Cubic Residue Symbols and Mod 3 Milnor Invariants in Eisenstein Integers — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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 *
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026cubic,
title = {{Cubic Residue Symbols and Mod 3 Milnor Invariants in Eisenstein Integers — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23430955},
url = {https://zenodo.org/records/23430955},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cubic Residue Symbols and Mod 3 Milnor Invariants in Eisenstein Integers — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23430955
UR - https://zenodo.org/records/23430955
AB - 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 * Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -