Gauge Couplings of the Standard Model from First Principles in the Octonionic Framework

We present a self-contained gauge-sector account of the octonionic programme, starting from the underlying 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 derivation has three steps. First, inside the visible bosonic sector we derive the broken-phase relation αsαem=16, from a single visible Yang--Mills coupling before symmetry breaking. The mechanism combines the standard visible charge-trace factor \( 8/3 \) with a six-direction support factor \( 6 \) on the real octonionic ladder space \( H_6 \). Second, we recall the 2022 Eur. Phys. J. Plus. paper [1], where the minimal visible charge quantum \( q_0=1/3 \) fixes the exponential seed A:=exp[q0(q0−38)]=exp[13(13−38)]. Combining this seed with the charged-sector datum \( 3/8 \) gives αs\thv(MZ)=964exp[23(13−38)]=0.11675418, while the broken-phase factor \( 16 \) then yields αem\thv(0)=91024exp[23(13−38)]=0.00729713629. Third, we briefly review the earlier spinorial derivation of the weak mixing angle~\cite{RajSinghBosonic}, which leads to 1=cos⁡(θW/2)2+sin⁡(θW/2),sin2⁡θW\thv=0.24969776. A key conceptual point is that the seed is attached to the \emph{minimal visible charge quantum} \( q_0=1/3 \), not to a specific particle species. The electron, whose charge is \( 1=3q_0 \), is not omitted: its contribution enters explicitly through the electromagnetic charge trace \( k_{\mathrm{em}}=8/3 \). In this form the derivation of $\alpha_{\mathrm{em}}$ is conceptually sharper than in the earlier Eur. Phys. J. Plus. presentation [1], because the factor\( 1/16 \) is no longer hidden in a length-identification step but is derived directly from the visible broken-phase gauge structure.