PM5M: A Formal Axiomatic Language for Probability Theory with Memory-Dependent Limit Theorems

DescriptionThis work presents the completed formal axiomatic language PM5M (“Probability with Memory, 5 Modules”) and its application to solving memory–dependent limit theorems for sequences of dependent events. PM5M extends classical probability theory by introducing explicit memory weights αn\alpha_nαn into the probabilistic framework, enabling rigorous handling of long–range dependencies and decaying influence of past events. The paper includes: Full syntax and semantics of PM5M in BNF form, with precise derivation rules and model–theoretic interpretation of memory. Weighted Borel–Cantelli–type theorems under Kochen–Stone–style covariance decay, yielding necessary and sufficient criteria for almost–sure infinitely–often occurrence in processes with memory. Reduction proofs showing that PM5M collapses to classical probability when αn\alpha_nαn is constant or equal to 1. Internal consistency analysis ensuring that the axioms are non–contradictory and compatible with standard probability theory. Empirical validation via Monte Carlo simulations for multiple dependency profiles (ρ\rhoρ) and weight/probability decay laws, confirming the theoretical predictions in both divergent and convergent regimes. This framework closes a long–standing gap in handling the interaction between event dependency and memory in probability theory. It is both formally checkable and empirically verifiable, making it applicable to theoretical research, stochastic process modeling, and AI systems that require probabilistic reasoning with memory effects.