Spectral Duality and Reset-Neutral Distributions in Random Walks with Multi-Site Geometric Resetting

We study the gambler's ruin problem for a biased random walk on $\{0,1,\dots,a\}$ under multi-site geometric resetting: at each time step, the walker is reset with probability $\gamma\in(0,1)$ to a random position drawn from a distribution $\pi$ over $m$ interior sites. Using renewal theory, we derive an exact closed-form expression for the ruin probability $q_z(\gamma)$, showing that the effect of $\pi$ is fully encoded in a single scalar quantity, the \emph{coupling constant} $C(\pi,\gamma)=\bar{u}_\pi/\bar{s}_\pi$. A spectral analysis via Doob symmetrization reveals the structure of this coupling. Our main result is a general criterion -- valid for any absorbed Markov chain admitting a spectral decomposition -- for the existence of a \emph{reset-neutral} distribution $\pi^*$ such that $C(\pi^*,\gamma)$ is independent of $\gamma$. This occurs under a spectral duality condition: there exists an involution $\sigma$ on the reset sites and $\nu$-independent weights $\kappa(z)$ such that $B_\nu(z) = \kappa(z)\,A_\nu(\sigma(z))$ for all spectral modes $\nu$. When this condition holds, the invariant value is $C^* = q_{a/2}^{(0)}$, the classical ruin probability from the midpoint, independent of the choice of symmetric reset sites or resetting rate. For the biased random walk, the condition reduces to the geometric symmetry $z_i + z_i' = a$. This result holds for any $a$, any number of reset sites $m$, and any bias $p\in(0,1)$. Both analytical and Monte Carlo simulations confirm the theory with high precision, including tests of spectrally neutral sites. Numerical results also reveal a phase-like structure in the space of reset distributions, with $\pi^*$ acting as a separatrix between monotone regimes.