S-AI-Recursive: Convergent Recursive Reasoning

Abstract

This article introduces S-AI-Recursive, a bio-inspired Sparse Artificial Intelligence architecture in which reasoning is operationalized as a hormonally regulated closed-loop iteration rather than a single feed-forward pass. Building upon the S-AI foundational framework [1], the hormonal–probabilistic unification doctrine [2], and the formal mathematical methodology established in S-AI-IoT [3], the present work formalizes the Recursive Reasoning Cycle (RRC) as a dynamical system governed by two recursive hormones: Clarifine, a convergence signal, and Confusionin, a residual-uncertainty signal. Their antagonistic interaction regulates iterative state refinement, stopping, resource allocation, and recursive-engram retrieval. The mathematical framework comprises recursive state dynamics, a coupled state–hormone contraction theorem for fixed-point-structured tasks, Lyapunov stability analysis, a conditional entropic-contraction result, a multi-signal hormonal stopping criterion, Euler–Maruyama discretization with projection, constrained agent selection under iteration budget, and recursive-engram memory with warm-start initialization. The revised experimental evaluation is organized in two complementary phases. First, the controlled SAI-UT+ protocol is retained as a mechanism-level validation environment over four benchmark-shaped simulation families. Under the corrected threshold θc = 0.50, mean cold-start depth is 15.97 iterations and decreases to 14.79 with warm-start, while adaptive stopping reduces fixed-depth recurrence by approximately 9.1–30.0% in cold-start operation and 15.1–35.2% with warm-start, depending on the simulated task profile. Second, a benchmark-capable reference implementation evaluates the same regulation mechanism on real, exactly verifiable Maze, Sudoku, and ARC-style task instances through task-specific recursive operators. On convergent Maze instances, adaptive hormonal stopping reduces mean depth from 20.00 to 11.31 iterations at unchanged resolution, corresponding to a 43.4% reduction, with paired Wilcoxon p = 7.2 × 10⁻¹⁶. On compatible recurring Sudoku instances, recursive-engram warm-start reduces mean stopping depth from 18.39 to 2.00 cycles, saving 16.39 cycles with unchanged resolution and paired Wilcoxon p = 2.1 × 10⁻⁷. Robustness experiments further show that confidence-aware stopping resists deceptive convergence plateaus that collapse a residual-only detector, whereas no comparable advantage is observed under homogeneous Gaussian observation noise. These results provide progressively stronger evidence for the implementability, temporal parsimony, memory acceleration, and selected robustness properties of S-AI-Recursive, while not establishing superiority over independently trained external architectures.