S-AI-Recursive: Convergent Recursive Reasoning

This article introduces S-AI-Recursive, a bio-inspired Sparse Artificial Intelligence architecture in which reasoning is implemented as a hormonally regulated closed-loop iterative process rather than a single feed-forward pass. The framework formalizes a Recursive Reasoning Cycle governed by two antagonistic regulatory signals: Clarifine, representing convergence, and Confusionin, representing residual uncertainty. Their interaction regulates iterative state refinement, stopping, computational-resource allocation, and recursive-engram memory retrieval. The mathematical formulation includes coupled state–hormone contraction for fixed-point-structured tasks, Lyapunov stability analysis, conditional entropy contraction, multi-signal adaptive stopping, projected Euler–Maruyama discretization, constrained agent selection, and warm-start initialization from previously stored reasoning trajectories. The experimental evaluation combines controlled mechanism-level simulations with benchmark-capable experiments on real, exactly verifiable reasoning tasks. On convergent Maze instances, adaptive hormonal stopping reduces mean recursive depth by approximately 43% relative to fixed-depth execution while preserving resolution accuracy. On compatible recurring Sudoku instances, recursive-engram warm-start reduces mean stopping depth from 18.4 to 2.0 cycles, saving more than 16 reasoning cycles without loss of resolution. Robustness experiments further show that confidence-aware stopping can resist deceptive convergence plateaus that cause premature termination in residual-only stopping policies, while no comparable advantage is observed under homogeneous Gaussian observation noise. A complementary evaluation on the public ProofWriter benchmark further clarifies both the capabilities and limitations of convergence-based regulation. On clean instances, the architecture achieves 100% accuracy while reducing the required number of reasoning iterations by 66%. Under deliberately deceptive “mirage” conditions, however, convergence detection alone can falsely accept up to 49% of cases, whereas the symbolic certification mechanism of S-AI-RLM maintains zero false acceptance. This distinction demonstrates that detecting convergence is not equivalent to certifying logical correctness. Overall, the results provide evidence that S-AI-Recursive can support adaptive temporal parsimony, memory-based acceleration, and robustness to selected forms of deceptive stabilization, while also identifying the regimes in which an additional symbolic correctness certificate remains necessary. The study therefore positions recursive convergence regulation and formal logical certification as complementary rather than interchangeable components of reliable reasoning architectures.