Hidden-state trajectory curvature and variance slope distinguish correct from failing multi-turn LLM-agent episodes before completion

- Liang, Yu, Nasiri & Garraghan (2026) track each turn's hidden state at a fixed intermediate layer at the final token of each user input (layer 22 for Qwen3-14B, layer 42 for Qwen3-32B, layer 16 for Llama-3.1-8B; Sec. 3.2) as a trajectory through activation space, defining temporal curvature (cosine of the angle between consecutive turn-to-turn displacement vectors) and variance slope (growth rate of trajectory dispersion) as two geometric summary statistics. [liang-etal-2026-geometry-of-divergence-multi-turn-reasoning] - Across four tasks (Math, Code, Airline, Retail) and the six task-model pairings reported in Fig. 2, the mean temporal curvature of failed episodes is lower than that of correct episodes (differences +0.030 to +0.148) and correct episodes have higher variance slope (the paper's wording: failed trajectories "contract onto a narrower, unproductive region"); the variance-slope difference is not significant for the two Airline pairings (p=0.30 and p=0.057), and the Code-correct group has a positive median curvature. The Fig. 2 axis labels assign the pairings to models: Math to Llama-3.1-8B, Code, Retail and Airline to Qwen3-14B, and Retail and Airline also to Qwen3-32B. [liang-etal-2026-geometry-of-divergence-multi-turn-reasoning] - Using these signals as thresholds to trigger thinking mode (Sec. 6, Qwen3-14B and Qwen3-32B on tau-Bench Airline/Retail) raises average task reward from 0.241 under Never-thinking to 0.396 (24.1% to 39.6%) and lowers mean token cost from 104.8k to 93.0k (11.2%), averaged over the four domain-model settings; gains over Always-thinking are smaller (e.g. 0.442 vs 0.384 on Retail/Qwen3-32B). [liang-etal-2026-geometry-of-divergence-multi-turn-reasoning]