Abstract
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.
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Jul 11, 2026cs.LG
Recursive reasoning models address structured problems by repeatedly updating latent states of small neural networks. However, their test-time scaling lacks a principled inference mechanism: increasing depth or stochastic breadth generates more trajectories without a clear criterion for selection, and existing methods predominantly rely on additional q-heads or heuristic voting. Here, we develop the Energy-guided Recursive Model (ERM), which introduces an intrinsic selection principle based on explicit Hopfield energies. ERM leverages Hopfield-type memories of valid local or global structures to define the selector over candidate trajectories. The resulting energy seamlessly integrates with energy-based techniques such as parallel tempering to enhance sampling efficiency and ranking. With
D=64 recurrent steps and
K=128 candidates, ERM reaches optimal solutions on Sudoku (
98.97%), Pencil Puzzle Bench (PPBench,
88.04%) and Maze (
99.30%), improving upon recent Probabilistic Tiny Recursive Model and Equilibrium Reasoners. These results suggest that incorporating explicit energy functions into recursive reasoning offers a principled path toward more effective inference.
Yifei Zhao, Ying Tang
May 24, 2026cs.AI
Recent work on recursive architectures has shown that tiny neural networks can be surprisingly powerful on structured reasoning tasks. The trick is to model reasoning trajectories with a latent dynamical system. We argue that the inference-time behaviour of these architectures is best understood as approximate inference over latent reasoning trajectories, with deterministic recursion as the one-particle, zero-noise limit. We make this view operational through guided stochastic exploration: stochastic perturbations of the reasoning dynamics propose neighbouring trajectories, and the model's existing early-stopping head reweights them online. The framework yields three label-free diagnostics: local stability, guide alignment, and cloud-token entropy. These predict, from inference traces alone, whether the procedure will help and which of its outputs to trust. On Sudoku-Extreme it lifts exact-solve accuracy from
85.9% to
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Andrew Corbett, Archit Sood, Anna Tzatzopoulou +2
May 19, 2026cs.AI
How should future neural reasoning systems implement extended computation? Recursive Reasoning Models (RRMs) offer a promising alternative to autoregressive sequence extension by performing iterative latent-state refinement with shared transition functions. Yet existing RRMs are largely deterministic, following a single latent trajectory and converging to a single prediction. We introduce Generative Recursive reAsoning Models (GRAM), a framework that turns recursive latent reasoning into probabilistic multi-trajectory computation. GRAM models reasoning as a stochastic latent trajectory, enabling multiple hypotheses, alternative solution strategies, and inference-time scaling through both recursive depth and parallel trajectory sampling. This yields a latent-variable generative model supporting conditional reasoning via
pθ(y∣x) and, with fixed or absent inputs, unconditional generation via
pθ(x). Trained with amortized variational inference, GRAM improves over deterministic recurrent and recursive baselines on structured reasoning and multi-solution constraint satisfaction tasks, while demonstrating an unconditional generation capability. https://ahn-ml.github.io/gram-website
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