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  "slug": "katthan-betti-poset-counterexample",
  "title": "The Betti poset does not determine the Stanley projective dimension of a monomial ideal",
  "shortTitle": "Same Betti poset. Different Stanley dimension.",
  "url": "https://evidencepress.org/releases/katthan-betti-poset-counterexample/",
  "oneLine": "An explicit six-generator pair has the same Betti poset over every field, but ideal Stanley projective dimensions three and two; six generators is minimal.",
  "abstract": "A computer-assisted candidate refutes the ideal assertion of Katthän's Betti-poset conjecture. Explicit ideals in five and seven variables have isomorphic Betti posets over every field and Stanley projective dimensions three and two, with Stanley depths two and five. Their lcm-lattices differ by one Betti-invisible meet-irreducible element. Positive decompositions and a repaired nine-point obstruction establish the separation; the complete 7,443 five-atom census gives generator minimality. An explicit 14-variable squarefree colon realisation places the same gap in one ring. The quotient assertion and Stanley's inequality for ideals are not refuted; both quotient projective-dimension invariants remain five.",
  "datePublished": "2026-10-14",
  "dateModified": "2026-10-14",
  "version": "2.0.0-candidate",
  "doi": "10.5281/zenodo.23362079",
  "doiUrl": "https://doi.org/10.5281/zenodo.23362079",
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  "authors": [
    "Anonymous"
  ],
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      "evidenceUrl": "https://zenodo.org/records/23362079",
      "note": "Published GitHub and Zenodo assets matched the frozen local package by provider digests and downloaded bytes; site readback is recorded separately."
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    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Full exact finite census, partitions, integral homology and colon reconstruction passed normally and with optimization; eleven semantic controls. Internal checking is not external reproduction.",
      "evidenceUrl": "https://github.com/ipitchford/katthan-betti-poset-counterexample/blob/v2.0.0-candidate/REPLAY_RECEIPT.md"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No unaffiliated reproduction established."
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    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Implementation diversity within the coordinated workflow is not unaffiliated reimplementation."
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    {
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      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof-assistant verification established."
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      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The supplied review's provenance does not establish external human specialist review."
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    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
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      "note": "Original public checker uses Python standard library; no hermetic environment or external reproduction claim."
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    "humanRole": "Publication authority; no human mathematical contribution asserted.",
    "reviewStatus": "Supplied referee-style material and coordinated internal checking; external human specialist review is not established."
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  "problem": {
    "name": "The ideal assertion of Katthän's Betti-poset conjecture",
    "url": "https://doi.org/10.1007/s40598-016-0039-5"
  },
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  "metadataProvenance": "Manuscript, supplied review and coordinated revision, 14 October 2026. Mathematical evidence, provider publication and canonical site readback remain separate gates.",
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  "keywords": [
    "Betti poset",
    "Stanley depth",
    "Stanley projective dimension",
    "monomial ideals",
    "lcm-lattice",
    "counterexample",
    "squarefree colon ideal"
  ],
  "keyResults": [
    "Six-generator ideals I and J share their Betti poset over every field but have ideal Stanley projective dimensions 3 and 2.",
    "The complete five-atom census has constant ideal Stanley projective dimension within every Betti-poset class, making six generators minimal.",
    "A squarefree ideal H in 14 variables and (H:z14) realise the same separation with Stanley depths 11 and 12; 14 variables is not claimed minimal.",
    "The quotients have equal Stanley projective dimension 5; neither Stanley's inequality for ideals nor the quotient Betti-poset assertion is disproved."
  ],
  "reviews": [],
  "evidencePackage": "Explicit generators, integral homology data, constructive Stanley decompositions, a repaired nine-point obstruction and the complete five-atom lattice census. Exact publication replay and its scope are recorded in the package; no formal verification or external human review is asserted.",
  "openProblems": [],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Katthän (2016). Betti posets and the Stanley depth. Arnold Mathematical Journal 2, 267–276. The conjecture, squarefree-colon formulation and restricted alternatives are distinct.",
      "url": "https://doi.org/10.1007/s40598-016-0039-5",
      "doi": "10.1007/s40598-016-0039-5"
    },
    {
      "citation": "Ichim, Katthän and Moyano-Fernández (2017). Stanley depth and the lcm-lattice. Journal of Combinatorial Theory, Series A 150, 295–322. Lattice invariance, realisations and monotonicity.",
      "url": "https://doi.org/10.1016/j.jcta.2017.03.005",
      "doi": "10.1016/j.jcta.2017.03.005"
    },
    {
      "citation": "Ichim, Katthän and Moyano-Fernández (2016). LCM-lattices and Stanley depth: A first computational approach. Experimental Mathematics 25, 46–53. Earlier lattice census and the quotient result for at most five generators.",
      "url": "https://doi.org/10.1080/10586458.2015.1005257",
      "doi": "10.1080/10586458.2015.1005257"
    },
    {
      "citation": "Herzog, Vladoiu and Zheng (2009). How to compute the Stanley depth of a monomial ideal. Journal of Algebra 322, 3151–3169. Characteristic-poset interval partitions.",
      "url": "https://doi.org/10.1016/j.jalgebra.2008.01.006",
      "doi": "10.1016/j.jalgebra.2008.01.006"
    },
    {
      "citation": "Duval, Goeckner, Klivans and Martin (2016). A non-partitionable Cohen–Macaulay simplicial complex. Advances in Mathematics 299, 381–395. Established quotient-side Stanley-conjecture context, not the assertion refuted here.",
      "url": "https://doi.org/10.1016/j.aim.2016.05.011",
      "doi": "10.1016/j.aim.2016.05.011"
    }
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