{
  "abstract": "FINDING: Dirichlet characters mod 3 connect to Eisenstein integers (ℤ[ω], ω = e^{2πi/3}), whose primes exhibit hexagonal lattice anisotropy; the arXiv paper links mod 3 triple Milnor invariants to cubic residue symbols in ℚ(√−3). | MATH: Eisenstein field ℚ(√−3); ring ℤ[ω] with ω²+ω+1=0; norm N(a+bω)=a²−ab+b²; cubic residue symbol (α/π)₃ ∈ {1, ω, ω²}; Dirichlet characters mod 3: χ(1)=1, χ(2)=−1 (trivial quadratic), and the nontrivial cubic character χ₃(n) = (n/3)₃ with values in {1, ω, ω²}; triple Milnor invariant μ̄(3) mod 3; Rédei triple symbol generalization. | CONNECTION: **Hexagonal lattice** — Eisenstein integers form the triangular/hexagonal lattice A₂, the root lattice of SU(3). The angular distribution of Eisenstein primes is **isotropic in the limit** (prime number theorem for ℚ(√−3) gives uniform angular density), but **finite‑N anisotropy** follows hexagonal symmetry: primes split into 6 residue classes mod 6ω, with angles clustered near multiples of 60° (π/3). This is the *\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ae5vf3r0wq7h9.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23430955",
  "fulltext_url": null,
  "id": "ae5vf3r0wq7h9",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23430955",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ae5vf3r0wq7h9.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T12:22:09Z",
      "native_id": "23430955",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23430955"
    }
  ],
  "subjects": [],
  "title": "Cubic Residue Symbols and Mod 3 Milnor Invariants in Eisenstein Integers — E8 Intelligence Research",
  "title_text": "Cubic Residue Symbols and Mod 3 Milnor Invariants in Eisenstein Integers — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-31T12:22:09Z",
  "url": "https://research.draconitic.com/paper/ae5vf3r0wq7h9",
  "venue": "Zenodo",
  "versions": []
}