Spectral duality structures and the Fisher--Rao geometry of reset distributions

Abstract

We study the geometry that spectral duality induces on the simplex of reset distributions for absorbed Markov processes with geometric resetting. The Fisher--Rao metric provides the intrinsic geometry: under the square-root embedding, the reset-neutral separatrix $\Sigma$ becomes a totally geodesic subsphere and a Fisher--Rao simplex of lower dimension. We then reduce the reset response to a finite structure: the response functionals $\psi(\gamma)$ span a subspace $V$ whose dimension $r$ equals the number of active orbits of the duality involution, while the local separatrix is its annihilator. At the vertices of the simplex we prove a sign theorem valid for every $r$, recovering the two-zone phenomenon of Paper~I. The invariant $r$ also resolves the global orientation principle conjectured in Paper~III. For $r=1$, all response functionals are collinear and, under a scalar sign condition satisfied by the canonical realisation, the response has a fixed sign on each side of $\Sigma$. For $r\ge2$, the response span has dimension at least two and the orientation can rotate; we give the criterion for the failure of a global sign law and exhibit counterexamples in the abstract class. The biased random walk with multi-site geometric resetting realises the whole construction explicitly. This is the fifth paper in a program connecting stochastic resetting with spectral theory and information geometry.