://link.springer.com/article/10.1007/BF02165411",
          "status": "published",
          "display": "≤ log₂ 7",
          "short": "Strassen"
        },
        {
          "date": "1990",
          "label": "1990",
          "value": 2.375477,
          "who": "Don Coppersmith and Shmuel Winograd",
          "title": "The Coppersmith–Winograd bound",
          "detail": "The Coppersmith–Winograd method pushed the exponent below 2.376. This chart uses 2.375477 from Tao’s numerical reference ledger and the journal publication year, 1990; conference presentation was in 1987.",
          "url": "https://www.sciencedirect.com/science/article/pii/S0747717108800132",
          "status": "published",
          "display": "≈ 2.375477",
          "short": "Coppersmith · Winograd",
          "supportUrl": "https://teorth.github.io/optimizationproblems/constants/15a.html"
        },
        {
          "date": "2014-01-30",
          "label": "2014",
          "value": 2.3728639,
          "who": "François Le Gall",
          "title": "Le Gall’s tensor analysis",
          "detail": "New analysis gave an exponent below 2.3728639. Posted and published at ISSAC in 2014.",
          "url": "https://arxiv.org/abs/1401.7714",
          "status": "published",
          "display": "< 2.3728639",
          "short": "Le Gall"
        },
        {
          "date": "2020-10-12",
          "label": "2020",
          "value": 2.3728596,
          "who": "Josh Alman and Virginia Vassilevska Williams",
          "title": "A refined laser method",
          "detail": "A stronger analysis lowered the exponent to 2.3728596. Preprint in 2020; SODA publication in 2021.",
          "url": "https://arxiv.org/abs/2010.05846",
          "status": "published",
          "display": "< 2.3728596",
          "short": "Alman · Williams"
        },
        {
          "date": "2022-10-18",
          "label": "2022",
          "value": 2.371866,
          "who": "Ran Duan, Hongxun Wu, Renfei Zhou",
          "title": "Asymmetric hashing improves the bound",
          "detail": "The asymmetric hashing approach improved the exponent to below 2.371866. Preprint in 2022; FOCS publication in 2023.",
          "url": "https://arxiv.org/abs/2210.10173",
          "status": "published",
          "display": "< 2.371866",
          "short": "Duan · Wu · Zhou"
        },
        {
          "date": "2023-07-16",
          "label": "2023",
          "value": 2.371552,
          "who": "Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou",
          "title": "An upper bound of 2.371552",
          "detail": "The authors report an upper bound of 2.371552. Preprint in 2023; SODA publication in 2024.",
          "url": "https://arxiv.org/abs/2307.07970",
          "status": "published",
          "display": "≤ 2.371552",
          "short": "Williams et al."
        },
        {
          "date": "2024-04-25",
          "label": "2024",
          "value": 2.371339,
          "who": "Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou",
          "title": "An upper bound of 2.371339",
          "detail": "Further refinements yield an exponent below 2.371339. First posted in April 2024 and published at SODA in 2025.",
          "url": "https://arxiv.org/abs/2404.16349",
          "status": "published",
          "display": "< 2.371339",
          "short": "Alman et al."
        },
        {
          "date": "2026-08-17",
          "label": "2026",
          "value": 2.371177,
          "who": "Emilien Dupont and collaborators",
          "title": "AlphaEvolve-assisted optimization",
          "detail": "The preprint combines gradient optimization with an algorithm refined by AlphaEvolve. It reports an exponent below 2.371177 and rigorous verification of the resulting bound.",
          "url": "https://arxiv.org/abs/2608.16884",
          "status": "preprint",
          "display": "< 2.371177",
          "short": "Dupont et al.",
          "ai": "AlphaEvolve refined the optimization algorithm after gradient-based optimization had already improved the existing bound. This is distinct from small-matrix records by AlphaTensor."
        }
      ]
    },
    {
      "id": "ramsey",
      "name": "Ramsey numbers",
      "kind": "range",
      "category": "04 / Narrowing the possibilities",
      "metric": "Possible values of R(5,5) in this snapshot",
      "question": "How big a party guarantees a group of five?",
      "explanation": "Assume acquaintance is mutual. How many guests guarantee five who all know one another—or five who are all strangers? That number is R(5,5).",
      "unit": "Guests · upper and lower bounds",
      "start": 1992,
      "end": 2026,
      "yearTicks": [
        1992,
        2000,
        2010,
        2017,
        2024
      ],
      "domain": [
        40,
        55
      ],
      "ticks": [
        40,
        43,
        46,
        49,
        52,
        55
      ],
      "chartDescription": "Upper and lower bounds on R(5,5). The upper bound descends from 53 to 46; the lower bound stays at 43.",
      "chartNote": "The shaded band contains the possible integer answers. Lower bound 43 dates to Exoo’s 1989 construction; this selected series begins in 1992.",
      "interpretation": "The answer is one of 43, 44, 45, or 46. The lower bound has stood since 1989. Almost all the narrowing shown here comes from above.",
      "milestones": [
        {
          "date": "1992",
          "label": "1992",
          "value": 53,
          "who": "Brendan McKay and Stanisław Radziszowski; lower bound by Geoffrey Exoo",
          "title": "Eleven possible answers",
          "detail": "The range is 43 through 53, inclusive. The authors’ later presentation records the 1992 upper bound and the subsequent improvements.",
          "url": "https://ggtw.ugent.be/2017/slides/McKay_GGTW2017.pdf",
          "status": "published",
          "lower": 43,
          "short": "McKay · Radziszowski"
        },
        {
          "date": "1994",
          "label": "1994",
          "value": 52,
          "who": "Brendan McKay and Stanisław Radziszowski",
          "title": "One more possibility eliminated",
          "detail": "The upper bound falls to 52. Historical date and value are recorded in McKay and Angeltveit’s 2017 presentation.",
          "url": "https://ggtw.ugent.be/2017/slides/McKay_GGTW2017.pdf",
          "status": "published",
          "lower": 43,
          "short": "McKay · Radziszowski"
        },
        {
          "date": "1995",
          "label": "1995",
          "value": 50,
          "who": "Brendan McKay and Stanisław Radziszowski",
          "title": "Below fifty-one",
          "detail": "The upper bound falls to 50, leaving eight possible integer answers. Historical date and value follow the authors’ presentation.",
          "url": "https://ggtw.ugent.be/2017/slides/McKay_GGTW2017.pdf",
          "status": "published",
          "lower": 43,
          "short": "McKay · Radziszowski"
        },
        {
          "date": "1997",
          "label": "1997",
          "value": 49,
          "who": "Brendan McKay and Stanisław Radziszowski",
          "title": "A bound that holds for twenty years",
          "detail": "The upper bound reaches 49. The next improvement in this selected history does not arrive until 2017.",
          "url": "https://ggtw.ugent.be/2017/slides/McKay_GGTW2017.pdf",
          "status": "published",
          "lower": 43,
          "short": "McKay · Radziszowski"
        },
        {
          "date": "2017-03-26",
          "label": "2017",
          "value": 48,
          "who": "Vigleik Angeltveit and Brendan McKay",
          "title": "Six possible answers remain",
          "detail": "The authors prove R(5,5) ≤ 48 with computer-assisted methods. The preprint appeared in 2017 and was published in 2018.",
          "url": "https://arxiv.org/abs/1703.08768",
          "status": "published",
          "lower": 43,
          "short": "Angeltveit · McKay"
        },
        {
          "date": "2024-09-24",
          "label": "2024",
          "value": 46,
          "who": "Vigleik Angeltveit and Brendan McKay",
          "title": "Down to four integers",
          "detail": "Mathematics, linear programming and extensive computation establish R(5,5) ≤ 46. The authors report independent replication. Preprint in 2024; journal publication in 2026.",
          "url": "https://arxiv.org/abs/2409.15709",
          "status": "published",
          "lower": 43,
          "short": "Angeltveit · McKay",
          "supportUrl": "https://onlinelibrary.wiley.com/doi/10.1002/jgt.70029"
        }
      ]
    },
    {
      "id": "perfect",
      "name": "Odd perfect numbers",
      "kind": "floor",
      "category": "05 / Excluding a larger range",
      "metric": "Selected lower bound on an odd perfect number",
      "question": "Can a perfect number be odd?",
      "explanation": "6 is perfect: its proper positive divisors, 1, 2 and 3, add up to 6. Every known perfect number is even. Any odd perfect number must exceed the lower bounds shown here.",
      "unit": "Excluded size · exponent of 10 (publication years)",
      "start": 1957,
      "end": 2020,
      "yearTicks": [
        1957,
        1973,
        1991,
        2012
      ],
      "domain": [
        0,
        2200
      ],
      "ticks": [
        0,
        500,
        1000,
        1500,
        2000
      ],
      "chartDescription": "Published lower bounds on the size of an odd perfect number, followed by an unpublished 2014 report. Vertical values are powers of ten.",
      "chartNote": "The vertical axis plots the exponent: 1,500 means 10^1500. Dates are publication years, except the 2014 author presentation. The undated 10^2200 extension is not plotted.",
      "interpretation": "Ruling out every candidate below 10^1500 still leaves infinitely many larger candidates. A growing exclusion bound does not tell us whether an odd perfect number exists.",
      "note": "Pascal Ochem’s <a href=\"https://www.lirmm.fr/~ochem/opn/\" target=\"_blank\" rel=\"noopener\">institutional project page</a> now reports a bound of 10^2200. Its announcement date was not verified, so it is not assigned a 2026 position on this timeline.",
      "milestones": [
        {
          "date": "1957",
          "label": "1957",
          "value": 20,
          "who": "Hans-Joachim Kanold",
          "title": "A lower bound of 10²⁰",
          "detail": "Kanold’s 1957 lower bound of 10^20 is recorded in the introduction to Hagis’s original 1973 paper. This milestone is sourced through that later mathematical paper.",
          "url": "https://community.ams.org/journals/mcom/1973-27-124/S0025-5718-1973-0325507-9/S0025-5718-1973-0325507-9.pdf",
          "status": "published",
          "short": "Kanold"
        },
        {
          "date": "1973",
          "label": "1973",
          "value": 50,
          "who": "Peter Hagis Jr.",
          "title": "A lower bound of 10⁵⁰",
          "detail": "Hagis proves that an odd perfect number must exceed 10^50. The date is the publication year; this selected timeline omits some intermediate bounds.",
          "url": "https://community.ams.org/journals/mcom/1973-27-124/S0025-5718-1973-0325507-9/S0025-5718-1973-0325507-9.pdf",
          "status": "published",
          "short": "Hagis"
        },
        {
          "date": "1989",
          "label": "1989",
          "value": 160,
          "who": "Richard Brent and Graeme Cohen",
          "title": "A lower bound of 10¹⁶⁰",
          "detail": "The published paper raises the lower bound to 10^160 using restrictions on the prime factors of an odd perfect number.",
          "url": "https://maths-people.anu.edu.au/~brent/pub/pub100.html",
          "status": "published",
          "short": "Brent · Cohen"
        },
        {
          "date": "1991",
          "label": "1991",
          "value": 300,
          "who": "Richard Brent, Graeme Cohen, Herman te Riele",
          "title": "A lower bound of 10³⁰⁰",
          "detail": "The journal paper proves the 10^300 lower bound. A full technical report had already appeared in October 1989; this chart consistently uses the publication year.",
          "url": "https://maths-people.anu.edu.au/~brent/pub/pub116.html",
          "status": "published",
          "short": "Brent · Cohen · te Riele"
        },
        {
          "date": "2012",
          "label": "2012",
          "value": 1500,
          "who": "Pascal Ochem and Michaël Rao",
          "title": "The published bound reaches 10¹⁵⁰⁰",
          "detail": "A computer-assisted proof establishes that any odd perfect number exceeds 10^1500. Published in Mathematics of Computation in 2012.",
          "url": "https://www.lirmm.fr/~ochem/opn/opn.pdf",
          "status": "published",
          "short": "Ochem · Rao"
        },
        {
          "date": "2014",
          "label": "2014",
          "value": 2000,
          "who": "Pascal Ochem and Michaël Rao",
          "title": "An unpublished bound of 10²⁰⁰⁰",
          "detail": "The authors’ dated Montpellier presentation reports 10^2000 and explicitly labels the extension unpublished. It is displayed as an author report, separately from the published bound.",
          "url": "https://www.lirmm.fr/~ochem/opn/opn_slide.pdf",
          "status": "report",
          "short": "Unpublished author report"
        }
      ]
    },
    {
      "id": "hadamard",
      "name": "Hadamard matrices",
      "kind": "grid",
      "category": "06 / Filling in the missing cases",
      "metric": "Unresolved orders through 2,000 in this snapshot",
      "question": "Can every multiple of four make a perfect ±1 grid?",
      "explanation": "A Hadamard matrix has entries +1 or −1 and mutually perpendicular rows. The conjecture asks for one at every positive order divisible by four.",
      "unit": "Unresolved matrix orders",
      "chartDescription": "A grid of 500 orders divisible by four, from 4 to 2000. Published snapshots leave 13, then 12 gaps; 2026 constructions are reported for the remaining 12.",
      "chartNote": "Only orders divisible by four, from 4 to 2,000, are shown. Hatched cells are reported constructions, not unresolved cases. The general conjecture concerns infinitely many orders.",
      "interpretation": "Filling every gap through 2,000 would settle a finite collection of cases. It would not prove the Hadamard conjecture for all multiples of four.",
      "note": "The <a href=\"https://ion.nechita.net/posts/new-hadamard-matrices/\" target=\"_blank\" rel=\"noopener\">independent August 13, 2026 check by Ion Nechita</a> covers order 668. Epoch reports all 12 constructions; this collection has not independently checked every matrix. AI attribution on Epoch’s page is provisional.",
      "milestones": [
        {
          "date": "2007-03-11",
          "label": "2007",
          "value": 13,
          "who": "Dragomir Ž. Đoković",
          "title": "Thirteen gaps below 2,000",
          "detail": "A construction of order 764 leaves the 13 orders listed in the paper’s introduction unresolved through 2,000. The original list gives each order divided by four.",
          "url": "https://arxiv.org/abs/math/0703312",
          "status": "construction",
          "unknown": [
            668,
            716,
            892,
            1004,
            1132,
            1244,
            1388,
            1436,
            1676,
            1772,
            1916,
            1948,
            1964
          ],
          "display": "13 gaps",
          "short": "After order 764"
        },
        {
          "date": "2013-01-16",
          "label": "2013",
          "value": 12,
          "who": "Dragomir Ž. Đoković, Oleg Golubitsky, Ilias S. Kotsireas",
          "title": "Order 1,004 is filled in",
          "detail": "The paper constructs order 1,004 and lists the remaining 12 gaps. Preprint in January 2013; journal publication in 2014.",
          "url": "https://arxiv.org/abs/1301.3671",
          "status": "construction",
          "unknown": [
            668,
            716,
            892,
            1132,
            1244,
            1388,
            1436,
            1676,
            1772,
            1916,
            1948,
            1964
          ],
          "display": "12 gaps",
          "short": "After order 1,004"
        },
        {
          "date": "2026-08",
          "label": "Aug 2026",
          "value": 0,
          "who": "Levent Alpöge and collaborators; Claude contribution reported by Epoch",
          "title": "Constructions reported for the remaining twelve",
          "detail": "Epoch’s problem page records constructions for all previously unknown admissible orders through 2,000. Ion Nechita independently checked the matrix of order 668; this is not a proof of the general conjecture.",
          "url": "https://epoch.ai/frontiermath/open-problems/hadamard",
          "status": "report",
          "unknown": [],
          "reported": [
            668,
            716,
            892,
            1132,
            1244,
            1388,
            1436,
            1676,
            1772,
            1916,
            1948,
            1964
          ],
          "display": "0 gaps",
          "short": "Reported constructions",
          "ai": "Epoch credits three human researchers working with an internal Claude model and explicitly marks the AI attribution provisional.",
          "supportUrl": "https://ion.nechita.net/posts/new-hadamard-matrices/"
        }
      ]
    }
  ],
  "updated": "2026-10-20"
}