Does transformer reasoning leave a geometric signature in representation space?

Most interpretability treats each layer as an independent snapshot — probing what a layer encodes. This work reframes the forward pass as a discrete population trajectory through a high-dimensional representation manifold and asks how representations travel, not what they hold. Measuring five intrinsic geometric metrics directly in the ambient space (trajectory length, curvature, semantic convergence, layerwise cosine similarity, stability) across GPT-2, TinyLlama, and Qwen2.5, two findings stand out. Semantically related prompts undergo statistically significant trajectory convergence in middle-to-late layers (convergence index 0.41–0.58, p<0.001), consistent with attractor-like dynamics. And reasoning/analogy tasks produce markedly higher mean curvature (0.71–0.83 rad) than lexical-variation tasks (0.27–0.31 rad).

The load-bearing claim is that mean curvature encodes computational complexity — the path a hard task carves through representation space bends more than an easy one's. This is a geometric signature of difficulty computed without any labeled features, which means it could serve as a task-difficulty or reasoning-effort readout independent of behavioral outcomes. It complements the structural argument that Why do transformers need explicit chain-of-thought reasoning?: if reasoning demands more geometric work within a fixed-depth pass, curvature is where that strain shows up. It also gives a manifold-level counterpart to How do transformers learn to reason across multiple steps? — both find that reasoning leaves a distinctive geometric trace, and the convergence-into-attractors picture suggests interventions on semantic content should target the elaboration phase rather than late output layers.