Alankar Chains: Stochastic Processes on Evolving Theory-Space

Classical learning and inference frameworks typically assume that optimization occurs within a fixed representational structure. In many scientific and computational settings, however, improvements in explanatory adequacy require structural modifications of the representational framework itself rather than merely parametric refinement within a predetermined model class. This paper introduces Alankar chains, a class of stochastic processes defined on evolving theory-space, that formalize such structurally adaptive dynamics. Within the Structural Descent Framework, a theory is represented as a pair T = (S, θ) consisting of a structural specification S and parameters θ internal to that structure. Alankar chains describe stochastic trajectories through theory-space in which parametric descent is occasionally interrupted by structurally gated transitions. The resulting process can be expressed as a mixture kernel combining continuous parameter updates with discrete structural jumps triggered by plateau conditions in the inadequacy functional. Several theoretical results are established. First, under mild boundedness conditions the inadequacy functional forms a supermartingale along admissible trajectories, yielding convergence properties for the parametric component of the process. Second, structural transitions occur sparsely through plateaugated burst events, implying that most updates remain within fixed structural fibers of theory-space. Third, when structural jumps occur with nonzero probability, the process escapes structural plateaus with probability one, ensuring that representational limitations of fixed model classes can be overcome asymptotically. These results provide a formal mathematical model for dynamic representational adaptation in scientific reasoning and machine learning systems. Alankar chains therefore supply a stochastic dynamical interpretation of structural descent, complementing previous work on Structural Bayesian inference and Gödel-type limitations of fixed theoretical systems. More broadly, the framework suggests that processes capable of controlled structural modification may play a fundamental role in the evolution of scientific theories and in the design of self-improving artificial intelligence systems.