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 $B_ν(z)=κ(z)\,A_ν(σ(z))$ is equivalent to a single symmetry of the resolvent $R(γ)=(I-(1-γ)P)^{-1}$: the intertwining relation $[Δ^2\mathcal{R},R(γ)]=0$, where $\mathcal{R}$ is the reflection operator of an involution $σ$ and $Δ=\operatorname{diag}(\sqrt{κ(z)})$; equivalently, $\widetilde{\mathcal{T}}=K^{-1/2}Δ^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\sqrt{K})$, with $K=κ(z)κ(σ(z))$, which depends only on the scalar $K$ --- not on the resetting rate $γ$, the reset distribution, or the particular chain. We characterize the class of $(σ,κ)$-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 $ψ(γ)=R(γ)(b^{(0)}-C^*b)$: its gauge-normalized form $Δ^{-1}ψ(γ)$ is antisymmetric under $σ$, it has an exact node at the fixed point of $σ$, and it governs the exact sign law $\operatorname{sgn}(C(π,γ)-C^*) =\operatorname{sgn}\langleπ,ψ(γ)\rangle$. 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.