D4-Octonion-Clifford Link to Quantum Codes and Toric Decoding — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 23, 2026• 0 citations

Abstract

FINDING: D4 symmetry and octonion algebra are linked via Clifford algebra Cl(0,7) to quantum correlation polytopes and error-correcting codes, with toric decoding and QLDPC codes as applications. | MATH: D4 root system (24 roots, Coxeter number 12, Weyl group order 192); octonion multiplication table defined by Fano plane; Clifford algebra Cl(0,7) parametrizes octonions via multivector products; toric code stabilizer generators (plaquette and vertex operators) on square lattice; QLDPC codes with sparse check matrices. | CONNECTION: D4 lattice is the densest sphere packing in 4D (kissing number 24); octonion norm 1 yields 7-sphere S7; toric code on torus has genus-dependent degeneracy; no direct golden ratio or base-60 link found. | DEPTH: 7 — Strong structural link between exceptional algebra (octonions), high-symmetry lattice (D4), and quantum error correction, but no explicit harmonic ratios or ancient geometric constants. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21503840

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D4-Octonion-Clifford Link to Quantum Codes and Toric Decoding — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 23, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.21503840

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Quantum Error Correction as Lattice Gauge Theory on the D4 Lattice — E8 Intelligence Research

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Toric Code Symmetry via p4m Group and Cohomology — E8 Intelligence Research

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Octonionic Projective Plane: Non-Associative Foundation for Quantum Mechanics — E8 Intelligence Research

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The Fano Plane, Octonion Non-Associativity, and the Peres-Mermin Proof of Contextuality — E8 Intelligence Research

Andrew Stewart Caldin

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Geometric Algebra Derivation of Tsirelson Bound via E8 and Octonions — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Tsirelson bound derivable from geometric algebra, with explicit link to E8 root system and octonionic structure in quantum correlations. | MATH: Tsirelson bound $C \leq 2\sqrt{2}$ for CHSH; geometric algebra formulation yields $C = 2\sqrt{2}$ via bivector/rotor structure; E8 root lattice $\Gamma_8$ with 240 roots, octonion multiplication table $e_i e_j = -\delta_{ij} + \epsilon_{ijk} e_k$ ; key ratio $\sqrt{2} \approx 1.414$ , related to $2\cos(\pi/4)$ . | CONNECTION: E8 root system encodes maximal Bell violation — $2\sqrt{2}$ emerges from the angle $\pi/4$ between measurement axes, which is the dihedral angle of the 4D hypercube; octonionic $G_2$ symmetry (automorphism group of octonions, 14-dim) stabilizes the correlation structure; ratio $1/\sqrt{2} \approx 0.707$ links to $\cos(\pi/4)$ , and $2\sqrt{2} \approx 2.828$ is $2 \times 1.414$ ; the E8 lattice has kissing number 240, and its Coxeter number 30 relates to $ 2\pi/30 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Tsirelson bound derivable from geometric algebra, with explicit link to E8 root system and octonionic structure in quantum correlations. | MATH: Tsirelson bound \( C \leq 2\sqrt{2} $ for CHSH; geometric algebra formulation yields $ C = 2\sqrt{2} $ via bivector/rotor structure; E8 root lattice $ \Gamma_8 $ with 240 roots, octonion multiplication table $ e_i e_j = -\delta_{ij} + \epsilon_{ijk} e_k $; key ratio $ \sqrt{2} \approx 1.414 $, related to $ 2\cos(\pi/4) $. | CONNECTION: E8 root system encodes maximal Bell violation — $ 2\sqrt{2} $ emerges from the angle $ \pi/4 $ between measurement axes, which is the dihedral angle of the 4D hypercube; octonionic $ G_2 $ symmetry (automorphism group of octonions, 14-dim) stabilizes the correlation structure; ratio $ 1/\sqrt{2} \approx 0.707 $ links to $ \cos(\pi/4) $, and $ 2\sqrt{2} \approx 2.828 $ is $ 2 \times 1.414 $; the E8 lattice has kissing number 240, and its Coxeter number 30 relates to \( 2\pi/30 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 29, 20260 citations

S \subset \mathcal{P}_n

(Pauli group), CSS code condition

H_X H_Z^T = 0

(mod 2); B2 root system:

\{\pm e_1 \pm e_2, \pm e_1, \pm e_2\}

→ D4 lattice (checkerboard, kissing number 24); plaquette operators

B_p = \prod_{\ell \in \partial p} \sigma_\ell^z

, vertex operators

A_v = \prod_{\ell \ni v} \sigma_\ell^x

; toric code:

|\psi\rangle = \sum_{g} |g\rangle

with

A_v|\psi\rangle = B_p|\psi\rangle = |\psi\rangle

. | CONNECTION: B2/D4 lattice is the root system of

SO(8)

— its Weyl group order 192, with Coxeter number 6. The ratio of distances between B2 root lengths (long:short =

\sqrt{2}:1

) yields

1/\sqrt{2} \approx 0.707

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

H = -\sum_v A_v - \sum_p B_p

, where

A_v = \prod_{i \in v} \sigma^x_i

and

B_p = \prod_{i \in p} \sigma^z_i

. Logical operators

X_L, Z_L

are non-contractible loops; on a genus-

g

surface, degeneracy

= 4^g

. For the p4m wallpaper group (square lattice with reflection/rotation symmetries), the code space is invariant under the group action; the anyonic excitations (e, m, ε) obey

\mathbb{Z}_2 \times \mathbb{Z}_2

fusion rules, and the braiding phase is

e^{i\pi} = -1

. | CONNECTION: The p4m group is the full symmetry group of the square lattice — its point group is

D_4

(order 8), with generators

r

(90° rotation) and

s

( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

H = -\sum_v A_v - \sum_p B_p

with

A_v = \prod_{i \in v} X_i

B_p = \prod_{i \in p} Z_i

, and logical operators

X_L, Z_L

as non-contractible loops on the torus. The p4m wallpaper group has point group

D_4

(order 8) and translation lattice

\mathbb{Z}^2

. Group cohomology

H^2(G, U(1))

classifies projective representations. | CONNECTION: p4m is a crystallographic group with 4-fold rotational symmetry and mirror lines — its fundamental domain has aspect ratio 1:1 (square lattice). The toric code lives on a square lattice, which is the p4m lattice. The toric code's anyonic excitations (e, m, ε) obey fusion Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

e_1, \dots, e_7

with

e_i^2 = -1

and

e_i e_j = \epsilon_{ijk} e_k

where

\epsilon_{ijk}

is given by the 7 lines (each a triple) of the Fano plane. The Peres-Mermin square is a 3×3 array of observables whose row and column products yield ±1, demonstrating contextuality via the Kochen-Specker theorem. No new constants or ratios appear. CONNECTION: The Fano plane is the projective plane of order 2, with 7 points and 7 lines, each line containing 3 points. This is a finite geometry with symmetry group PSL(2,7) of order 168. No direct link to golden ratio, base-60, or crystallographic root systems. However, the octonion algebra is intimately tied to the exceptional Lie group

G_2

(automorphism group of octonions) and the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Tsirelson bound derivable from geometric algebra, with explicit link to E8 root system and octonionic structure in quantum correlations. | MATH: Tsirelson bound

C \leq 2\sqrt{2}

for CHSH; geometric algebra formulation yields

C = 2\sqrt{2}

via bivector/rotor structure; E8 root lattice

\Gamma_8

with 240 roots, octonion multiplication table

e_i e_j = -\delta_{ij} + \epsilon_{ijk} e_k

; key ratio

\sqrt{2} \approx 1.414

, related to

2\cos(\pi/4)

. | CONNECTION: E8 root system encodes maximal Bell violation —

2\sqrt{2}

emerges from the angle

\pi/4

between measurement axes, which is the dihedral angle of the 4D hypercube; octonionic

G_2

symmetry (automorphism group of octonions, 14-dim) stabilizes the correlation structure; ratio

1/\sqrt{2} \approx 0.707

links to

\cos(\pi/4)

, and

2\sqrt{2} \approx 2.828

2 \times 1.414

; the E8 lattice has kissing number 240, and its Coxeter number 30 relates to \( 2\pi/30 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com