Separatrix structure and the geometry of reset distributions

We study the geometry of reset distributions for absorbed Markov processes with geometric resetting, working at an abstract level that isolates the structural mechanism underlying reset-neutral invariance. We show that the spectral duality endows the simplex of reset distributions with a non-trivial spectral response geometry: the simplex ${\mathrm{\Delta }}_{m-1}$ carries a foliation by level sets of the coupling functional $C$, organized around a critical manifold $\mathrm{\Sigma }$ that acts as a global orientation boundary for the reset response. Under four structural conditions (S1)–(S4) on the coupling functional, we establish the existence and explicit characterization of the separatrix $\mathrm{\Sigma }$, derive the invariant value $C^{\ast }=1\mathrm{/}(1+\sqrt{K})$, identify a projective structure in the spectral coefficients, and prove a global sign principle in the two-site case. The linear functional $\psi (\gamma )$ emerges as a global orientation field: numerical evidence suggests $\mathrm{sgn}({\mathrm{\partial }}_{\gamma }C)=\mathrm{sgn}⟨\pi -{\pi }^{\ast },\psi (\gamma )⟩$ for all $\pi \in {\mathrm{\Delta }}_{m-1}^{\circ }$. The biased random walk with multi-site geometric resetting provides a canonical realization. This is the third paper in a program connecting stochastic resetting with spectral theory and information geometry.