Resolvent intertwining and spectral duality in Markov chains with geometric resetting

Abstract

We uncover the resolvent origin of the spectral duality governing reset-neutral distributions in Markov chains with geometric resetting. Starting from the abstract conditions of Paper~III, we show that the spectral duality $B_\nu(z)=\kappa(z)\,A_\nu(\sigma(z))$ is equivalent to a single symmetry of the resolvent $R(\gamma)=(I-(1-\gamma)P)^{-1}$: the intertwining relation $[\Delta^2\mathcal{R},R(\gamma)]=0$, where $\mathcal{R}$ is the reflection operator of an involution $\sigma$ and $\Delta=\operatorname{diag}(\sqrt{\kappa(z)})$; equivalently, $\widetilde{\mathcal{T}}=K^{-1/2}\Delta^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\sqrt{K})$, with $K=\kappa(z)\kappa(\sigma(z))$, which depends only on the scalar $K$ --- not on the resetting rate $\gamma$, the reset distribution, or the particular chain. We characterize the class of $(\sigma,\kappa)$-reversible chains, encompassing both the biased random walk and genuinely non-homogeneous dynamics sharing the same $C^*$; a Doob $h$-transform realizes the duality $K\mapsto1/K$, hence $C^*\mapsto1-C^*$, with fixed point $C^*=1/2$. The orientation field admits the explicit resolvent representation $\psi(\gamma)=R(\gamma)(b^{(0)}-C^*b)$: its gauge-normalized form $\Delta^{-1}\psi(\gamma)$ is antisymmetric under $\sigma$, it has an exact node at the fixed point of $\sigma$, and it governs the exact sign law $\operatorname{sgn}(C(\pi,\gamma)-C^*) =\operatorname{sgn}\langle\pi,\psi(\gamma)\rangle$. Numerical experiments confirm the theory to machine precision. These results establish the operator-theoretic foundation of the spectral duality of Paper~III and provide the bridge to the information-geometric framework of Paper~V.