LLM Reasoning as Trajectories: Step-Specific Representation Geometry and Correctness Signals

Lihao Sun; Hang Dong; Bo Qiao; Qingwei Lin; Dongmei Zhang; Saravan Rajmohan

cs.CLApr 7, 2026

LLM Reasoning as Trajectories: Step-Specific Representation Geometry and Correctness Signals

Authors: Lihao Sun, Hang Dong, Bo Qiao, Qingwei Lin, Dongmei Zhang, Saravan Rajmohan

Organizations: Microsoft

Abstract

This work characterizes large language models' chain-of-thought generation as a structured trajectory through representation space. We show that mathematical reasoning traverses functionally ordered, step-specific subspaces that become increasingly separable with layer depth. This structure already exists in base models, while reasoning training primarily accelerates convergence toward termination-related subspaces rather than introducing new representational organization. While early reasoning steps follow similar trajectories, correct and incorrect solutions diverge systematically at late stages. This late-stage divergence enables mid-reasoning prediction of final-answer correctness with ROC-AUC up to 0.87. Furthermore, we introduce trajectory-based steering, an inference-time intervention framework that enables reasoning correction and length control based on derived ideal trajectories. Together, these results establish reasoning trajectories as a geometric lens for interpreting, predicting, and controlling LLM reasoning behavior.

Figures & tables

Figure 1: Step-specific representation structure across layers and reasoning steps. (a) t-SNE visualization of activations preceding Step markers (Instruct model, GSM8K test split). Step-specific regions become more separated with layer depth. (b) Layer-wise linear probe accuracy for step identity prediction. $x$ -axis denotes the layer from which activations are extracted, and the $y$ -axis reports test accuracy of a linear classifier trained on these activations as input, with the step number as the label. Early steps are separable from shallow layers; later steps require deeper layers.

Probe	Eval.	Step				Final Ans.
From	On	2	3	4	5	Marker
Instruct	R1-Dist.	0.99 L18	0.93 L19	0.87 L12	0.91 L18	0.87 L23
	Base	1.00 L12	0.97 L08	0.95 L08	0.93 L18	0.97 L21
R1-Dist.	Instruct	1.00 L03	0.92 L08	0.88 L07	0.93 L27	0.98 L19
	Base	1.00 L06	0.94 L04	0.89 L30	0.91 L08	0.96 L19
Base	Instruct	1.00 L21	0.98 L18	0.97 L18	0.97 L23	1.00 L31

Table 1: Cross-model transfer of step-specific linear probes. Each entry is the best accuracy across layers; subscripts indicate the peak layer. Step 1 omitted (1.00 for all pairs).

Figure 2: Trajectory divergence and mid-reasoning correctness signals (Instruct model, GSM8K). (a) Late between-step activation distances diverge between correct and incorrect trajectories, while early transitions remain similar; † marks transitions where 95% CIs of correct and incorrect distances do not overlap. (b) Late-step trajectory features predict correctness with average AUC $\approx 0.83$ (peak $0.87$ at layer 29) vs. $\approx 0.63$ for early-step features. Curves are from a single random seed; see Table 3 for seed-averaged values.

Format Category	Count	%	Segments
Step X:	4,820	64.5	4.6 steps
\n\n -sep. paragraphs	838	11.2	5.6 paragraphs
\n -separated lines	687	9.2	7.6 lines
Single block	662	8.9	7.7 sentences
Numbered list (1./1))	467	6.2	4.4 items

Table 2: Response format distribution under minimal freeform prompting on GSM8K (Instruct model). Without explicit formatting instructions, the model spontaneously adopts Step X: formatting in

64.5\%

of responses. Segment counts report per-response means.

Method	Best-layer AUC
Step-count only	$0.649\pm 0.021$
LogitLens (best config)	$0.765\pm 0.027$
Trajectory features (ours)	$0.852\pm 0.039$

Table 3: Correctness predictor comparison. Trajectory features substantially outperform both a step-count-only baseline and logit-level features (entropy, answer-marker token rank, top-1 probability) at step boundaries. Values report best-layer AUC averaged over three seeds.

Intervention	Always	Gated	Gain vs Always
Step	$-1.59$	$+0.91$	$+2.50$
Check	$-11.70$	$+0.23$	$+11.90$
Wait	$-30.50$	$-0.68$	$+29.80$
Hmm	$-36.00$	$-0.61$	$+35.41$
Prolong (Last)	$+0.45$	$+0.76$	$+0.31$
Prolong (Mid)	$+0.38$	$+0.76$	$+0.38$

Table 4: Unconditional vs. error-targeted interventions on GSM8K (Instruct model). Values report absolute accuracy change (

\%

) vs. baseline. Unconditional interventions often degrade performance, while predictor-gated interventions (

|\alpha|=0.05

) on only

12\%

of examples yield additional gains.

Figure 3: Trajectory-based control of reasoning behavior. ( a ) Correctness steering on GSM8K, stratified by original step count; values in $\%$ . ( b ) Reasoning length control via the termination subspace as a function of steering strength $|\alpha|$ .

Eval. Dataset	Step					Ans.
	1	2	3	4	5	Marker
MATH-500	1.00	1.00	0.98	0.91	0.85	0.99
MMLU	1.00	1.00	0.98	0.92	0.89	0.98
Freeform GSM8K	0.93	0.84	0.83	0.84	0.88	0.92

Table 5: Cross-dataset and cross-format transfer of step probes from fixed-form GSM8K. Best-layer probe accuracy on held-out datasets. Structural geometry transfers robustly across tasks and formats.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Complete t-SNE visualizations of step-aligned hidden states extracted immediately before Step markers across three training regimes on the GSM8K test split. Each point corresponds to a residual-stream activation at a reasoning-step boundary. While all models exhibit step-structured organization, R1-Distilled and Instruct models show earlier and more pronounced separation between steps, whereas the Base model exhibits weaker but still discernible structure.

Figure 5: Layer-wise linear probe accuracy for predicting reasoning step identity. In each sub-figure, the $x$ -axis denotes the layer from which activations are extracted, and the $y$ -axis reports test accuracy of a linear classifier trained on these activations as input, with the reasoning step number (or the final answer marker) as the label. Early steps are linearly separable from shallow layers with near-ceiling accuracy, whereas later steps require deeper layers to become separable. Differences across training regimes indicate that instruction tuning and reasoning distillation shift the linear accessibility of step information.

Probe	Eval.	Step					Final Ans.
From	On	1	2	3	4	5	Marker
Instruct	Base	0.9976	0.9689	0.9182	0.8099	0.8785	0.8800
	R1-Dist.	0.9998	0.9753	0.8579	0.7705	0.8511	0.7826
R1-Dist.	Base	0.9977	0.9381	0.8651	0.8344	0.8516	0.9102
	Instruct	1.0000	0.9057	0.7988	0.8065	0.8730	0.8819
Base	Instruct	1.0000	0.9900	0.9063	0.8714	0.8794	0.9197

Appendix

Table 6: Average cross-model transfer accuracy of step-specific linear probes. Each entry reports the average linear-probe accuracy across all layers when a classifier trained on step-specific activations from one model is evaluated on another model. Results are averaged across transfer directions for each probe–evaluation pair. Unlike Table 1 , Step 1 is included, showing near-perfect transferability across models.

Category	Step 1	Step 2	Step 3	Step 4	Step 5	Answer
Step X:	0.96 (0.98 @ L1)	0.83 (0.85 @ L19)	0.81 (0.84 @ L28)	0.72 (0.83 @ L10)	0.71 (0.88 @ L30)	0.86 (0.93 @ L20)
Non- Step X:	0.85 (0.86 @ L1)	0.82 (0.85 @ L18)	0.79 (0.84 @ L7)	0.77 (0.85 @ L10)	0.75 (0.88 @ L25)	0.86 (0.92 @ L10)
Numbered list	0.82 (0.91 @ L1)	0.83 (0.89 @ L18)	0.81 (0.83 @ L30)	0.75 (0.85 @ L10)	0.73 (0.89 @ L24)	0.86 (0.95 @ L10)
\n\n paragraphs	0.82 (0.82 @ L1)	0.75 (0.81 @ L14)	0.79 (0.83 @ L30)	0.82 (0.86 @ L10)	0.81 (0.91 @ L22)	0.81 (0.96 @ L23)
\n lines	0.87 (0.87 @ L0)	0.82 (0.87 @ L18)	0.77 (0.86 @ L0)	0.75 (0.86 @ L10)	0.79 (0.90 @ L30)	0.89 (0.94 @ L10)
Single block	0.87 (0.88 @ L1)	0.87 (0.87 @ L18)	0.81 (0.87 @ L30)	0.74 (0.85 @ L30)	0.67 (0.85 @ L31)	0.87 (0.94 @ L6)

Appendix

Table 7: Per-category freeform probe transfer accuracy. Each cell reports average accuracy across 32 layers, with the best single-layer accuracy and corresponding layer in parentheses. Probes trained on fixed-form Step X: activations transfer to all freeform format categories, including those with no Step markers.

Component	Result
Linear probes (best-layer acc.)	$\leq\pm 0.001$ across all steps
Correctness pred. (Final State)	$0.837\pm 0.016$ (L29)
Correctness pred. (Final State PCA)	$0.859\pm 0.023$ (L19)
Correctness pred. (Late Steps PCA)	$0.853\pm 0.038$ (L21)

Appendix

Table 8: Seed-variance analysis for key components. Linear probe accuracies are highly stable; correctness predictor AUCs show modest variance consistent with the smaller effective sample size of incorrect examples.

Operation	Per-call	Total
PCA projection + divergence check	${\sim}12.9\,\mu$ s	${\sim}81\,\mu$ s
Low-rank update ( $r{=}32$ )	${\sim}10.8\,\mu$ s	${\sim}24\,\mu$ s
Trajectory steering (combined)	—	${\sim}105\,\mu$ s
Prolong (per-token $\times$ all timesteps)	${\sim}1.12\,\mu$ s	${\sim}3{,}371\,\mu$ s

Appendix

Table 9: Steering computation cost per problem on a single NVIDIA H200 GPU with Llama-3.1-8B-Instruct. Trajectory steering at step boundaries is approximately

32\times

cheaper than per-token vector addition. Totals assume

{\sim}7

step boundaries and

{\sim}3{,}000

generated tokens, respectively.

	Instruct	Base	R1-Distill
Mean $K$	4.5	6.6	6.9
Median $K$	4	6	5
$K\in[3,5]$	70.5%	33.4%	52.1%

Appendix

Table 10: Reasoning step-count distributions on GSM8K across training regimes. Instruct produces substantially shorter chains than Base and R1-Distill, yet step-specific probes transfer across models with high accuracy, indicating that the geometry is tied to reasoning progress rather than absolute step position.

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Search for Truth from Reasoning: A Dynamic Representation Editing Framework for Steering LLM Trajectories

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National Engineering Research Center of Software Engineering, Peking University · School of Computer Science, Peking University · Key Laboratory of High Confidence Software Technologies, Ministry of Education+1

Language Model Steering Representation Geometry in Language Models LLM Reasoning

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Current approaches to enhance Large Language Model (LLM) reasoning, such as Chain-of-Thought and "Wait" prompts, primarily encourage models to think more, yet often fail to guide them toward Truth. While Representation Editing (RepE) offers a intrinsic control, its application to dynamic reasoning trajectories remains underexplored. In this work, we bridge this gap by investigating the geometry of truth within unfolding reasoning chains. We uncover three critical insights: (1) Truth is encoded at the sentence level and is entangled with latent reasoning patterns; (2) Effective intervention follows an Uncertainty Principle and a Decay Effect, requiring localization to early, high-entropy forks; (3) Naive steering vectors suffer from noise, risking collateral damage to correct trajectories. Based on these findings, we propose DynaSteer, a dynamic RepE framework. DynaSteer employs pattern clustering to disentangle reasoning manifolds and utilizes Fisher-LDA to project purified truth. By dynamically monitoring lookahead entropy, it selectively steers and rolls back trajectories only when necessary. Comprehensive experimental results on several MATH benchmark verify the effectiveness of DynaSteer, and experiments on out-of-domain coding tasks further confirm its generalization ability. Our code is publicly available at https://github.com/tianlwang/DynaSteer.

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