Gauge couplings of the Standard Model in the octonionic framework: a broken-phase mechanism for $\alpha_s/\alpha_{\mathrm{em}}=16$

We present a consolidated gauge-sector account of the octonionic programme, starting from the trace-dynamics Lagrangian and ending with closed-form expressions for the strong and electromagnetic couplings, together with a brief review of the weak mixing angle. The main new step is a broken-phase support mechanism on the real octonionic ladder space $H_6$ which, under a specific support hypothesis, gives \begin{equation} \frac{\alpha_s}{\alpha_{\mathrm{em}}}=16 \end{equation} from a common visible Yang--Mills coupling. We then combine this relation with the 2022 Eur. Phys. J. Plus seed [1] \begin{equation} A:=\exp\!\left[q_0\!\left(q_0-\sqrt{\frac38}\right)\right],\qquad q_0=\frac13, \end{equation} to obtain \begin{equation} \alpha_s^{\mathrm{th}}(M_Z)=\frac{9}{64}\exp\!\left[\frac23\!\left(\frac13-\sqrt{\frac38}\right)\right]=0.11675418, \end{equation} \begin{equation} \alpha_{\mathrm{em}}^{\mathrm{th}}(0)=\frac{9}{1024}\exp\!\left[\frac23\!\left(\frac13-\sqrt{\frac38}\right)\right]=0.00729713629. \end{equation} The electromagnetic formula is algebraically the same as in the earlier paper [1], but its factor $1/16$ is now attached to an explicit broken-phase gauge normalization rather than to a length-identification step. A key conceptual point is that the seed is tied to the minimal visible charge quantum $q_0=1/3$, not to a specific particle species: the electron, whose charge is $1=3q_0$, enters later through the electromagnetic charge trace $k_{\mathrm{em}}=8/3$. We also review the earlier spinorial derivation of the weak mixing angle [2], which yields $\sin^2\theta_W^{\mathrm{th}}=0.24969776$, and assess it separately. The strong and electromagnetic results are numerically close to experiment; the weak-angle comparison is substantially less successful.