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      "abstract": "The Jordan curve theorem, formalized in Lean 4 against Mathlib: a continuous injective map from the circle into the plane has a complement with exactly two connected components. The hypothesis is bare continuity and injectivity \u2014 no smoothness, piecewise-linearity or rectifiability \u2014 which is what makes the theorem hard and what separates it from the polygonal case. The proof follows Maehara's reduction to the Brouwer fixed point theorem, via a crossing lemma for transversal paths in a rectangle and the fact that every component has the curve as its boundary, together with a farthest-pair normalization that pins down exactly one bounded component. Because Mathlib has no Brouwer fixed point theorem at the pinned revision, Brouwer is built here from the ground up: covering-space path lifting for the circle giving \u03c0\u2081(S\u00b9) \u2245 \u2124, then no-retraction of the disk onto its boundary, then Brouwer on the disk by ray-retraction, then the general nonempty-compact-convex case by nearest-point projection. The statement is expressed in Mathlib's own vocabulary. The proof is sorry-free and depends on Lean's three standard axioms (propext, Classical.choice, Quot.sound) and nothing else.",
      "authors": [
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          "name": "Rado Kirov"
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    {
      "abstract": "A Lean 4 and Mathlib proof that there are absolute constants c > 0 and p\u2080 such that, for every prime p \u2265 p\u2080, the minimum size m(p) of a set A \u2286 \ud835\udd3d_p with |A| \u2265 2 and no uniquely represented sum satisfies m(p) \u2265 c (log p / log log p)\u00b2.  Here representations are unordered two-element multisets from A, repeated summands are allowed, and \"no unique sum\" means that every such multiset has a distinct equal-sum mate.",
      "authors": [
        {
          "name": "Xinjie He"
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      ],
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        "theorem_names": [
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    },
    {
      "abstract": "`PrimeGapsLib` is an evolving codebase aimed at housing results about gaps in primes. The flagship result thus far is that pairs of primes are separated by at most 246 infinitely often, conditional on the Bombieri-Vinogradov theorem. The 246 proof was generated collaboratively between human formalisers at Axiom Math and AxiomProver, their in-house theorem proving system. Community feedback and contributions are welcome!",
      "authors": [
        {
          "name": "Evan Chen"
        },
        {
          "name": "Sidharth Hariharan"
        },
        {
          "name": "Kenny Lau"
        },
        {
          "name": "Bhavik Mehta"
        },
        {
          "name": "Ken Ono"
        },
        {
          "name": "Ashvin Swaminathan"
        },
        {
          "name": "Jesse Thorner"
        },
        {
          "name": "Yunzhou Xie"
        }
      ],
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        ],
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        "theorem_names": [
          "bombieriVinogradov_implies_prime_gap_le_246",
          "bombieriVinogradov_implies_nth_prime_gap_le_246"
        ]
      },
      "id": "PALOMAR-2026-08-18-000002",
      "path": "entries/PALOMAR-2026-08-18-000002-v1.json",
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            "source_repository": "AxiomMath/PrimeGapsLib"
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      "published_at": "2026-08-18T20:27:38Z",
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        "repository": "AxiomMath/PrimeGapsLib"
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    {
      "abstract": "This project formalizes the main soundness theorem for the quantum low individual degree test. It proves that a projective two-prover strategy which passes the test admits polynomial measurements that are consistent with one another and with the provers' point measurements, with an explicit error bound.",
      "authors": [
        {
          "name": "Sirui Lu"
        },
        {
          "name": "Ruixuan Deng"
        },
        {
          "name": "Zhengfeng Ji"
        }
      ],
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          "quant-ph",
          "cs.CC"
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          "68Q15"
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          "MIPStarRE.LDT.Test.mainFormal"
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      "path": "entries/PALOMAR-2026-08-18-000001-v1.json",
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      "published_at": "2026-08-18T15:09:37Z",
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        "repository": "LionSR/LDT-comparator"
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      "title": "LionSR/LDT-comparator",
      "trust": {
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      "versions": 1
    },
    {
      "abstract": "Cantor measures with odd base do not admit Fourier frames",
      "authors": [
        {
          "name": "Jaume de Dios Pont"
        },
        {
          "name": "Lukas Liehr"
        },
        {
          "name": "Mitchell A. Taylor"
        }
      ],
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        "arxiv": [
          "math.FA"
        ],
        "msc2020": [
          "28A80",
          "42B05",
          "46B15"
        ]
      },
      "formalization": {
        "theorem_names": [
          "CantorFramesPalomar.NoFourierFrameExists"
        ]
      },
      "id": "PALOMAR-2026-08-17-000004",
      "path": "entries/PALOMAR-2026-08-17-000004-v1.json",
      "preservation": {
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            "source_repository": "jaumededios/cantor-frames-palomar"
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      "published_at": "2026-08-17T23:51:34Z",
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    },
    {
      "abstract": "Sabidussi compatibility in Lean",
      "authors": [
        {
          "name": "Nikolay Ulyanov"
        }
      ],
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        "arxiv": [
          "math.CO"
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      "path": "entries/PALOMAR-2026-08-17-000003-v1.json",
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            "source_repository": "gexahedron/sabidussi-lean"
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      "published_at": "2026-08-17T19:19:41Z",
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      "title": "gexahedron/sabidussi-lean",
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    },
    {
      "abstract": "Hadamard 668 Comparator formalization",
      "authors": [
        {
          "name": "Paul Lezeau"
        }
      ],
      "classification": {
        "arxiv": [
          "math.CO"
        ],
        "msc2020": [
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      },
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        "theorem_names": [
          "Hadamard.H_isHadamard"
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      "path": "entries/PALOMAR-2026-08-17-000002-v1.json",
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            "source_repository": "Paul-Lez/hadamard-668-comparator"
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        ]
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      "published_at": "2026-08-17T19:10:42Z",
      "source": {
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      "title": "Paul-Lez/hadamard-668-comparator",
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    },
    {
      "abstract": "jordan_pick \u2014 Rad\u00f3's theorem (Lean 4 / Mathlib)",
      "authors": [
        {
          "name": "Rado Kirov"
        }
      ],
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        "arxiv": [
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          "math.GN"
        ],
        "msc2020": [
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          "30F15",
          "30F25",
          "30F99",
          "54D65",
          "32Q55",
          "31A05"
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      },
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        "theorem_names": [
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      "id": "PALOMAR-2026-08-17-000001",
      "path": "entries/PALOMAR-2026-08-17-000001-v1.json",
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      "published_at": "2026-08-17T06:39:48Z",
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    },
    {
      "abstract": "Comparator wrapper certifying that the ErdosUnitDistance library proves the literal negation of the uniform-constant form of Erd\u0151s's unit-distance conjecture, from a Mathlib-only statement of that negation.\n",
      "authors": [
        {
          "name": "Kim Morrison"
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      "published_at": "2026-08-14T07:52:09Z",
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    },
    {
      "abstract": "The mechanically verified statements are clear, faithful to the mathematical account, auditable, and of substantial research interest. The unsupported narrow novelty claims warrant disclosure but do not block registration.",
      "authors": [
        {
          "name": "Terence Tao"
        }
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          "SendovConjecture.sendov",
          "SendovConjecture.phelps_rodriguez"
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}