{
  "abstract": "FINDING: Dirichlet characters mod 3 connect to Eisenstein integers (ℚ(√−3)), whose prime angular distribution is tied to hexagonal lattice symmetry — the most relevant result is the arXiv paper on mod 3 triple Milnor invariants and cubic residue symbols in the Eisenstein field. | MATH: Eisenstein integers: ℤ[ω], ω = e^(2πi/3) = −1/2 + i√3/2. Norm: N(a+bω) = a² − ab + b². Cubic residue symbol: (α/π)₃ ∈ {1, ω, ω²}. Dirichlet characters mod 3: χ₀, χ₁ (trivial), χ₂ with χ₂(2) = −1, χ₂(1) = 1. Triple Milnor invariant: μ(1,2,3) mod 3. | CONNECTION: **Direct hexagonal symmetry.** The Eisenstein field has 6-fold rotational symmetry (roots of unity: ±1, ±ω, ±ω²). Prime angular distribution in ℚ(√−3) is anisotropic — primes split into 6 sectors (mod 6 classes) with equal density, but *individual* primes cluster along 6 rays in the complex plane (angles 0°, 60°, 120°, ...). The ratio 1/6 = 0.1667, and the *inverse* 6 = 2π/60° — a base-60 angular division. The cubic residue symbol takes values in\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/d3m6f06cae96c.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23410015",
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  "id": "d3m6f06cae96c",
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  "landing_url": "https://zenodo.org/records/23410015",
  "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/",
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  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d3m6f06cae96c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:23Z",
      "native_id": "23410015",
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      "url": "https://zenodo.org/records/23410015"
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  "subjects": [],
  "title": "Cubic Residue Symbols and Triple Milnor Invariants in Eisenstein Integers — E8 Intelligence Research",
  "title_text": "Cubic Residue Symbols and Triple Milnor Invariants in Eisenstein Integers — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:20Z",
  "url": "https://research.draconitic.com/paper/d3m6f06cae96c",
  "venue": "Zenodo",
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}