S-AI-RLM: A Recursive Logic Machine

BACKGROUND — Contemporary artificial intelligence systems based on probabilistic language models achieve remarkable empirical performance but lack formal guarantees of termination, logical correctness, and reasoning stability. Existing neuro-symbolic architectures restore decidability but abandon adaptive regulation. Satisfying decidability, dynamic convergence, hormonal metacognitive control, and modular sparse orchestration simultaneously has remained an open architectural challenge; no unified framework to our knowledge has yet met all four requirements. METHODS — This article introduces S-AI-RLM, an extension of the Sparse Artificial Intelligence paradigm that integrates three formally coupled layers: (i) a recursive symbolic core grounded in recursive language theory, implementing a total symbolic decision function that halts and returns a certified answer on every valid input; (ii) a hormonally regulated dynamical system governed by six artificial hormones whose global asymptotic stability is established by a Lyapunov argument; and (iii) a triadic metacognitive regime — Accept, Clarify, or Reject (ACR) — formalised as an optimal control policy. The architecture is realised through twelve agents organised in three functional layers and coupled to a large language model via a semantic translation module with formally characterised fidelity and polynomial-time computability. RESULTS — Five theorems are established: global asymptotic stability of the hormonal regulatory field (Theorem 1), formal equivalence between hormonal convergence and totality of the ACR stopping procedure (Theorem 2), finite-time termination with an explicit three-term bound (Theorem 3), a quadruple equivalence of Lyapunov stability, entropic contraction, symbolic coherence, and totality of the ACR procedure (Theorem 4), and Lyapunov stability of the full hybrid system (Theorem 5). Experimental validation on our evaluation testbench across four standard reasoning benchmarks and against four reference systems confirms a decidability rate of 100% on the evaluated suite, a decision accuracy exceeding 84%, monotonic entropy reduction across iterations, and a frugality index of at least 0.71 — with fewer than ten million parameters. CONCLUSIONS — S-AI-RLM demonstrates that an intelligent system can simultaneously be adaptive, formally decidable, convergent, and explainable. The framework defines a new class of AI architectures grounded in the principle: not self-improving intelligence, but self-converging, regulated, and decidable intelligence.