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      "identifier": "lemmata:2610.00001",
      "url": "https://lemmata.iol.berlin/abs/2610.00001",
      "html_url": "https://lemmata.iol.berlin/html/2610.00001",
      "title": "Composition of implications",
      "category": "logic",
      "category_title": "Logic",
      "tags": [
        "order"
      ],
      "contributors": [
        "Sebastian Pokutta"
      ],
      "first_published": "2026-10-19T19:34:05+02:00",
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        {
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      "acknowledgements": null,
      "bibtex": "@misc{lemmata:2610.00001,\n  title        = {Composition of implications},\n  author       = {Sebastian Pokutta},\n  howpublished = {Lemmata, lemmata:2610.00001, \\url{https://lemmata.iol.berlin/abs/2610.00001}},\n  year         = {2026},\n  url          = {https://lemmata.iol.berlin/abs/2610.00001},\n  note         = {Version 29ff25c1aac1; formal statements checked in Lean 4}\n}"
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      "identifier": "lemmata:2610.00002",
      "url": "https://lemmata.iol.berlin/abs/2610.00002",
      "html_url": "https://lemmata.iol.berlin/html/2610.00002",
      "title": "A Lagrangian duality gap that survives closed convexification",
      "category": "optimization",
      "category_title": "Optimization",
      "tags": [],
      "contributors": [
        "Nico Pelleriti"
      ],
      "first_published": "2026-10-19T23:13:32+02:00",
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      "ai_use": null,
      "acknowledgements": null,
      "bibtex": "@misc{lemmata:2610.00002,\n  title        = {A Lagrangian duality gap that survives closed convexification},\n  author       = {Nico Pelleriti},\n  howpublished = {Lemmata, lemmata:2610.00002, \\url{https://lemmata.iol.berlin/abs/2610.00002}},\n  year         = {2026},\n  url          = {https://lemmata.iol.berlin/abs/2610.00002},\n  note         = {Version 29ff25c1aac1; formal statements checked in Lean 4}\n}"
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      "identifier": "lemmata:2610.00003",
      "url": "https://lemmata.iol.berlin/abs/2610.00003",
      "html_url": "https://lemmata.iol.berlin/html/2610.00003",
      "title": "Nonnegative rank of the spanning-tree slack matrix",
      "category": "optimization",
      "category_title": "Optimization",
      "tags": [],
      "contributors": [
        "Sebastian Pokutta"
      ],
      "first_published": "2026-10-19T22:15:57+00:00",
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        {
          "name": "Lemmata.SpanningTreeMinSplit.fullSlack_rank_bound",
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          "name": "Lemmata.SpanningTreeMinSplit.fullSlack_factorization",
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      "ai_use": "Obtained as auxiliary result when using GPT-6 Astra to formalize other research.",
      "acknowledgements": "Research reported in this paper was partially supported by the Berlin Mathematics Research Center MATH$^+$ (EXC-2046/2, project ID 390685689), funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy. This work has been partially supported by OpenAI's Academic Researchers program.",
      "bibtex": "@misc{lemmata:2610.00003,\n  title        = {Nonnegative rank of the spanning-tree slack matrix},\n  author       = {Sebastian Pokutta},\n  howpublished = {Lemmata, lemmata:2610.00003, \\url{https://lemmata.iol.berlin/abs/2610.00003}},\n  year         = {2026},\n  url          = {https://lemmata.iol.berlin/abs/2610.00003},\n  note         = {Version 29ff25c1aac1; formal statements checked in Lean 4}\n}"
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      "identifier": "lemmata:2610.00004",
      "url": "https://lemmata.iol.berlin/abs/2610.00004",
      "html_url": "https://lemmata.iol.berlin/html/2610.00004",
      "title": "LP hardness of approximation for polynomial-time solvable problems",
      "category": "optimization",
      "category_title": "Optimization",
      "tags": [],
      "contributors": [
        "Sebastian Pokutta"
      ],
      "first_published": "2026-10-20T17:22:19+02:00",
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            "propext",
            "Classical.choice",
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          "source_file": "lean/Lemmata/LPHardnessInP/Formulation/Main.lean"
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          "name": "Lemmata.LPHardnessInP.Matching.matching_minimum",
          "kind": "theorem",
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            "propext",
            "Classical.choice",
            "Quot.sound"
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          "name": "Lemmata.LPHardnessInP.Matching.matching_lp_lower_bound",
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          "name": "Lemmata.LPHardnessInP.CostTransfer.cut_metric_properties",
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          "name": "Lemmata.LPHardnessInP.Joins.vjoin_minimum",
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            "propext",
            "Classical.choice",
            "Quot.sound"
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            "Quot.sound"
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            "propext",
            "Classical.choice",
            "Quot.sound"
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        },
        {
          "name": "Lemmata.LPHardnessInP.ParityExtension.ConcreteJSYRectangleData.corruption_bound",
          "kind": "theorem",
          "label": "thm-lp-hardness-in-p-formal-corruption",
          "status": "checked",
          "axioms": [
            "propext",
            "Classical.choice",
            "Quot.sound"
          ],
          "source_file": "lean/Lemmata/LPHardnessInP/ParityExtension/ConcreteData.lean"
        },
        {
          "name": "Lemmata.LPHardnessInP.ParityExtension.pairing_factorization_le",
          "kind": "theorem",
          "label": "thm-lp-hardness-in-p-separation",
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            "propext",
            "Classical.choice",
            "Quot.sound"
          ],
          "source_file": "lean/Lemmata/LPHardnessInP/ParityExtension/Separation.lean"
        },
        {
          "name": "Lemmata.LPHardnessInP.ParityExtension.calibrated_witness_gap",
          "kind": "theorem",
          "label": "prop-lp-hardness-in-p-parameters",
          "status": "checked",
          "axioms": [
            "propext",
            "Classical.choice",
            "Quot.sound"
          ],
          "source_file": "lean/Lemmata/LPHardnessInP/ParityExtension/Parameters.lean"
        },
        {
          "name": "Lemmata.LPHardnessInP.ParityExtension.calibrated_shifted_rank_from_jsy",
          "kind": "theorem",
          "label": "thm-lp-hardness-in-p-finite-extension",
          "status": "checked",
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            "propext",
            "Classical.choice",
            "Quot.sound"
          ],
          "source_file": "lean/Lemmata/LPHardnessInP/ParityExtension/Main.lean"
        },
        {
          "name": "Lemmata.LPHardnessInP.ParityExtension.parity_lp_lower_bound_from_jsy",
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            "propext",
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          "name": "Lemmata.LPHardnessInP.ParityExtension.shifted_rank_mono",
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            "Classical.choice",
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          "source_file": "lean/Lemmata/LPHardnessInP/ParityExtension/Padding.lean"
        }
      ],
      "arxiv": [],
      "license": "CC-BY-4.0",
      "license_url": "https://creativecommons.org/licenses/by/4.0/",
      "ai_use": "The exposition and Lean formalization were developed with OpenAI Codex. The published matching rank theorem, the earlier parity partition and entropy assertions, and affine Farkas certificate existence remain external. The new parity extension is formalized conditionally on precise hypotheses; no custom axioms are introduced.",
      "acknowledgements": "Research reported in this paper was partially supported by the Berlin Mathematics Research Center MATH$^+$ (EXC-2046/2, project ID 390685689), funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy. This work has been partially supported by OpenAI's Academic Researchers program.",
      "bibtex": "@misc{lemmata:2610.00004,\n  title        = {LP hardness of approximation for polynomial-time solvable problems},\n  author       = {Sebastian Pokutta},\n  howpublished = {Lemmata, lemmata:2610.00004, \\url{https://lemmata.iol.berlin/abs/2610.00004}},\n  year         = {2026},\n  url          = {https://lemmata.iol.berlin/abs/2610.00004},\n  note         = {Version 29ff25c1aac1; formal statements checked in Lean 4}\n}"
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}