This article introduces S-AI-Recursive, a bio-inspired Sparse Artificial Intelligence architecture in which reasoning is operationalized as a hormonal 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 novel hormones: Clarifine, a convergence signal, and Confusionin, an uncertainty detector, whose antagonistic regulation drives iterative state refinement toward a stable cognitive equilibrium. The complete mathematical framework is developed, including recursive state dynamics, Lyapunov stability proof, entropic contraction theorem, hormonal stopping criterion with finite-time termination guarantee, Euler-Maruyama discretization with projection, primal-dual agent selection under iteration budget, and recursive engram memory with warm-start acceleration. Experimental validation on the SAI-UT+ testbench demonstrates that S-AI-Recursive achieves competitive reasoning performance on abstract and symbolic benchmarks with fewer than ten million parameters, confirming the central principle of temporal parsimony: iterative cognitive depth substitutes for architectural width.

翻译：本文提出S-AI-Recursive，一种受生物启发的稀疏人工智能架构，其中推理被建模为激素驱动的闭环迭代过程，而非单次前馈传递。该工作基于S-AI基础框架[1]、激素-概率统一学说[2]及S-AI-IoT[3]中建立的严谨数学方法论，将递归推理循环形式化为由两种新型激素调控的动态系统：Clarifine（收敛信号）与Confusionin（不确定性检测器），二者通过拮抗调节驱动迭代状态精炼至稳定的认知均衡。我们构建了完整数学框架，包括递归状态动力学、Lyapunov稳定性证明、熵收缩定理、具有有限时间终止保障的激素停止准则、带投影的Euler-Maruyama离散化、迭代预算约束下的原始-对偶智能体选择，以及带热启动加速的递归记忆印迹。在SAI-UT+测试基准上的实验表明，S-AI-Recursive以不到一千万参数在抽象与符号推理基准上达到有竞争力的推理性能，验证了时间简约性的核心原则：迭代认知深度可替代架构宽度。