Hesitation Has a Geometry: Entropy-Trained Hyperbolic Probes for Sparse Activation Steering

Zeyong Zhang; Tung Sum Thomas Kwok; Tengfei Ma; Mengjia Xu

cs.LGOct 1, 2026

Hesitation Has a Geometry: Entropy-Trained Hyperbolic Probes for Sparse Activation Steering

Authors: Zeyong Zhang, Tung Sum Thomas Kwok, Tengfei Ma, Mengjia Xu

Organizations: Department of Data Science New Jersey Institute of Technology Newark, NJ, USA · Department of Statistics and Data Science University of California, Los Angeles Los Angeles, CA, USA · Department of Biomedical Informatics Stony Brook University Stony Brook, NY, USA

Abstract

When a large language model solves a mathematical problem, its reasoning is largely hierarchical, and the solution often branches at a few tokens where the next-token entropy is high. Such tree-like structure embeds in hyperbolic space with far lower distortion than in Euclidean space. Activation steering, however, usually edits the hidden states of a pretrained model by adding one fixed Euclidean vector at every token, even though most tokens of a solution are already determined by the context. We propose Hyperbolic Entropy Steering (HEST), which embeds the hidden states in the Poincaré ball with a lightweight probe whose only label is the model's own next-token entropy. Where this entropy exceeds a threshold, HEST moves the embedded state along the geodesic of steepest descent of a readout of the probe and maps the change back to the hidden state. For the Busemann readout of a learned ideal point, we prove that a step of fixed length lowers it by the same amount at every state. On three instruction-tuned models from the Qwen2.5-Math and Llama-3.1 families, HEST with the Busemann readout improves greedy accuracy on MATH-500 and GSM8K in five of six settings, by up to 1.8 points, whereas a contrastive steering vector added at every token lowers accuracy. With a Euclidean probe trained in the same way, this gain disappears on Qwen2.5-Math-1.5B-Instruct. The gains are largest on problems where the model hesitates often, and accuracy on the remaining problems is almost unchanged.

Figures & tables

Figure 1: HEST steers only the tokens at which the model hesitates, along a geodesic in the space of a hyperbolic probe. (a) Fixed-vector steering adds the same vector

\alpha v

at every token. (b, c) HEST maps hidden states into the Poincaré ball with a probe trained on next-token entropy. The readout

s

is either the Busemann function of the ideal point

\xi

or the distance

r

to the origin

O

, and the arrows show the steering direction

u

. (d) HEST steers only tokens whose entropy

H_{i}

exceeds the threshold

\theta

. (e) At such a token,

h

is mapped to

z

, moved by hyperbolic distance

t

along the geodesic in the direction

u

, and mapped back. In (e),

\lVert\cdot\rVert

is the Euclidean norm. The point clouds and step lengths in (b, c) are schematic, and Appendix O shows real embedded states.

	HEST , Busemann ( $t^{\star}=0.5$ )			HEST , radius ( $t^{\star}=0.25$ )
$t$	MATH-500	GSM8K	OlympiadBench	MATH-500	GSM8K	OlympiadBench
0.1	74.80 (+0.60)	84.53 ( $-$ 0.30)	41.39 (0.00)	74.20 (0.00)	84.53 ( $-$ 0.30)	41.04 ( $-$ 0.35)
0.25	75.00 (+0.80)	84.84 (0.00)	41.39 (0.00)	75.20 (+1.00)	84.53 ( $-$ 0.30)	40.87 ( $-$ 0.52)
0.5	76.00 (+1.80)	85.37 (+0.53)	40.00 ( $-$ 1.39)	74.40 (+0.20)	84.38 ( $-$ 0.45)	42.26 (+0.87)
1	75.40 (+1.20)	85.29 (+0.45)	41.22 ( $-$ 0.17)	73.40 ( $-$ 0.80)	84.15 ( $-$ 0.68)	42.61 (+1.22)

Table 1: Accuracy (%) of Qwen2.5-Math-1.5B-Instruct and change from greedy decoding (points) at four step lengths

t

. Greedy decoding:

74.20

on MATH-500,

84.84

on GSM8K and

41.39

on OlympiadBench. All other models use the step lengths

t^{\star}

. Bold: best accuracy per benchmark.

Model	Dataset	$\theta$	Greedy	Busemann	radius	CAA	Steered (%)
				HEST
Qwen2.5-Math-1.5B	MATH-500	1.0	74.20	76.00 (+1.80)	75.20 (+1.00)	70.00 ( $-$ 4.20)	1.1
Qwen2.5-Math-1.5B	GSM8K	1.0	84.84	85.37 (+0.53)	84.53 ( $-$ 0.30)	84.23 ( $-$ 0.61)	1.6
Llama-3.1-8B	MATH-500	2.0	50.80	52.00 (+1.20)	51.00 (+0.20)	38.20 ( $-$ 12.60)	0.8
Llama-3.1-8B	GSM8K	1.4	84.61	84.31 ( $-$ 0.30)	84.53 ( $-$ 0.08)	80.06 ( $-$ 4.55)	3.0
Qwen2.5-Math-7B	MATH-500	1.0	84.00	84.80 (+0.80)	84.20 (+0.20)	82.60 ( $-$ 1.40)	1.0

Table 2: Accuracy (%) and change from greedy decoding (points).

\theta

: gate threshold. Steered: median percentage of tokens steered per solution (Busemann). Bold: best accuracy per row.

Figure 2: Accuracy changes little on most problems, and HEST gains most where the greedy solution hesitates often. Problems are grouped by the number of hesitation points in the greedy solution. Left: greedy accuracy. Right: change in accuracy with HEST , Busemann readout,

t=0.5

Panel of Figure 3 — Figure 3: One steered token changes the course of the solution. Top: next-token entropy along the greedy solution, in which $3$ of $189$ tokens exceed $\theta=1$ . Bottom: the two continuations from the first of them. HEST steers $12$ of the $583$ tokens of its own solution. Appendix P gives both solutions.

(a) Positions	Steered	Step	Accuracy
Hesitation points ( HEST )	$1.1\%$	$0.06$	$\mathbf{76.00}$ ( $+1.80$ )
Every token	$100\%$	$0.07$	$74.00$ ( $-0.20$ )
Step boundaries	$1.4\%$	$0.07$	$73.80$ ( $-0.40$ )
CAA, every token	$100\%$	$0.33$	$70.00$ ( $-4.20$ )

Table 3: Ablations on Qwen2.5-Math-1.5B-Instruct with

\theta=1

and layer

19

, in accuracy (%) and change from greedy decoding (points). (a) Where to steer, on MATH-500 with the Busemann readout,

t=0.5

. Steered: median fraction of steered tokens. Step: median

\lVert\Delta h\rVert/\lVert h\rVert

. CAA has no fallback. (b) Which direction, with the gate and step size of HEST . (c) Which geometry, where the Euclidean probe has

c\to 0

and the same median edit size. Bold marks the best of each comparison.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Direction	Positions	Training signal	Geometry	Evaluation	Math
CAA ( Rimsky et al., 2024 )	fixed, difference of means	all tokens after the prompt	contrastive pairs	Euclidean	behavioral questions, open-ended generation, MMLU, TruthfulQA	–
ITI ( Li et al., 2023 )	fixed, top attention heads	every generated token	labeled truthful and untruthful answers	Euclidean	TruthfulQA, NQ, TriviaQA, MMLU	–
CLAS ( Hsu et al., 2026 )	per-layer direction, coefficient linear in the activation	all tokens	probe direction, prompt and completion pairs	Euclidean	$11$ tasks including GSM8K	GSM8K, to elicit step-by-step solutions
ReFT ( Wu et al., 2024 )	learned low-rank edit of a subspace	fixed prefix and suffix positions of the prompt	task supervision, frozen model	Euclidean	commonsense and arithmetic reasoning, instruction following, GLUE	GSM8K and three others, after task finetuning
HyperSteer ( Sun et al., 2025 )	generated by a hypernetwork from the steering prompt and the input	all tokens	AxBench steering prompts, language-modeling loss	Euclidean	AxBench concepts	–
AcT ( Rodriguez et al., 2025 )	coordinate-wise affine maps from optimal transport	all tokens	source and target sentence sets	Euclidean	toxicity, concept induction, TruthfulQA, text-to-image models	–

Appendix

Table 4: Steering methods compared along the design choices of HEST . Positions : the tokens at which the edit is applied. Math : whether accuracy on mathematical benchmarks is reported, and in which setting. Each entry was checked against the original paper.

			Busemann			radius
Model	Probe data	States	Pair acc.	$\rho$	$c$	Pair acc.	$\rho$	$c$
Qwen2.5-Math-1.5B	MATH	50,717	0.880	0.554	0.74	0.842	0.472	0.53
Qwen2.5-Math-1.5B	GSM8K	27,212	0.855	0.501	0.71	0.829	0.429	0.40
Qwen2.5-Math-1.5B	OlympiadBench	87,813	0.900	0.547	0.78	0.876	0.478	0.38
Llama-3.1-8B	MATH	41,873	0.903	0.747	0.85	0.894	0.711	0.72
Llama-3.1-8B	GSM8K	15,821	0.871	0.706	0.91	0.868	0.698	0.73

Appendix

Table 5: Held-out quality of the probes used in the main results. States come from the greedy solutions of

100

training problems, and

20

of these problems are held out for checkpoint selection. Pair accuracy is the fraction of held-out pairs with an entropy gap above

0.3

nats that the readout orders correctly, and it also selects the checkpoint.

\rho

is the Spearman correlation between the readout and the entropy on held-out states, and

c

is the learned curvature magnitude.

$\theta$	Tokens (%)	Per solution	Profile (%)	$L_{1}$
Llama-3.1-8B, MATH-500
1.0	6.45	48.3	7/24/69	130
1.4	3.31	24.8	33/29/39	79
1.5	2.79	20.9	40/26/34	64
1.6	2.34	17.5	47/25/28	50
1.7	1.95	14.6	53/25/22	38

Appendix

Table 6: Gate-threshold calibration. For each candidate

\theta

, the columns give the percentage of greedy tokens with

H>\theta

, the mean number of such tokens per solution, and the percentage of solutions with

0

–

10

11

–

24

and at least

25

of them.

L_{1}

is the distance to the reference profile of Qwen2.5-Math-1.5B-Instruct at

\theta=1

72/18/10

on MATH-500 and

76/19/4

on GSM8K, which counts its steered tokens per solution (Appendix G ). The selected threshold (bold) minimizes

L_{1}

. Entropies come from one teacher-forced pass over each model’s greedy solutions and use no labels. For Qwen2.5-Math-7B on GSM8K, the

9

greedy solutions that reached the length limit are excluded.

Readout	$t$	$\Delta$	F/B	Cap	$\Delta$	F/B	Cap	$\Delta$	F/B	Cap
		MATH-500			GSM8K			OlympiadBench
Busemann	0.1	+0.60	7/4	6	$-$ 0.30	11/15	15	0.00	11/11	12
Busemann	0.25	+0.80	9/5	6	0.00	19/19	12	0.00	17/17	12
Busemann	0.5	+1.80	16/7	6	+0.53	23/16	14	$-$ 1.39	15/23	13
Busemann	1	+1.20	17/11	6	+0.45	31/25	8	$-$ 0.17	29/30	18
radius	0.1	0.00	6/6	2	$-$ 0.30	9/13	16	$-$ 0.35	8/10	13

Appendix

Table 7: Qwen2.5-Math-1.5B-Instruct, full results.

\Delta

: change from greedy decoding in points. F/B: problems fixed (greedy wrong, HEST right) and broken (greedy right, HEST wrong). Cap: problems whose steered decode reached the 3072-token limit and fell back to the greedy answer. The largest change in each column is in bold.

Model	Dataset	$\ell$	$\theta$	Greedy	Accuracy	F/B	Cap	Accuracy	F/B	Cap
					HEST , Busemann $t=0.5$			HEST , radius $t=0.25$
Qwen2.5-Math-1.5B	MATH-500	19	1.0	74.20	76.00 (+1.80)	16/7	6	75.20 (+1.00)	12/7	7
Qwen2.5-Math-1.5B	GSM8K	19	1.0	84.84	85.37 (+0.53)	23/16	14	84.53 ( $-$ 0.30)	12/16	14
Llama-3.1-8B	MATH-500	13	2.0	50.80	52.00 (+1.20)	17/11	78	51.00 (+0.20)	12/11	64
Llama-3.1-8B	GSM8K	13	1.4	84.61	84.31 ( $-$ 0.30)	18/22	2	84.53 ( $-$ 0.08)	19/20	5
Qwen2.5-Math-7B	MATH-500	19	1.0	84.00	84.80 (+0.80)	8/4	13	84.20 (+0.20)	5/4	14

Appendix

Table 8: All models, full results for the two HEST configurations of Table 2 . Columns as in Table 7 ,

\ell

is the steered layer, and the best accuracy in each row is in bold.

Model	Dataset	Greedy	CAA	F/B	Cap	Length	Changed (%)	With fallback
Qwen2.5-Math-1.5B	MATH-500	74.20	70.00 ( $-$ 4.20)	26/47	36	1.26	100	72.40 ( $-$ 1.80)
Qwen2.5-Math-1.5B	GSM8K	84.84	84.23 ( $-$ 0.61)	58/66	8	0.91	99	84.38 ( $-$ 0.45)
Llama-3.1-8B	MATH-500	50.80	38.20 ( $-$ 12.60)	31/94	131	1.33	100	44.20 ( $-$ 6.60)
Llama-3.1-8B	GSM8K	84.61	80.06 ( $-$ 4.55)	56/116	7	0.87	100	80.21 ( $-$ 4.40)
Qwen2.5-Math-7B	MATH-500	84.00	82.60 ( $-$ 1.40)	19/26	10	0.99	97	83.20 ( $-$ 0.80)
Qwen2.5-Math-7B	GSM8K	95.38	95.00 ( $-$ 0.38)	15/20	6	0.95	98	95.00 ( $-$ 0.38)

Appendix

Table 9: CAA baseline in detail. CAA adds the raw difference-of-means vector (multiplier

1

) at every generated token and is reported without the fallback. Cap: decodes that reached the 3072-token limit. Length: mean solution length relative to greedy decoding. Changed: percentage of solutions whose text differs from greedy decoding. The last column applies the same fallback as HEST and is given only as a diagnostic.

Model	Data	Setting	$\Delta$	%	$g$	$f$	$b$	$\Delta$	%	$g$	$f$	$b$	$\Delta$	%	$g$	$f$	$b$
			0–10					11–24					$\geq 25$
Qwen2.5-Math-1.5B	MATH-500	Busemann 0.5	+0.55	73	0.89	0.10	0.01	+5.32	19	0.41	0.15	0.08	+4.65	9	0.19	0.11	0.25
Qwen2.5-Math-1.5B	MATH-500	Busemann 0.25	$-$ 0.55	73	0.89	0.00	0.01	+2.13	19	0.41	0.07	0.05	+9.30	9	0.19	0.14	0.13
Qwen2.5-Math-1.5B	MATH-500	radius 0.25	$-$ 1.10	73	0.89	0.00	0.01	+3.19	19	0.41	0.09	0.05	+13.95	9	0.19	0.20	0.13
Qwen2.5-Math-1.5B	GSM8K	Busemann 0.5	+0.10	76	0.91	0.05	0.00	+1.16	20	0.72	0.16	0.05	+4.69	5	0.44	0.17	0.11
Qwen2.5-Math-1.5B	GSM8K	Busemann 0.25	$-$ 0.40	76	0.91	0.04	0.01	+0.78	20	0.72	0.12	0.04	+3.13	5	0.44	0.17	0.14

Appendix

Table 10: Change from greedy decoding by the number of hesitation points in the greedy solution. For each group:

\Delta

in points, the percentage of problems in the group (%), the greedy accuracy

g

, the fixed rate

f

among greedy-wrong problems and the broken rate

b

among greedy-right problems, so that

\Delta=100\,[(1-g)f-gb]

for the unrounded rates.

Model	Data	Setting	$\Delta$	$n$	$g$	$f$	$b$	$\Delta$	$n$	$g$	$f$	$b$	$\Delta$	$n$	$g$	$f$	$b$
			$<2\%$					$2$ – $10\%$					$>10\%$
Qwen2.5-Math-1.5B	MATH-500	Busemann 0.5	+1.58	380	0.84	0.15	0.01	+2.50	120	0.44	0.10	0.08	–
Qwen2.5-Math-1.5B	MATH-500	Busemann 0.25	0.00	380	0.84	0.03	0.01	+3.33	120	0.44	0.10	0.06	–
Qwen2.5-Math-1.5B	MATH-500	radius 0.25	$-$ 0.26	380	0.84	0.06	0.02	+5.00	120	0.44	0.12	0.04	–
Qwen2.5-Math-1.5B	GSM8K	Busemann 0.5	+0.26	764	0.92	0.05	0.00	+1.31	535	0.76	0.15	0.03	$-$ 10.00	20	0.55	0.11	0.27
Qwen2.5-Math-1.5B	GSM8K	Busemann 0.25	$-$ 0.13	764	0.92	0.03	0.00	+0.19	535	0.76	0.12	0.03	0.00	20	0.55	0.22	0.18

Appendix

Table 11: Change from greedy decoding by hesitation density, the fraction of tokens in the greedy solution with

H>\theta

. Columns as in Table 10 , except that the second column of each group is the number of problems. These groups were defined after the results of Table 10 were known.

Model	Dataset	Median	Mean	90th pct.	All	Excl. capped	solution	token (%)	$n$	Steered (%)
		Per problem (%)			Pooled (%)		Steered per	No steered	Capped
Qwen2.5-Math-1.5B	MATH-500	1.07	1.58	3.57	1.55	1.61	8.9	3.4	6	3
Qwen2.5-Math-1.5B	GSM8K	1.64	2.37	5.31	2.34	2.45	7.8	5.7	14	6
Qwen2.5-Math-1.5B	OlympiadBench	2.05	2.87	6.27	2.46	2.60	20.4	0.9	13	3
Llama-3.1-8B	MATH-500	0.80	1.32	3.20	1.01	1.42	8.3	17.4	78	41
Llama-3.1-8B	GSM8K	3.00	3.80	7.49	4.18	4.22	7.7	2.9	2	1

Appendix

Table 12: How many tokens HEST steers (Busemann,

t=0.5

). Per problem, the steered fraction is the number of steered tokens divided by the length of the steered solution. Pooled is the total number of steered tokens over the total number of generated tokens. Steered per solution: mean number of steered tokens. No steered token: percentage of solutions in which no token is steered. Capped: steered decodes that reached the 3072-token limit, and the percentage of all steered tokens that fall in them. A few capped decodes repeat themselves and pass the threshold many times, which inflates the pooled fraction for Qwen2.5-Math-7B. The median is not affected.

Readout	Pairwise cos.	PC 1	PCs 1–5	PCs for 50%	Size	Cos. to mean
Busemann, $t=0.5$	$0.210$ $(0.113,\,0.334)$	$4.7\%$	$17.6\%$	$32$	$0.060$	$0.474$
radius, $t=0.25$	$0.251$ $(0.153,\,0.377)$	$5.6\%$	$18.8\%$	$27$	$0.017$	$0.510$

Appendix

Table 13: Diversity of the HEST edits on

400

hesitation points of Qwen2.5-Math-1.5B-Instruct. The columns give the median pairwise cosine between edits, with the 10th and 90th percentiles in parentheses, the variance explained by the first principal component and by the first five, the number of components needed for half of the variance, the median relative size

\lVert\Delta h\rVert/\lVert h\rVert

, and the median cosine between each edit and the mean edit.

	Variant	Accuracy	F/B	Cap	No fallback	Steered	Edits	Step	Length
Positions MATH-500	Hesitation points ( HEST )	76.00 (+1.80)	16/7	6	76.00	1.1 / 1.6	5	0.064	472
	Every token	74.00 ( $-$ 0.20)	20/21	7	73.80	100 / 100	488	0.067	488
	Step boundaries	73.80 ( $-$ 0.40)	5/7	6	73.80	1.4 / 1.4	6	0.072	480
Direction MATH-500	Fixed	76.00 (+1.80)	13/4	6	76.00	1.1 / 1.6	5	0.060	458
Direction MATH-500	Reverse	74.60 (+0.40)	9/7	7	74.20	1.1 / 1.5	5	0.068	468
Direction GSM8K	Probe ( HEST )	85.37 (+0.53)	23/16	14	85.37	1.6 / 2.3	5	–	290

Appendix

Table 14: Full statistics of the ablations on Qwen2.5-Math-1.5B-Instruct, Busemann readout,

t=0.5

. Accuracy in % with the change from greedy decoding in points, with the fallback and without it. F/B: problems fixed and broken. Cap: steered decodes that reach the limit of

3072

tokens. Steered: median and pooled fraction of steered tokens in %. Edits: median number of steered tokens per solution. Step: median over problems of the mean

\lVert\Delta h\rVert/\lVert h\rVert

of a solution. For HEST on MATH-500 it is measured at the hesitation points of the greedy solutions, and it was not logged for HEST on GSM8K. Length: median number of tokens, against

470

for greedy decoding on MATH-500 and

289

on GSM8K.

	Fixed		Broken
Direction	MATH-500	GSM8K	MATH-500	GSM8K
Probe ( HEST )	0.154	0.204	0.042	0.038
Fixed	0.127	0.155	0.022	0.043
Reverse	0.088	0.130	0.044	0.037

Appendix

Table 15: Changed solutions by direction, with the gate and the step size of HEST . Fixed: fraction of the greedy-wrong problems with a changed solution that the variant fixes. Broken: fraction of the greedy-right problems with a changed solution that it breaks.

Readout	Geometry	$c$	Pair acc.	$\rho$	$t$	Step	Accuracy	F/B	Changed	Fix rate	Break rate	Paired
MATH-500 , greedy accuracy $74.20$
Busemann	Hyperbolic ( HEST )	0.74	0.880	0.554	0.5	0.060	76.00 (+1.80)	16/7	270	0.154	0.042	–
Busemann	Euclidean	0	0.895	0.589	0.613	0.060	73.40 ( $-$ 0.80)	10/14	245	0.106	0.093	23/10
radius	Hyperbolic ( HEST )	0.53	0.842	0.472	0.25	0.017	75.20 (+1.00)	12/7	181	0.156	0.067	–
radius	Euclidean	0	0.849	0.493	0.252	0.017	74.40 (+0.20)	6/5	185	0.077	0.047	12/8
GSM8K , greedy accuracy $84.84$

Appendix

Table 16: Euclidean probes against the probes of HEST on Qwen2.5-Math-1.5B-Instruct, with

\theta=1

and layer

19

c

: curvature magnitude, which is

0

for the Euclidean probes. Pair acc. and

\rho

: held-out quality as in Table 5 . Step: median

\lVert\Delta h\rVert/\lVert h\rVert

on the hesitation points of the probe-training solutions, which the step length

t

of the Euclidean probe matches. Accuracy in % with the change from greedy decoding in points, and the better accuracy of each pair is in bold. F/B: problems fixed and broken. Changed: solutions whose text differs from greedy decoding. Fix rate: fraction of the changed greedy-wrong solutions that become right. Break rate: fraction of the changed greedy-right solutions that become wrong. Paired: problems on which only HEST is right and on which only the Euclidean probe is right.

Figure 4: Held-out states of Qwen2.5-Math-1.5B-Instruct on MATH-500, embedded by the Busemann probe (left) and the radius probe (right). Each state $y=\sqrt{c}\,z$ is drawn at its norm and at its angle to the ideal point $\xi$ (left) or to the first principal direction (right), which keeps its readout exact. Rotations about the horizontal axis leave the readout unchanged, so a half disk shows all states. Dashed curves are level sets of the readout at its $10$ th, $50$ th and $90$ th percentiles. Darker points have higher next-token entropy.

Figure 5: Held-out states of every model and dataset, drawn as in Figure 4 . Top: Busemann probes. Bottom: radius probes.

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LLM Fine-Tuning Efficient Language Model Reasoning Structured Reasoning LLM Reasoning Structured Reasoning in Language Models

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Multi-step reasoning remains a central challenge for large language models: single-pass generation is efficient but lacks accuracy; tree-search methods explore multiple paths but are computation-heavy. We address this gap by distilling reasoning progress into a hyperbolic geometric signal that guides step-by-step generation. Our approach is motivated by a structural observation: in combinatorial reasoning trees, solution-bearing states are few while dead ends are exponentially numerous. The hyperbolic space matches this asymmetry, with compact volume near the origin and exponentially expanding capacity toward the boundary, so that distance-to-origin naturally encodes solution proximity while angular separation distinguishes branches requiring different next operations. We train a lightweight head to project LLM hidden states into this space, then fine-tune a low-rank adapter interactively on its own reasoning attempts to act on the injected signal. Across multiple benchmarks, the geometric signal yields consistent gains, with larger improvements on deeper reasoning chains. Our code is publicly available at https://github.com/yuyuliu11037/HyperGuide.

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