Nuclear $\gamma$-Ray Cascades as Markov Processes

A framework for computing $\gamma$-ray feeding probabilities in nuclear decay schemes based on absorbing Markov chains is presented. In this approach, excited nuclear states are treated as transient states and long-lived levels as absorbing states, allowing feeding fractions to be obtained exactly from the transition matrix. Experimental uncertainties are propagated via Monte Carlo sampling from Dirichlet distributions, which naturally maintains the physical constraint of unit normalization for branching-ratio vectors. This framework is applied to the key $E_r= 92$ keV resonance in the $^{25}$Mg(p,$\gamma$)$^{26}$Al reaction ($E_x = 6398$ keV), which governs the production of $^{26}$Al in hydrogen-burning environments. Combining multiple experimental datasets within a Hierarchical Bayesian framework, a ground-state feeding probability of $f_0 = 0.68 \pm 0.06~(1\sigma) \pm 0.13~(2\sigma)$ is found, and for the first time the dominant $\gamma$-decay transitions contributing to its uncertainty are identified. The formalism reproduces traditional cascade calculations while providing analytic sensitivity information and a transparent uncertainty decomposition. This approach offers a general and computationally efficient tool for propagating nuclear-structure uncertainties to astrophysical reaction rates and can be readily extended to other nuclei.