The Density of Thought: Predicting and Steering Reasoning in Large Language Models

Tanush Kalhan; Anupam Chattopadhyay; Sriram Ranga; Amritendu Mukherjee

cs.AIOct 15, 2026

The Density of Thought: Predicting and Steering Reasoning in Large Language Models

Authors: Tanush Kalhan, Anupam Chattopadhyay, Sriram Ranga, Amritendu Mukherjee

Organizations: Plaksha University · Nanyang Technological University · Interview Street Technologies Pvt. Ltd. (HackerRank) · Indian Statistical Institute

Abstract

We investigate whether variation in information density across chain-of-thought (CoT) reasoning steps predicts answer correctness and can serve as a controllable inference-time target. We prompt frozen pretrained models to solve problems via CoT and measure the density of each reasoning step with two independent metrics: LP variance (how unevenly information is distributed across steps, computed as the variance in mean per-token negative log-likelihood) and the Caveman compression ratio. We then test whether density correlates with, and can be steered to improve, answer correctness. Experiments across six model scales, two training paradigms, and 60 GSM8K problems yield four main findings. First, lower LP variance predicts correctness in supervised fine-tuned (SFT) models, but this relationship weakens with scale and reverses in reinforcement-learning (RL) distilled models. Training paradigm therefore appears more influential than model size in determining density behaviour. Second, synthetic experiments reveal an intrinsic complexity preference: models trained on uniformly distributed data learn complexity-favouring distributions 2.0 to 2.6 times more accurately than simplicity-favouring ones. Third, direct inference-time density steering reduces accuracy from 60.0% to 8.3%, indicating that natural density patterns are functionally important to reasoning. Fourth, pre-generation nudging combining hint injection with contrastive KV-cache steering improves accuracy by 13.3 percentage points, consistent across both metrics. Finally, we introduce Entropy-Gated Density Routing (EGDR), a self-supervised method using token-level Shannon entropy to gate proportional KVcache adjustments during generation, improving accuracy by 10.0 percentage points without any external guidance.

Figures & tables

Figure 1: Overview of the paper’s argument. (A) The complexity preference gap

\Delta

grows 21.3% from GPT-2 XL (2019) to Mistral 7B (2023), driven by training quality rather than scale alone. (B) The density–correctness relationship depends on training paradigm: SFT models show uniform density in correct solutions (consistent with UID); RL-distilled models show structured non-uniformity. This determines the beneficial direction of any density-based intervention. (C) Naïve density steering collapses accuracy by 51.7 pp; principled pre-generation nudging (D: Hint+Cache) achieves

+13.3

pp; EGDR achieves

+10.0

pp without any external hint.

Comparison	Base $\Delta$	Max $\Delta$	Change
GPT-2 (124M $\rightarrow$ 1.5B)	0.3190	0.3378	$+4.8\%$
Pythia (70M $\rightarrow$ 2.8B, ctrl.)	0.2935	0.3323	$+13.2\%$
Cross-gen. (2019 $\rightarrow$ 2023)	0.3343	0.4054	$+21.3\%^{***}$

Table 1: Complexity preference gap

\Delta

across model families. The cross-generational effect (

+21.3\%

) substantially exceeds the controlled within-family scaling effect (

+13.2\%

over

40\times

parameters in Pythia), implicating training methodology, not scale, as the primary driver. Full per-model results in Table 8 .

Corpus	KL (avg)	Bias	Spear. $\rho$	KL ratio
Uniform	0.0038	$+0.094$	$0.60^{*}$	–
Linear	0.0409	$-0.50$	$0.964^{***}$	$1.0\times$
Inverse	0.0967	$+0.77$	0.21 n.s.	$2.4\times$

Table 2: Synthetic experiment results (averaged across 2M–50M models). The Uniform KL (0.0038) replicates Ranga et al. (10) . The KL asymmetry ratio (Inverse/Linear

=2.4\times

) is consistent across all individual model sizes (range: 2.0–2.6

\times

Model	Paradigm	Acc.	$\Delta_{\mathrm{LP}}$	Regime
Qwen2.5-0.5B	SFT	26.7%	$-0.029$	Reversed
Qwen2.5-1.5B	SFT	51.7%	$+0.020$	UID
Qwen2.5-3B	SFT	60.0%	$-0.003$	Collapsed
Qwen2.5-7B	SFT	63.3%	$-0.001$	Collapsed
R1-Distill-1.5B	RL	66.7%	$+0.013$	Dual-density
R1-Distill-7B	RL	76.7%	$+0.014$	Dual-density

Table 3: Density–correctness relationship across model families. LP gap

=\mathrm{Var}(C)-\mathrm{Var}(W)

. Architecture, vocabulary, tokeniser, and inference procedure are identical between SFT and RL families at each scale; the reversal is attributable to training paradigm.

Condition	Accuracy	vs. Baseline
Baseline (greedy CoT)	60.0%	–
High-density prompting	50.0%	$-10.0$ pp
Adaptive mid-step steering	8.3%	$-51.7$ pp

Table 4: Naïve density steering (Qwen2.5-3B,

N{=}60

). Adaptive mid-step injection causes catastrophic accuracy collapse.

Cond.	Acc.	vs. A	$\Delta_{\mathrm{LP}}$	$\Delta_{\mathrm{Cav}}$	$p$
A: Baseline	51.7%	–	$+0.002$	$+0.006$	n.s.
B: Hint	46.7%	$-5.0$ pp	$+0.017$	$-0.008$	0.017
C: Cache	53.3%	$+1.6$ pp	$+0.005$	$-0.027$	n.s.
D: Hint+Cache	65.0%	$+13.3$ pp	$+0.004$	$-0.007$	n.s.

Table 5: Principled nudging results (Qwen2.5-1.5B-Instruct,

N{=}60

\Delta_{\mathrm{LP}}{=}\mathrm{Var}(C){-}\mathrm{Var}(W)

;

\Delta_{\mathrm{Cav}}{=}\mathrm{Cav}(C){-}\mathrm{Cav}(W)

(negative = correct solutions denser). Both signals agree in direction across all conditions.

Figure 2: EGDR applies a piecewise KV cache correction at every generated token based on the model’s own Shannon entropy

H_{t}

. High-entropy tokens (computationally dense moments,

H_{t}>1.5\bar{H}

) receive a positive correction

+\alpha\mathbf{d}

, reinforcing the reasoning trajectory. Low-entropy tokens (predictable filler,

H_{t}<0.6\bar{H}

) receive a small negative correction

-0.3\alpha\mathbf{d}

. This self-supervised gating requires no external input and never interrupts step-level generation. At

\alpha{=}0.30

, EGDR achieves

+10.0

pp accuracy on GSM8K with the most statistically significant density footprint of any evaluated condition (

p{=}0.019

$\alpha$	Accuracy	vs. A	vs. D
0.05	58.3%	$+6.6$ pp	$-6.7$ pp
0.10	55.0%	$+3.3$ pp	$-10.0$ pp
0.15	56.7%	$+5.0$ pp	$-8.3$ pp
0.20	58.3%	$+6.6$ pp	$-6.7$ pp
0.30	61.7%	$+10.0$ pp	$-3.3$ pp

Table 6: EGDR coefficient grid search (Qwen2.5-1.5B-Instruct,

N{=}60

). Accuracy increases monotonically; absence of degradation at

\alpha{=}0.30

suggests the optimum lies above the tested range.

Cond.	Acc.	vs. A	$\Delta_{H}$	$\Delta_{\mathrm{Cav}}$	$p$
A: Baseline	51.7%	–	–	$+0.006$	–
B: Hint	46.7%	$-5.0$ pp	–	$-0.008$	–
C: Cache	53.3%	$+1.6$ pp	–	$-0.027$	–
D: Hint+Cache	65.0%	$+13.3$ pp	–	$-0.007$	–
F: EGDR	61.7%	$+10.0$ pp	$-0.053$	$-0.033$	0.019
G: Hint+EGDR	63.3%	$+11.6$ pp	$-0.041$	$-0.024$	0.021

Table 7: Complete results across all conditions.

\Delta_{H}

: mean entropy(correct)

-

mean entropy(wrong); negative = correct solutions more confident per token. EGDR (F) is the only condition reaching

p<0.05

\Delta_{H}

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Model	Yr	$\Delta$	Change	KS (next)
GPT-2 Small (124M)	2019	0.3190	–	$p<0.01$
GPT-2 Medium (355M)	2019	0.3378	$+5.9\%$	n.s.
GPT-2 Large (774M)	2019	0.3343	$+4.8\%$	n.s.
GPT-2 XL (1.5B)	2019	0.3343	$+4.8\%$	–
Pythia 70M	2021	0.2935	–	$p<0.01$
Pythia 160M	2021	0.3063	$+4.4\%$	$p<0.01$

Appendix

Table 8: Per-model complexity preference gap

\Delta

. The Pythia two-stage plateau (rapid growth to 410M, near-stasis at 1B–2.8B) suggests a scaling law for complexity preference with a ceiling around 1B parameters in this architecture.

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