{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0007",
  "title": "A Counterexample to a Linear-Junta Inverse Theorem for Spherical Triangles",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct smooth, real, mean-zero, 1-bounded functions on high-dimensional spheres whose average product over zero-sum unit triangles tends to 7/16384, while their correlation with every function of m linear coordinates is O(sqrt(m/d)) when m=o(d). The bound is uniform over all choices of coordinates and all test functions of L2 norm at most one. The construction is a centered cosine of a balanced, rescaled quadratic form. A conditional coupling proves the projection estimate, and a multivariate Gaussian limit gives the positive triangle average and a nonzero limiting variance. This supplies a complete counterexample to the dimension-free quantitative interpretation of Dmitrii Zakharov's spherical-triangle inverse question, AIM Problem 2.4 / UnsolvedMath AIM-COMBINATORICS-0007. It does not settle the related zero-sum-free hemisphere conjecture or the planar and finite-field analogues. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.PR",
    "math.CA"
  ],
  "keywords": [
    "spherical triangle",
    "inverse theorem",
    "linear junta",
    "quadratic form",
    "conditional coupling",
    "high-dimensional sphere"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0007/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0007/paper.pdf?v=ebabb716e2c9",
  "doi": "10.5281/zenodo.23314092",
  "zenodo_record_url": "https://zenodo.org/records/23314092",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the dimension-free quantitative interpretation of the spherical-triangle linear-junta inverse question. No closure is claimed for a purely qualitative nonzero-correlation interpretation, the zero-sum-free hemisphere conjecture, or the planar and finite-field variants. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "ebabb716e2c9256c52f62b18d3454d83b16a9e0fe0849e1727efcc9e39829477"
    },
    "source.zip": {
      "sha256": "eb4e657ae07d2da25b8531bfb7ee795ff536ed6010a215b8fa6a9a6ff83180ab"
    },
    "verification_report.md": {
      "sha256": "0cd7207a1a5a15162536ce5646e6f79a9f44fd61abdec06457d9d5acf054390b"
    }
  },
  "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."
}