{
  "schema_version": 1,
  "problem_number": "AIM-COMPUTATION-0073",
  "title": "Strict Likelihood Multimodality and Septic Equations for Two-Component Birkhoff Models",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give a positive integer dataset on S_4 whose likelihood on the closed two-component Birkhoff model has at least three distinct strict local maxima in distribution space. The dataset has 2013 observations and all three probability vectors have full support. One component of each displayed mode is a boundary component. An exact nonnegative-decomposition certificate and a compactness argument rule out the possibility that the modes are artifacts of a parameter chart or component-label switching. Separately, for every n >= 4, unequal positive parity-constant counts give at least three distinct global maximizers in the closed two-component model. We also determine the first nonzero homogeneous equation degree of the complex S_4 secant variety: it is seven. A 96-term septic and its 24 symmetry images supply local equations at the displayed smooth point. We do not determine the global defining ideal or assert modes with both component matrices strictly positive.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.ST",
    "math.AG",
    "math.CO"
  ],
  "keywords": [
    "Birkhoff model",
    "likelihood multimodality",
    "secant variety",
    "septic equations",
    "algebraic statistics",
    "AIM-COMPUTATION-0073",
    "math.ST",
    "math.AG",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-computation-0073/",
  "pdf_url": "https://eulersolve.org/papers/aim-computation-0073/paper.pdf?v=db5197c6bdc1",
  "doi": "10.5281/zenodo.23019398",
  "zenodo_record_url": "https://zenodo.org/records/23019398",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The strict distribution-space modes belong to the closed statistical two-component model; one component is on the boundary, although all probabilities are positive. Strict modes with both component matrices positive are not established. Isolation of the separate parity global maximizers and a global defining ideal or radical are not claimed. These are completed scoped theorems, not a resolution of every part of AIM-COMPUTATION-0073. Classical methods and the upstream dimension computation are credited. Self-audited and unrefereed; no independent certification, formal verification or absolute priority claim.",
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    "source.zip": {
      "sha256": "02750b869e071234fe2e8bfe47bcfedff9187e35578c2d31c71acd175cdbed36"
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    "verification_report.md": {
      "sha256": "ca348ee5595a149a705697ae67ce6d807a0d1f6bbd25fc0a66d6f58562b7c86b"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}