A Möbius-Like Twist in Channel Space: Nontrivial 𝑍 2 Holonomy from the Non-Split Fine Step in the R 8 Bank-Leaf Architecture

## Abstract This note studies a focused stability question inside the R^8 bank-leaf architecture: can a channel-role flip help keep the bank on the protective 24-channel branch and away from the phase-locked 8-channel collapse? We model the protective configuration as a cluster-splay orbit on \(\mathbb Z_8\times\mathbb Z_3\), with eight dephased triad-centers and balanced \(120^\circ\) triads. A natural Lyapunov functional is introduced, \[\mathcal V[z]=\alpha\sum_{t=0}^{7}|s_t|^2+\beta\sum_{t=0}^{7}|q_{t+1}-\xi q_t|^2,\] where \(s_t\) detects local triad collapse and \(q_t\) is the cubic lift of the triad-center phase. Its global minima are exactly the protective cluster-splay orbits. For the benchmark choice \(\alpha=\beta=1\), the Hessian at a representative minimum has one neutral global-clock mode and 23 positive modes, with nonzero spectrum \[3\ (\times 16),\quad 3.51471863\ (\times 2),\quad 12\ (\times 2),\quad 20.48528137\ (\times 2),\quad 24\ (\times 1).\] A reproducible synthetic scout with 50 random starts and plain gradient descent reaches the global protective minimum in only \(8/50\) trajectories, the rest falling into a small set of metastable traps. We then test an idealized corrective anti-locking flip: after partial relaxation, any defective triad is rebalanced to the nearest balanced \(120^\circ\) configuration around its instantaneous cubic-center phase. On the first 30 starts of the same seed, plain descent reaches the global minimum in \(5/30\) cases, whereas one corrective rebalance flip followed by descent reaches it in \(30/30\) cases. This does not yet identify the microscopic \(\mathbb Z_2\) holonomy flip with the toy correction map. It does show that a good flip can act as a stabilizer if and only if it lowers the Lyapunov functional, preserves the \(C_8\times C_3\) offset pattern, and reduces coherent dwell time in the average-mode block. A complementary compression family that drives the triads toward the average mode raises \(\mathcal V\) sharply, confirming that a bad flip is pumping rather than damping.