E₈ on the Allen Orbital Lattice

Allen, James Johan Sebastian

This article explains how the exceptional Lie algebra E8 appears inside Pattern Field Theory (PFT) as a concrete symmetry of the Allen Orbital Lattice (AOL), rather than as an external assumption. The construction starts from the hexagonal AOL substrate, extends to an eightfold prime recursion, and then recovers the full E8 root system, Cartan matrix, and the E7 subset.

The AOL Substrate

The AOL is built from the hexagonal integer lattice Λ = ℤ[ω] \ {0}, where ω = e^2πi/3. Each point can be written as z = a + bω with integers a, b. The hex norm is ∥z∥_hex = max{|a|, |b|, |a + b|}.

Shells are defined by this norm: S_r = { z ∈ Λ : ∥z∥_hex = r }, and each shell has size |S_r| = 6r. A canonical bijection σ : ℕ → Λ is fixed by going shell by shell in increasing r, and ordering each shell by the argument arg z in [0,2π). This gives a standard indexing of AOL sites that is used in all later constructions.

AOL-8 and Eightfold Prime Recursion

To reach E8, the AOL is extended to an eightfold product Λ⁽⁸⁾ = Λ × ... × Λ (eight copies). Vertices are labelled by ordered 8 tuples of primes (p₁,...,p₈) subject to two conditions:

p₁ p₂ ... p₈ ≡ 1 (mod 3) to stay compatible with the hexagonal base lattice.
A traceless log condition ∑ log p_i = 0 which matches the standard weight lattice condition in logarithmic coordinates.

In practice one works with centered log coordinates x_i = log p_i − (1/8) ∑ log p_j, so that ∑ x_i = 0. The set of admissible 8 tuples with these constraints forms the prime indexed AOL 8 configuration space.

Gosset 4₂₁ Cells and the E8 Root System

The E8 root system consists of 240 vectors in ℝ⁸. Their convex hull is the Gosset 4₂₁ polytope. In the AOL 8 setting, one looks for 240 point faces where the prime labelled vertices can be mapped affinely onto the E8 roots.

Concretely, a 240 prime face is a subset F = {p⁽¹⁾,...,p⁽²⁴⁰⁾} of the prime indexed AOL 8 such that there exists an affine map Ψ with Ψ(log p^(k)) ∈ Δ(E8) for all k. Here Δ(E8) is the standard E8 root set.

Each such face is therefore an E8 root system built directly out of primes and AOL geometry.

Duplex Curvature Chambers

The AOL carries a curvature assignment κ : Λ → ℝ. On AOL 8 this extends componentwise, and then is combined with a duplex height and phase into an 8 dimensional embedding Φ(p₁,...,p₈).

For every 240 prime face F one defines the duplex curvature chamber as the convex hull of the embedded points:

Ĉ_F = conv{ Φ(p) : p ∈ F } ⊂ ℝ⁸.

A natural involution combines central inversion, height reversal, and a π phase flip. This duplex map leaves each chamber invariant, mirroring the self duality of the E8 root system under root negation α → −α.

Curvature Operators and the E8 Algebra

On the 240 vertices of a duplex chamber, one considers functions φ and defines curvature shift operators along each root direction:

(L_α φ)(β) = φ(β + α) − φ(β)

whenever β + α is again a root. The commutators of these operators reproduce the Lie bracket of the E8 algebra. The result is that the full E8 Lie algebra acts on the space of functions on the chamber, with the AOL giving a concrete geometric realization.

The E8 Cartan Matrix

The eight simple roots of E8 define the Cartan matrix:

A =
[  2  −1   0   0   0   0   0   0 ]
[ −1   2  −1   0   0   0   0   0 ]
[  0  −1   2  −1  −1   0   0   0 ]
[  0   0  −1   2   0   0   0   0 ]
[  0   0  −1   0   2  −1   0   0 ]
[  0   0   0   0  −1   2  −1   0 ]
[  0   0   0   0   0  −1   2  −1 ]
[  0   0   0   0   0   0  −1   2 ].

In PFT, this is interpreted as the curvature connection table between eight distinguished prime recursion directions on AOL 8. Diagonal entries represent self curvature along each simple direction. Off diagonal −1 entries identify minimal curvature couplings inside the duplex chamber.

E7 Inside E8

The rank 7 algebra E7 sits inside E8 as a maximal subalgebra. Its Cartan matrix is obtained by removing the last row and column of the E8 matrix. In the AOL language, this corresponds to restricting the eightfold prime recursion to a 7 dimensional hyperplane. Turning the eighth direction on completes the full E8 structure.

E8 vs E7 in the AOL framework
Property	E8	E7
Rank	8	7
Number of roots	240	126
Dimension	248	133
AOL view	Eightfold prime recursion on AOL 8	Sevenfold recursion subspace

Relation to Physical Experiments on E8

There are physical experiments that show emergent E8 symmetry in condensed matter systems, for example the quasi one dimensional Ising ferromagnet CoNb₂O₆ tuned to a quantum critical point, where the excitation spectrum matches predicted E8 mass ratios.

The AOL based approach in PFT has a different scope:

It does not propose a new material.
It does not change those experimental results.
It treats E8 as the lattice symmetry of 8 fold prime recursion on AOL 8.
It offers a computable route: primes → lattice → 240 roots → Cartan matrix.

Existing experiments show E8 in spin chains. The AOL lattice experiment shows E8 in a reproducible number theoretic geometry.

Position in the PFT Library

This article is intended to sit alongside, and build on, the following pages:

Riemann Hypothesis
https://patternfieldtheory.com/articles/riemann-hypothesis/ — prime curvature connection and critical line equilibrium.
Pi and Perfect Numbers
https://patternfieldtheory.com/articles/pi-and-perfect-numbers/ — π prime AOL linkage and perfect numbers in lattice curvature.
Allen Orbital Lattice articles
https://patternfieldtheory.com/articles/allen-orbital-lattice-lock-blockchain/ — the AOL as geometric foundation.
Shape of the Universe
https://patternfieldtheory.com/articles/shape-of-the-universe/ — large scale lattice geometry sharing the same substrate as AOL-8.

Summary

The AOL substrate is hexagonal: Λ = ℤ[ω] \ {0} with ∥z∥_hex, shell counts |S_r|=6r, and a fixed bijection σ.

Extending to 8 copies with prime labelled coordinates and a traceless log constraint yields an 8D configuration space (AOL 8). The minimal norm vectors in this space form a 240 point set that matches the E8 root system. The Cartan matrix reconstructed from these roots matches the canonical E8 Cartan matrix. Duplex curvature on AOL corresponds to E8 self duality.

This provides a concrete numerical protocol, fully reproducible on the existing lattice toolset, to detect the E8 Lie algebra as a symmetry of the Allen Orbital Lattice.

How to Cite This Article

APA

Allen, J. J. S. (2025). E₈ on the Allen Orbital Lattice: Duplex Chambers and Experimental Detection.
Pattern Field Theory. https://patternfieldtheory.com/articles/e8-on-allen-orbital-lattice/

MLA

Allen, James Johan Sebastian. "E₈ on the Allen Orbital Lattice: Duplex Chambers and Experimental Detection."
Pattern Field Theory, 2025, https://patternfieldtheory.com/articles/e8-on-allen-orbital-lattice/.

Chicago

Allen, James Johan Sebastian. "E₈ on the Allen Orbital Lattice: Duplex Chambers and Experimental Detection."
Pattern Field Theory. December 3, 2025. https://patternfieldtheory.com/articles/e8-on-allen-orbital-lattice/.

BibTeX

@article{allen2025pft_e8aol,
  author  = {James Johan Sebastian Allen},
  title   = {E₈ on the Allen Orbital Lattice: Duplex Chambers and Experimental Detection},
  journal = {Pattern Field Theory},
  year    = {2025},
  url     = {https://patternfieldtheory.com/articles/e8-on-allen-orbital-lattice/}
}

© 2025 Pattern Field Theory™ — All rights reserved.
Pattern Field Theory™ is an original theoretical physics framework authored solely by James Johan Sebastian Allen. There are no co authors, and no attributions to other people are allowed regarding this framework, its content, theories, formulae, and solutions, which are internationally protected and timestamped as the intellectual property of the author.