Non-Abelian Spin Counting of Ordered Stochastic Trajectories: Reentrant Finite-Time Chern Numbers

Abstract

Conventional full counting statistics assigns commuting phases to integrated stochastic currents and therefore resolves net transport but not, in general, the temporal ordering of events associated with different cycles. We introduce a non-Abelian counting construction in which crossings of two fundamental cycles rotate an auxiliary spin about different axes. The resulting ordered trajectory statistic has an exact finite-dimensional evolution equation for its first moment. The two rotation angles form a counting torus, and whenever the mean spin is nonzero its normalized direction defines a map $\mathbb T^2\to\mathbb S^2$, equivalently a complex eigenline bundle with a Chern number. Our main analytical result is a Chern--parity correspondence. Reflection symmetry equips this eigenline with a real structure and expresses $C_T\bmod2$ through first Stiefel--Whitney classes on the circles fixed by reflection. When the transverse polarization has no additional zeros along the reflection-fixed circles, these classes reduce to ordinary current-parity statistics at the four high-symmetry counting points. For a five-state nonequilibrium figure-eight network, varying only the observation time produces four polarization-gap closings and the reentrant sequence $C_T=0\to-1\to0\to-1\to0$. Every transition occurs at $(π,π)$ and coincides with a sign reversal of $\mathbb E[(-1)^{Q_1+Q_2}]$, while the full integer Chern number is obtained independently from the two-dimensional spin texture. Finite observation time can therefore organize a fixed stochastic process into distinct topological sectors of its ordered path ensemble.