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Geometry of Divergence: Tracking Hidden-State Trajectories for Adaptive Multi-Turn Reasoning
Jie Liang, Zhengxin Yu, Hamid Nasiri, Peter Garraghan
TL;DR
Long multi-turn interactions can destabilize an LLM agent’s representation of accumulated task information, making useful reasoning difficult to distinguish from representation drift. The paper models this process through hidden-state trajectory geometry, identifies geometric separation between correct and incorrect episodes, and uses that signal to time adaptive reasoning, improving τ-Bench success while lowering token cost.
Problem
Existing methods do not reliably distinguish useful hidden-state updates from representation drift as accumulated multi-turn context can corrupt task-relevant constraints.
Method
The paper models multi-turn reasoning as a turn-aligned hidden-state trajectory and characterizes it with temporal curvature and variance slope for online intervention.
Results
Across four tasks and three LLMs, the geometric signals separate correct from incorrect episodes before termination; geometry-conditioned triggering raises τ-Bench success from 24.1% to 39.6% and reduces token cost by 11.2%.
Takeaways & Limitations
Trajectory geometry can serve as an online control signal for timing adaptive reasoning, with deliberation gains depending on when reasoning is triggered rather than only on its total amount.
Takeaways & Limitations
The analysis requires turn-aligned hidden states because incorrect episodes are systematically longer, and trajectory length mechanically increases curvature and accumulated variance.
Abstract
from arXiv · showhide
LLM agents need to sustain goal-consistent reasoning across long multi-turn interactions under strict resource constraints. However, as the multi-turn context accumulates, it can destabilize the underlying LLM's internal representation of task-relevant information from earlier turns, blurring the boundary between constructive reasoning and representation drift. We formulate multi-turn reasoning as a hidden-state trajectory of the underlying LLM that is characterized via two complementary signals: temporal curvature that captures the directional consistency of turn-to-turn updates, and variance slope which measures the expansion or contraction of the exploration space. Across four tasks and three underlying LLMs, we observed that these geometric signals distinguish between correct and incorrect episodes prior to completion. We further decompose each episode into three-action chains formed from four actions (Read, Write, Respond, Transfer) and show that separability is action-dependent, with different signals distinguishing various chain patterns. Our experiments demonstrate that trajectory geometry can identify critical turns in the reasoning process, increasing task success rates on $τ$-Bench from 24.1% to 39.6% while reducing token cost by 11.2%.
1 Introduction
The paper frames multi-turn reasoning as a hidden-state trajectory whose geometric changes can reveal divergence from task goals and guide timely intervention. It addresses limitations of fixed-effort reasoning and heuristic context engineering by tracking when correct and incorrect episodes separate.
- Motivation: Multi-turn interactions incrementally refine task requirements, making sustained goal-consistent reasoning difficult under limited model size and context windows.Users add constraints across turns based on intermediate feedback, so the agent must preserve accumulated task information.
- Motivation: Existing context-engineering methods constrain inference heuristically but do not resolve the mismatch between agent-level task completion and turn-level response optimization.The paper also notes that current reasoning models use fixed reasoning effort rather than adapting effort to each turn.
- Motivation: Unnecessary reasoning can add semantic noise and irrelevant counterfactuals that corrupt the internal representation of accumulated constraints.This corruption makes it difficult to distinguish useful hidden-state updates from representation drift and to determine when intervention is needed.
- Approach: The paper formulates multi-turn reasoning as an observable hidden-state trajectory and tracks where correct and incorrect episodes diverge to trigger adaptive reasoning before failure.The proposed intervention targets turns where failure is imminent.
- Contributions: Two geometric signals characterize the trajectory: temporal curvature and variance slope, evaluated across four tasks and three underlying LLMs.The study also decomposes episodes into three-action chains and reports strongest discriminative structure in information-output chains.
- Contributions: 24.1% to 39.6%: geometry-conditioned adaptive reasoning raises τ-Bench task success while reducing token cost by 11.2%.The policy triggers reasoning at turns identified as critical by trajectory geometry.
2 Related Work
Related work uses internal LLM representations to monitor quality, detect hallucinations, and control deliberation, but these approaches leave open how hidden-state trajectories evolve across multi-turn interactions. This paper distinguishes itself by using cross-turn geometry for online control rather than isolated-response detection.
- Hidden-state monitoring: Prior work shows that hidden states can predict claim validity, encode truthfulness, and support factuality improvements through activation interventions.These results motivate monitoring internal representations as signals of output quality.
- Hidden-state monitoring: Hidden representations support hallucination detection through covariance spectra, generation-time probes, attention features, and efficient semantic-entropy approximations.Truthfulness information can also be concentrated in specific tokens and predict upcoming error types.
- Hidden-state monitoring: These internal-state signals do not transfer universally across tasks, consistent with the paper’s task-dependent variance-slope behavior.The related work therefore provides precedent for internal monitoring while highlighting limits on generalization across tasks.
- Closest comparison: The ICR Probe tracks cross-layer hidden-state dynamics for hallucination detection, whereas this paper analyzes trajectories across turns and uses geometry as an online control signal.ICR Probe analyzes individual responses in isolation and is therefore the closest comparison identified by the paper.
- Adaptive deliberation: Prior deliberation methods address overthinking and interaction costs by selecting thinking modes or triggering chain-of-thought, reflecting that extended reasoning is not uniformly beneficial.Excessive deliberation can displace environment interaction and lower task success in agentic settings.
3 Hidden State Trajectory Formulation
The paper models multi-turn agent reasoning as a hidden-state trajectory shaped by user inputs, outputs, and environment interactions. It characterizes trajectory changes through directional consistency and dispersion growth or contraction across turns.
- 3.1 Multi-Turn Reasoning in Agent Tasks: Each interaction contains a system prompt, alternating user inputs, model outputs, and potentially environment-mediated tool interactions.Tool invocations are appended to the context and condition subsequent turns.
- 3.2 Hidden States in Multi-Turn Reasoning: At each turn, the model records a final-user-token hidden state from a fixed intermediate probe layer, forming the trajectory H.The selected probe layers are layer 22 for Qwen3-14B, layer 42 for Qwen3-32B, and layer 16 for Llama-3.1-8B.
- 3.2 Hidden States in Multi-Turn Reasoning: The hidden-state trajectory is modeled residually, with each displacement representing the representation change induced by the current turn and accumulated context.The residual relation is h_t = h_{t−1} + v_t.
- 3.3 Temporal Curvature: Temporal curvature measures the cosine similarity between consecutive displacement vectors to quantify directional consistency across turns.Values near 1 indicate consistent accumulation, near 0 indicate orthogonal updates, and near −1 indicate abrupt reversal.
- 3.4 Variance Slope: For a prefix containing λ + 1 hidden states, dispersion V(λ) is the trace of the sample covariance across those states.The trace sums per-coordinate variances, using normalization by λ for the unbiased estimator.
- 3.4 Variance Slope: Variance slope β_t is the ordinary least-squares slope of the dispersion sequence against prefix length from λ = 1 through t.It summarizes how the exploration space changes as turns accumulate.
- 3.4 Variance Slope: Positive β_t indicates progressively spreading representations, whereas negative β_t indicates contracting dispersion across accumulated prefixes.The slope uses the means of prefix lengths and dispersion values in its regression.
4 Empirical Analysis
Across four tasks and three LLMs, trajectory geometry separates correct from incorrect episodes: failures show more directional reversal and contraction, while correct episodes preserve broader exploration. Hidden states are turn-aligned to control trajectory-length confounding, with metrics computed from the aligned sequence.
- Experimental Setup: Four tasks and three LLMs are evaluated using ground-truth correctness, average reward, token cost, and significance tests.The tasks span Math, Code, Airline, and Retail; inference follows official decoding configurations on vLLM.
- Temporal Curvature: κ is compared across six task–model pairings using per-episode mean curvature.Figure 2 compares task–LLM pairings on the horizontal axis and mean κ on the vertical axis.
- Temporal Curvature: Correct−Incorrect mean-κ differences are +0.047 for Math, +0.148 for Code, +0.030 for Retail (14B), +0.049 for Airline (14B), +0.035 for Retail (32B), and +0.088 for Airline (32B).Failure groups have consistently lower means, indicating more frequent directional reversals.
- Turn Alignment: Turn alignment uses one hidden state per turn, reducing trajectory-length confounding because incorrect episodes are systematically longer.The state is taken from the final token of the user input before the agent acts, and both metrics use identical turn indices.
5 Probing Trajectory Divergence through Action Chains
The paper constructs three-action chains from Read, Write, Respond, and Transfer actions to examine how hidden-state trajectory geometry separates correct and incorrect episodes. Separability varies by task and chain pattern, with information-output and action-switching chains showing the strongest discrimination.
- Action Chain Construction: Three-action chains are constructed using sliding windows over consecutive assistant actions, with each chain containing Read, Write, Respond, or Transfer behaviors.Read-only calls query the environment, Write calls alter database state, Respond replies to the user, and Transfer escalates to a human.
- Separability of Hidden-State Trajectories: Trajectory separability varies with the distribution of agent actions across tasks.The two geometric probes provide different levels of discrimination depending on task category and action distribution.
- Separability of Hidden-State Trajectories: Temporal curvature and variance slope are complementary probes whose relative discriminative power changes across Retail and Airline.Variance slope is the primary signal in Retail, while Airline shows stronger variance-slope and temporal-curvature discrimination.
- Action Chain Patterns: Information-output and action-switching chains show significant discrimination in both Retail and Airline.Highlighted Retail chains are IDs 2, 6, 10, and 17; highlighted Airline chains are IDs 2, 5, 10, 12, 19, and 20.
- Action Chain Patterns: Respond→Respond→Respond (ID 2) has the highest error concentration and the most pronounced geometric discriminative structure among information-output chains.In PCA space, correct-episode scatter gradually shifts from the Write/Transfer region toward the edge of the Respond cluster across turns.
6 Geometry-Triggered Adaptive Reasoning
The section evaluates adaptive reasoning triggers that use fixed policies, predicted actions, learned classifiers, or hidden-state geometry to decide when to enter thinking mode. Geometry-conditioned triggers achieve strong reward-token trade-offs, while deliberation timing and action choice vary by task.
- Trigger policies: Geometry-conditioned policies trigger thinking when temporal curvature κt falls below τκ or variance slope βt exceeds τβ.
- Timing of deliberation: Deliberation timing matters: Warm-up reaches 0.361 versus Always-thinking’s 0.307 on Retail with Qwen3-14B at lower token cost.On Qwen3-32B, Warm-up reaches 0.405 versus Always-thinking’s 0.384 while using 12.2k fewer tokens, but Airline performance reverses this pattern.
- Action-conditioned policies: Action-conditioned triggers are task-dependent: Read reaches 0.393 on Retail, whereas Respond reaches 0.372 on Airline with Qwen3-14B.The best action differs across domains, so task-specific error distributions must be identified in advance.
- Geometry-conditioned policies: κ < −0.20 ties Learned-HST on Retail 14B at 0.397 versus 0.398 while lowering mean token cost from 104.4k to 101.0k.On Retail 32B, κ < −0.15 reaches 0.442, above Learned-HST’s 0.405 and 27% above Never-thinking.
- Geometry-conditioned policies: Thresholds are unimodal, transferable, and require no additional training, with recalibration guided by correct−incorrect curvature gaps.
- Reward and efficiency: Across four settings, geometry-conditioned triggers raise task success from 24.1% to 39.6% while reducing mean token cost by 11.2%.They occupy the favorable reward-token corner, attaining the highest task reward among the lowest token costs.
7 Conclusion
The paper represents multi-turn reasoning as a hidden-state trajectory characterized by temporal curvature and variance slope. These signals separate correct from incorrect episodes before termination and support adaptive reasoning at critical turns, improving τ-Bench success while reducing token cost.
- Across four tasks and three LLMs, temporal curvature and variance slope separate correct from incorrect episodes before termination.Failure trajectories exhibit directional reversals and premature convergence in hidden states.
- Three-action-chain decomposition shows that separability is action-dependent and concentrated in information-output chains.
- Trajectory geometry acts as an online control signal, raising τ-Bench task success from 24.1% to 39.6% while reducing token cost by 11.2%.