{
  "schema_version": 1,
  "problem_number": "AMR-014-0007",
  "title": "A Geometric Count of Fourth-Power Decompositions of Binary Octics",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A general complex binary octic has exactly 76 decompositions as a sum of three fourth powers of quadratic forms, modulo permutations and independent fourth-root-of-unity rescalings. This preprint gives a geometric proof of the count, previously obtained numerically by Kowalczyk and Vill. Restriction to a conic realizes the problem as a projection of the degree-112 third secant variety of the quartic Veronese surface. Its scheme-theoretic base is a ribbon on a rational quartic, with nilpotent line bundle of degree -4. The intersection correction is 112 - 64 + 28 = 76. The ordered affine map has degree 29184, and quotienting only by permutations gives 4864.\n\nThe complete theorem treats the perfect pair (k,d)=(4,2), not the full perfect-pair Problem E represented by AMR-014-0007 in UnsolvedMath. Real decompositions and closed-form construction of every summand are outside its scope. The numerical value and conic-restriction viewpoint are credited to prior work; no absolute priority is claimed.\n\nThe package contains the six-page English manuscript, source and a portable exact symbolic checker. AI assistance was used for literature navigation, calculations, proof development and drafting. The author takes responsibility for the claims. This is a self-audited, unrefereed preprint, not independently reviewed or proof-assistant formalized.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "keywords": [
    "binary octic",
    "Waring decomposition",
    "secant variety",
    "catalecticant",
    "ribbon",
    "Segre class",
    "AMR-014-0007"
  ],
  "manuscript_version_date": "2026-10-07",
  "publication_date": "2026-10-07",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-07",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-014-0007/",
  "pdf_url": "https://eulersolve.org/papers/amr-014-0007/paper.pdf?v=4a151154a6a0",
  "doi": "10.5281/zenodo.23202935",
  "zenodo_record_url": "https://zenodo.org/records/23202935",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete geometric proof that a general complex binary octic has 76 decompositions as three fourth powers of quadratics, modulo permutations and independent fourth roots of unity; ordered degree 29184 and permutation-only degree 4864. This treats only (k,d)=(4,2), not the general perfect-pair Problem E recorded as AMR-014-0007. Prior numerical 76 and the conic restriction viewpoint are credited to Kowalczyk and Vill. Real decompositions and closed-form construction of all summands are outside scope. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "4a151154a6a044cd9c4a1207487c744c6380c7786bc420d18886f8ab3ff00ec7"
    },
    "source.zip": {
      "sha256": "3bd58a9f4a3b1e7499e0b1c99d25c353801ec4c54a4f3a5e0083a0e5ace5cd8d"
    },
    "verification_report.md": {
      "sha256": "e818b09ab5a3803c2ee608854493654ad7d0b2bdb7ffe72be9405a05577180da"
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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."
}