Arrow: $J_3(\mathbb{O}) \to G_2$

What is there

$G_2$ is the automorphism group of the octonions $\mathbb{O}$. Since $J_3(\mathbb{O})$ is built from octonions, $G_2$ acts on $J_3(\mathbb{O})$ by acting on the octonionic entries. The full automorphism group of $J_3(\mathbb{O})$ is $F_4$, and $G_2 \subset F_4 \subset E_6$ is the relevant embedding chain.

Arrow type note: This arrow goes from an algebra ($J_3(\mathbb{O})$) back to a group ($G_2 \subset F_4 = \mathrm{Aut}(J_3(\mathbb{O}))$). It is not a reduction of $J_3(\mathbb{O})$ to $G_2$ in the quotient sense — it is the selection of a distinguished subgroup of the full automorphism structure. See the note on arrow types in master.md.

The $\sqrt{3}$ coarse-graining claim

$G_2$ reduces 7 imaginary octonionic dimensions to rank 2. The triple bond in the $G_2$ Dynkin diagram encodes a root length ratio of $\sqrt{3}$ — the only such ratio in any exceptional Lie algebra. The proposal is that this ratio is the algebraic fingerprint of the coarse-graining from 7 octonionic dimensions to the effective rank-2 description. It is forced by the dimension count, not chosen.

The Cartan matrix of $G_2$ is:

Under $G_2 \to SU(3)$ via $u$-selection, the short root generators become $SU(3)$ Cartan generators. The coupling ratio inherited through Killing form normalization is:

Whether this ratio survives to observable couplings is the key calculation.

The Leech connection

The off-diagonal sector of $J_3(\mathbb{O})$ over integral octonions contains a sublattice isometric to the Leech lattice (Baez/Egan). This is established mathematics, not a dimensional coincidence:

This is the strongest established result in the program. See statics.md for the full 24+3 split and Leech tier structure.

Gaps

• $G_2$ sits inside $F_4$ which sits inside $E_6$. The specific embedding chain and how it interacts with the $u$-selection needs to be made explicit.
• The Killing form calculation is now done; what remains is the physical interpretation of the resulting normalization data and whether any observable scale ratio survives beyond generator normalization.

Current reading after the Killing form calculation

The explicit Killing form normalization does not feed through to a distinct Standard Model coupling ratio: the embedding index is $j_{SU(3)\subset G_2}=1$, so the group-level coupling matches at the embedding scale. The surviving structural content is that the $G_2$ root ratio and the Weinberg-angle $\sqrt{3}$ both trace back to the same $N_c=3$ count, rather than giving two independent predictions. See dynamics.md.

Status