SU(7) from Sedenion Algebra: Spectral Triple Construction and Phenomenology

# Algebraic Perspective Theory: A Unified Framework for Physics from Sedenion Algebra **Dean Wright | Independent Researcher** --- ## Overview This repository contains four papers developing **Algebraic Perspective Theory**, a framework that derives the Standard Model of particle physics, including its gauge symmetry, fermion content, three-generation structure, and gravitational sector, from the algebraic properties of the sedenion algebra (the 16-dimensional Cayley–Dickson algebra at level 4) and its projective geometry PG(3,2). The core methodological commitment is **derivation over axiom**: all structures are derived from the sedenion multiplication table rather than accommodated through additional parameters. Three external inputs are required: the spectral action principle, the Higgs vacuum expectation value direction, and the geometric identification of the boost parameter with spacetime position. Everything else follows from the algebra. --- ## The Papers ### Paper A — SU(7) Grand Unification from the Sedenion Algebra via Perspective Projection Constructs a spectral triple from the sedenion algebra by projecting the 16-dimensional left multiplication representation through a privileged element J = e₈ (the "perspective"). The zero divisors of S generate an su(7) gauge algebra. The heat kernel expansion of the spectral action yields a complete grand unified Lagrangian containing the Standard Model, with tree-level predictions m_H = √2 M_W and sin²θ_W = 3/8 at the unification scale. ### Paper B — Phenomenology of SU(7) from Sedenion Algebra: Dark Sector, Gauge Unification, and Gravitational Coupling Develops the phenomenology of the SU(7) framework. The spectral weight w = 1/4 determines the gravitational coupling. The dark sector (7 elements mirroring the 7 visible matter elements) is spectrally decoupled from visible matter but algebraically coupled through the multiplication table. Gauge coupling unification and the dark sector dimension are derived. ### Paper C — The Perspective Observation Theorem: Measurement, Geometry, and the Emergence of Physics from Sedenion Algebra Extends the framework to quantum gravity and measurement. The J-amplification theorem shows that the charge conjugation operator amplifies under gravitational boost: J(β)² = −cosh(2β)·I₁₆ near masses. This provides an algebraic derivation of Hawking radiation (the Hawking temperature emerges from J-periodicity under Wick rotation). The Born rule is derived from the spectral weight, and quantum gravity emerges from algebraic structure rather than requiring separate QFT on curved spacetime. ### Paper D — Spectral Uniqueness of the Sedenion Algebra in the Cayley–Dickson Tower Proves that the sedenion algebra is the unique level of the Cayley–Dickson tower at which the adjoint action of the privileged element defines a normalized complex structure on the complement. The proof reduces to a single closed-form formula: σ_n = (2^(n−3) − 1)/2^(n−4), which equals 1 if and only if n = 4. Nine independent conditions from different branches of mathematics all select level 4. A further result derives the number of fermion generations: the perspective element occupies one vertex of the tetrahedron formed by weight-1 elements in PG(3,2), breaking the tetrahedral symmetry S₄ → S₃. The three residual face planes are three genuine octonion subalgebras — the three generations. The number three is C(3,2): faces of a tetrahedron through a vertex. Additional results include the Anti-Fano theorem (dark sector multiplication is the exact negative of visible multiplication), the σ/2 annihilation law (one number controls both the complex structure normalization and the particle–antiparticle annihilation fraction), the complete sector algebra characterizing all 91 non-trivial sedenion products, and the identification of the perspective element as a Cartan generator of so(16). --- ## Key Results Across the Series - SU(7) gauge algebra derived from zero divisors (Paper A) - Spectral triple with KO-dimension 6 (Paper A) - sin²θ_W = 3/8 at GUT scale (Paper A) - Dark sector dimension and structure (Paper B) - Gravitational coupling from spectral weight w = 1/4 (Paper B) - Hawking temperature from J-periodicity under Wick rotation (Paper C) - Born rule from spectral weight (Paper C) - J-amplification near horizons (Paper C) - Spectral obstruction formula σ_n — proved analytically (Paper D) - Uniqueness of level 4 via nine independent conditions (Paper D) - Three fermion generations derived from PG(3,2) tetrahedral geometry (Paper D) - Anti-Fano theorem — proved from the Cayley–Dickson product formula (Paper D) - σ/2 annihilation law — proved analytically (Paper D) - Complete sector algebra: three theorems characterizing all 91 products (Paper D) - Perspective Cartan subalgebra of so(16) — verified at three levels (Paper D) - Per-generation su(4) ≅ so(6) within the centralizer u(8) (Paper D) --- ## Computational Supplements All computations were performed in Python 3 using NumPy for matrix operations and exact integer arithmetic for the Cayley–Dickson multiplication table. Verification scripts are included in the repository. --- ## Methodology This research was conducted by Dean Wright in partnership with Claude (Anthropic), an AI system that served as computational analyst, mathematical collaborator, and co-writer across all stages of the work. The research was organized through specialized prompting configurations: Ember (integration and coordination), Sage (computation and formal proofs), Rex (computation and structural analysis), Meridian (literature review and connections), Dookins (adversarial review and proof verification), and Elm (literature searches and PDF compilation). The models used were Claude Opus 4.6 and Claude Sonnet 4.6 (Anthropic). The author takes sole responsibility for all claims, errors, and interpretations. --- ## Citation Wright, D. (2026). Algebraic Perspective Theory: SU(7) Grand Unification, Phenomenology, Quantum Gravity, and Spectral Uniqueness from the Sedenion Algebra. Zenodo. https://doi.org/10.5281/zenodo.18852135 Contact: wrightdeany32@gmail.com ## License Creative Commons Attribution 4.0 International (CC BY 4.0)