Manifold Agentic Reasoning: Extending Agentic POMDPs and Post-Training Reasoning to Riemannian State and Reasoning Spaces

Abstract

Agentic reasoning systems increasingly interact with environments whose states are only partially observed, dynamically evolving, and constrained by physical, biological, or logical structure. Existing agentic reasoning frameworks often model internal reasoning, tool use, and post-training adaptation using flat latent representations and struggle in curved manifold space environments. However, many scientific and embodied domains naturally lie on curved state spaces, including tissue geometry, developmental trajectories, protein conformations, robotic configuration spaces, and constrained physical systems. We introduce Manifold Agentic Reasoning, a geometric framework that extends agentic reasoning from Euclidean latent spaces to Riemannian manifolds. In the proposed framework, observations are encoded as manifold-valued states, memory is retrieved by geodesic similarity, candidate hypotheses are generated in tangent spaces, predicted transitions are projected by exponential maps, and decisions are admitted through verification-gated commitment or repaired by manifold self-correction. We further extend the framework to graph-agentic manifold reasoning, where node states live on manifolds and neighbor information is transported by logarithmic maps before attention-based aggregation. Manifold agent reasoning moves AI past brittle, prompt-chained templates to solve four critical production flaws: silent hallucinations and reasoning drift, brittle tool and context misuse, the black-box evaluation problem and stiff behavior profiles. To evaluate the framework, we introduce a Curved Tissue Manipulation and Recovery benchmark in which an agent must repair damaged tissue on a curved manifold. Simulated results show that the full manifold-agent substantially outperforms both a baseline reasoning agent and a full flat-agent reasoning system, achieving higher recovery success, lower geodesic shape error, lower pattern error, and fewer invalid transitions. Curvature and ablation studies indicate that the performance gain is driven by geometry-aware reasoning, verification, memory, and self-repair. These results suggest that manifold-aware agentic reasoning provides a principled foundation for reliable scientific and embodied AI systems operating in curved, constrained, or mechanistic domains.