The Macklin-Dickson Triad Construction: Spinor Resonance, Contact Completion, and the Third Octonion

The Macklin-Dickson Triad Construction: Spinor Resonance, Contact Completion, and the Third Octonion

Authors

Matthew Wayne Macklin, Sophia N.G. Macklin

Description

The Cayley-Dickson doubling procedure constructs each normed division algebra from the previous one — reals to complexes to quaternions to octonions — but says nothing about how three copies of the same algebra interact. Fairlie and Manogue (1987) attempted to address this through triality: Spin(8) has three inequivalent 8-dimensional representations (vector, left spinor, right spinor), and one could hope that fixing two octonions via triality forces the third. It doesn't work. Triality is a symmetry, not a generator — it shuffles what already exists but cannot create new algebraic content.

This paper replaces the triality fixed-point approach with a resonant construction. Two octonionic structures, generated by independent Cayley-Dickson doublings, are coupled through spinor resonance on the Spin(8) triality bundle. The third octonion is not fixed by symmetry but generated by the contact-completed LRC flow on the diagonal of the exceptional Jordan algebra J₃(𝕆).

The contact completion is the central technical result. The bare LRC system L̇ = RC, Ṙ = LC, Ċ = LR preserves volume (Nambu bracket) but blows up in finite time at t* = φ when initialised above the golden orbit. Adding a self-concordant barrier term — the same logarithmic barrier used in interior-point optimisation — gives the contact-completed flow L̇ = RC + L − 1, which has a unique globally stable orbit at amplitude φ⁻¹, selected by Hopf normal form analysis. The golden ratio is not assumed; it falls out of the bifurcation structure.

Five specific corrections to earlier formulations are documented explicitly, because the history of what broke matters as much as what survived: (1) the symmetric point L = R = C = φ⁻¹ is an orbit amplitude, not a fixed point; (2) the bare system blows up at t* = φ, motivating the contact completion; (3) the Hopf normal form selects φ⁻¹ uniquely through ṙ = σr − σφ²r³; (4) the pentagon identity Z₅(1/2) = 1 − Re(R₁) identifies the Fibonacci anyon braiding phase; and (5) the field-theoretic uniqueness Q(√5) = Q(ζ₅ + ζ₅⁻¹) locks the construction to the prime p = 5.

The construction produces a third octonion that is algebraically distinct from the two inputs, exists as a global section of the triality bundle if and only if the adelic product over Q(√5) converges, and carries the non-associativity of the octonionic algebra as the geometric cost of information crossing between the two generating structures. The Macklin-Dickson triad is to Cayley-Dickson doubling what triadic LRC generation is to binary tensor products.

License

Creative Commons Attribution 4.0 International (CC BY 4.0)

Resource Type

Preprint

Related Identifiers

"Semantic Algebra of a Bootstrapped Universe" (DOI: 10.5281/zenodo.19378099)

"A Pentagon Identity in the Riemann Zeta Euler Product and Trans-Series Structure of the Feigenbaum Gap"

"The Pentagon-Fibonacci Anyon Bridge"

Communities

Mathematical Physics, Algebra