Dynamics

Core Dynamical Ingredients

• Evolution in full space $T1 \oplus T2$
• Transport operator acting in $T1$
• Off-diagonal coupling between $T1$ and $T2$
• Reduced evolution for the projected sector
• Effective transport-diffusion after coarse-graining

The $G_2$ Coarse-Graining and Scale Fixing

The $\sqrt{3}$ root ratio in $G_2$ is the algebraic record of coarse-graining from 7 imaginary octonionic dimensions to rank 2. The Cartan matrix of $G_2$ (with rows/columns indexed by short root $\alpha_1$, long root $\alpha_2$) is:

$SU(3) \subset G_2$ via the long roots. Positive long roots ${\alpha_2,\; 3\alpha_1+\alpha_2,\; 3\alpha_1+2\alpha_2}$ form the $A_2 = SU(3)$ subsystem. Positive short roots ${\alpha_1,\; \alpha_1+\alpha_2,\; 2\alpha_1+\alpha_2}$ transform as $\mathbf{3}\oplus\bar{\mathbf{3}}$ under this $SU(3)$ — the “extra matter” sector, not the $SU(3)$ Cartan generators.

The $SU(3)$ (long-root) coupling is $1/\sqrt{3}$ of the short-root coupling within $G_2$.

Embedding index: $j_{SU(3) \subset G_2} = [T(\mathbf{8}) + T(\mathbf{3}) + T(\bar{\mathbf{3}})]/T_{G_2}(\mathbf{14}) = [3 + \tfrac{1}{2} + \tfrac{1}{2}]/4 = 1$. So the overall coupling $g_{SU(3)} = g_{G_2}$ at the $G_2$ breaking scale — the $1/\sqrt{3}$ is a generator-normalization ratio within $G_2$, not a group-level coupling mismatch.

Does $1/\sqrt{3}$ appear in SM coupling ratios? No — structural reason identified.

• $g_3/g_2$ at $M_{GUT} = 1$ (by $j=1$ embedding indices, all SM groups). No $\sqrt{3}$.
• $g’/g_2 = \sqrt{3/5}$ from $\sin^2\theta_W = 3/8$. Contains $\sqrt{3}$, but this is the octonionic 3+2 split $\sqrt{3}$, not the $G_2$ root ratio $\sqrt{3}$.

• Both $\sqrt{3}$’s share the same origin: $N_c = 3$. The $G_2$ root squared ratio $

  \alpha_L

  ^2/

  \alpha_S

  ^2 = 3 = N_c$ (because $G_2 \supset SU(3)$ via long roots, and $N_c = 3$ determines the root ratio). The Weinberg angle uses $N_c = 3$ via the 3+2 counting. They are two projections of the same $N_c$ fact, not independent $\sqrt{3}$ appearances.

Observable consequence of $1/\sqrt{3}$: If the short-root generators of $G_2$ (transforming as $\mathbf{3}\oplus\bar{\mathbf{3}}$ of $SU(3)$, i.e., leptoquark-type bosons) are massive at the $G_2$-breaking scale, their coupling to SM particles is $\sqrt{3}$ times the QCD coupling at that scale. This would be observable in leptoquark searches — not in SM coupling ratios among $g_3, g_2, g_1$.

Renormalization Reframed

• $SO(2,4)$: free dilatation generator → RG running → UV divergences are artifacts of unresolved scale freedom
• Vector selection + double cover → $Spin(2,3)$: scale physically fixed by geometry
• RG “running” becomes a one-time geometric fact rather than ongoing scale dependence
• UV divergences in standard QFT may be artifacts of using $SO(2,4)$ language when the actual theory has already fixed its scale via $u$

Reduced Dynamics

Under weak coupling, fast hidden-sector relaxation, and Markov coarse-graining, the reduced observable evolution is time-local with:

where $m$ is the sector-mixing scale and $\gamma$ is the hidden-sector relaxation rate.

Mass as Sector Mixing

$m$ is read as an effective mass parameter because it controls the first departure from purely ballistic visible transport. Diffusion $D \sim m^2/\gamma$ is the observable shadow of hidden-sector excursions. Higgs-mediated mass generation may be what opens access away from pure $T1$ propagation — the Higgs sets the scale at which the system first mixes between T1 and T2 channels.

Status