From Z$_{24}$ Branch Measure to C$_3$ Triadic Sign A branch-selected update of spontaneous electroweak baryogenesis in the R$^8$ framework

## Abstract The earlier R^8 baryogenesis stack already contained a viable single-branch mechanism: a drifting \(V_{24}(\theta,T)\), a microphysical trigger-response route that avoids a strict Hubble ceiling, and a sphaleron projector proportional to the anomaly coefficient \(A_W\). The present note updates that picture in the light of the newer branch-selection work. The correct effective source is no longer a single drifting angle by itself, but a branch-selected ensemble average over the locked \(\mathbb Z_{24}\) minima, \[Y_B^{\rm net}=\sum_{k=0}^{23} p_k\,Y_B^{(k)},\qquadp_k\propto \exp\!\big[-V_{\rm eff}(\theta_k;T)/T_{\rm eff}\big],\qquad\theta_k=\frac{k\pi}{12}.\] The key update is that concentration on the dominant quartet \(k\in\{4,8,16,20\}\) is necessary but not sufficient: without a channel selector the two CP-conjugate pairs \((4,16)\) and \((8,20)\) still cancel in the branch-averaged source. In the quartet-dominated regime, however, the sign-carrying factor collapses from the full \(\mathbb Z_{24}\) base to the two nontrivial \(C_3\) classes, \[\mathcal B(T)=\sum_{k=0}^{23} p_k(T)\sin\!\left(k\frac{\pi}{6}\right)\approx\frac{\sqrt3}{2}\Big[(p_4+p_{16})-(p_8+p_{20})\Big],\] so that the branch measure lives on the \(\mathbb Z_{24}\) base while the baryon-sign channel lives on the triadic fibre. A synthetic scout with a toy branch-selection potential confirms the update sharply: for \(\beta_0=0\) the quartet weight already reaches \(0.869268\) but the reduced source vanishes, whereas for \(\beta_0=+0.5\) one finds \(P_{\rm quartet}=0.915736\), \(P_{\rm m}=0.887942\), \(P_{\rm a}=0.027794\), and \(\widetilde Y_B=0.408514\); the sign flips exactly for \(\beta_0\to-\beta_0\). The mechanism is therefore upgraded from a single-branch spontaneous baryogenesis story to a branch-selected spontaneous baryogenesis story, with the main open tasks now reduced to full cosmological normalization and a microscopic derivation of the selector dynamics.