Arrow: $E_6 \to J_3(\mathbb{O})$

What is there

This arrow is the strongest in the chain. $E_6$ is precisely the group of determinant-preserving linear maps on $J_3(\mathbb{O})$. The connection is canonical and bidirectional: define $J_3(\mathbb{O})$ and derive $E_6$ as its symmetry group, or start with $E_6$ and derive $J_3(\mathbb{O})$ as its natural representation.

Steel man

This arrow runs in both directions without loss. It is not a choice — it is a mathematical identity between two objects. No additional justification is needed; the connection is part of the definition of the exceptional Jordan algebra.

Gap

The choice between $J_3(\mathbb{O})$ and $J_3(\mathbb{O}_\mathbb{C})$ remains open. Different real forms of $E_6$ correspond to different Jordan algebras, and the physical selection between them is still a bridge burden. The specific real form needed for a Lorentzian spacetime signature has not been derived from within the chain.

Status