S-AI-RLM: A Recursive Logic Machine

Abstract

S-AI-RLM introduces a Recursive Logic Machine within the Sparse Artificial Intelligence paradigm to address a central limitation of contemporary probabilistic reasoning systems: strong empirical performance does not by itself provide guarantees of termination, logical correctness, reasoning stability, or certified commitment. The architecture explicitly separates three components: an LLM-free coupled cognitive–hormonal dynamical core governed by six artificial hormones; a recursive symbolic layer implementing a total characteristic function and an explicit coherence certificate on a declared decidable domain; and an optional probabilistic proposer connected through semantic translation and treated only as a bounded peripheral perturbation. A triadic Accept–Clarify–Reject (ACR) regime coordinates stabilization and certification, while twelve specialized agents provide inference, verification, orchestration, memory, governance, and metacognitive control under a sparse minimum-coverage principle.Five theoretical results characterize the framework. They establish conditional convergence of the coupled cognitive–hormonal dynamics under contraction and small-gain assumptions; relate joint convergence to totality of the internally certified ACR stopping procedure under explicit compatibility conditions while keeping symbolic Turing-decidability as a separate property; establish finite-time internal stopping under strict stop/certificate compatibility; derive conditional entropy and symbolic-coherence consequences; and characterize optional LLM interventions as bounded perturbations, with exact convergence recovered when these perturbations cease or are summable. The formulation therefore explicitly distinguishes dynamical stabilization, algorithmic termination, and logical correctness rather than treating them as equivalent notions.Experimental validation is organized in successive levels. An original controlled pre-validation on 6,000 benchmark-shaped synthetic instances across five random seeds exercised all H1–H7 measurement paths; on four synthetic decidable-core families, the simulated S-AI-RLM policy reached 92.33% decision accuracy (95% CI: 91.46–93.13%) with a 100% valid-decision rate. The expanded executable validation contains 40 automated tests, all passing, and nine experiment families covering both the shared recursive mechanism and the RLM-specific certification layer. On exactly verifiable procedural-entailment, inconsistency, and finite-CSP instances, the implemented symbolic core achieved 100% observed decision accuracy on the tested samples, with sound Accept and contradiction-Reject decisions in their applicable regimes. Certification-aware adaptive stopping preserved observed accuracy while substantially reducing reasoning depth relative to fixed-depth execution.A central certificate ablation isolates the difference between convergence monitoring and certified commitment. When residual and entropy signals are simultaneously corrupted by deceptive false-confidence mirages, a Recursive-style C1–C3 stopping rule falsely accepts approximately 5–11% of tested procedural-entailment instances, whereas the complete S-AI-RLM C1–C4 certification rule maintains a 0% observed false-acceptance rate.Public-benchmark validation was additionally performed on ProofWriter without modifying the symbolic engine. Within the declared unary positive Horn fragment, 5,364 ProofWriter test instances were loaded, and the grounding procedure combined with the total characteristic function reproduced the benchmark Gold labels exactly. On a balanced sample of 400 ProofWriter test instances, the complete triadic symbolic engine obtained 100% observed accuracy. On 200 multi-step ProofWriter instances exposed to the same false-confidence mirage protocol, the Recursive-style stopping rule falsely certified incorrect answers in approximately 14–18% of corrupted cases, whereas the complete RLM certificate again maintained 0% observed false acceptance. The paired difference remained statistically significant after Holm correction at every non-zero tested mirage level. These experiments provide public-benchmark evidence that explicit symbolic certification can prevent commitments that instantaneous convergence-related signals alone may permit under deceptive conditions.Experiments on the shared recursive mechanism further showed that adaptive hormonal stopping reduced Maze iterations by approximately 43% while preserving resolution, and that compatible warm-start initialization reduced reasoning depth by approximately 16 cycles on recurring converged Sudoku instances. Conversely, a controlled homogeneous Gaussian-noise experiment found no advantage of the hormonal rule over a residual-only detector, providing an explicit negative result and delimiting the conditions under which multi-signal regulation is beneficial.