Hallucination Basins: A Dynamic Framework for Understanding and Controlling LLM Hallucinations

Kalyan Cherukuri; Lav R. Varshney

cs.CLApr 6, 2026

Hallucination Basins: A Dynamic Framework for Understanding and Controlling LLM Hallucinations

Authors: Kalyan Cherukuri, Lav R. Varshney

Organizations: Illinois Mathematics and Science Academy, Illinois, USA · AI Innovation Institute, Stony Brook University, New York, USA

Abstract

Large language models (LLMs) hallucinate: they produce fluent outputs that are factually incorrect. We present a geometric dynamical systems framework in which hallucinations arise from task-dependent basin structure in latent space. Using autoregressive hidden-state trajectories across multiple open-source models and benchmarks, we find that separability is strongly task-dependent rather than universal: factoid settings can show clearer basin separation, whereas summarization and misconception-heavy settings are typically less stable and often overlap. We formalize this behavior with task-complexity and multi-basin theorems, characterize basin emergence in L-layer transformers, and show that geometry-aware steering can reduce hallucination probability without retraining.

Figures & tables

Symbol	Definition
$h^{(\ell)}$	Hidden state at layer $\ell\in\{0,\ldots,L\}$
$d$	Dimension of hidden states ( $h^{(\ell)}\in\mathbb{R}^{d}$ )
$\mu^{(\ell)}$	Reference/centroid state at layer $\ell$ (basin center)
$\mathcal{B}^{(\ell)}(r)$	Basin of attraction: $\{h:\\|h-\mu^{(\ell)}\\|_{2}\leq r\}$
$J_{\ell}$	Jacobian $\partial f_{\ell}/\partial h$ at layer $\ell$
$\rho(\cdot)$	Spectral radius (largest eigenvalue magnitude)

Table 1: Notation summary

Factoid: Point Attractors ( $\rho_{\text{var}}\gg 1$ )
Model	Dataset	Task Type	$\boldsymbol{\rho_{\text{var}}}$	Basin Sep $d$
Llama-1B	HaluEval QA	Factoid	4.55	2.89
Llama-1B	MuSiQue	Factoid	10.00	3.40
Qwen-1.5B	HaluEval QA	Factoid	5.56	32.83
Gemma-2B	HaluEval QA	Factoid	1.82	58.91
Summarization: High-Dimensional Manifolds ( $\rho_{\text{var}}\approx 1$ )

Table 2: Variance Analysis by Task Type. Factoid tasks show up to 4

\times

variance expansion (factual states occupy larger volume), while summarization maintains parity (high-dimensional manifolds). Basin separation

d

measured as

\|\mu_{\text{fact}}-\mu_{\text{hall}}\|_{2}

\rho_{\text{var}}=\text{Var}[\text{factual}]/\text{Var}[\text{hallucinated}]

Figure 1: Task-Dependent Basin Geometry. Llama-3.2-3b’s performance on various tasks and 3D PCA projected outputs. (a) shows performance on MuSiQue, (b) shows performance on HaluEvalQA, (c) shows performance on HaluEvalSummarization, (d) shows performance on TruthfulQA.

Model	Data	Lay	Centroid (95% CI)	Maha (95% CI)	$N$	B?
gemma-2-2b	FEVER	$L_{14}$	$0.515\ (0.489,0.548)$	$0.514\ (0.487,0.535)$	$9999$	×
gemma-2-2b	HaluEval_qa	$L_{14}$	$0.727\ (0.703,0.744)$	$0.725\ (0.710,0.745)$	$20000$	✓
gemma-2-2b	HaluEval_summ	$L_{20}$	$0.508\ (0.491,0.519)$	$0.479\ (0.457,0.502)$	$20000$	×
gemma-2-2b	MuSiQue	$L_{26}$	$0.912\ (0.894,0.932)$	$0.926\ (0.909,0.944)$	$4834$	✓
gemma-2-2b	TruthfulQA	$L_{14}$	$0.607\ (0.547,0.662)$	$0.597\ (0.535,0.651)$	$1580$	×
llama-3.2-1b	FEVER	$L_{8}$	$0.670\ (0.641,0.700)$	$0.680\ (0.659,0.702)$	$9999$	×

Table 3: Centroid and Mahalanobis (Maha) AUROC reported with 95% CI. Note: Lay indicates Layer with highest AUROC,

N

indicates number of data samples, and

B?

indicates whether a basin exists or not. For larger models, due to computational constraints, this was limited to

N=1000

Figure 2: Causal Intervention: Factual $\to$ Basin. (Left) Dose-response curve fold increase in hallucination probability as factual hidden states are in-model steered toward the hallucination centroid (interpolation strength $\alpha$ on the horizontal axis). Right: bar plot comparing the maximum fold increase produced by steering along the basin direction versus two controls (random direction and an orthogonal direction). See Appendix D.3 and F.1 .

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

\sigma_{\text{hall}}^{2}\leftarrow\frac{1}{|H_{\text{hall}}|}\sum_{h\in H_{\text{hall}}}\|h-\mu_{\text{ref}}\|^{2}

Appendix

Algorithm 1 Multi-Basin Detection for Misconceptions

Panel of Figure 3 — Figure 3: Multi-basin Voronoi structure across models on TruthfulQA. Each panel shows distinct hallucination basins corresponding to different misconception modes.

Figure 4: Efficacy of Algorithm 2 in hallucination reduction as a function of the steering strength $\lambda$ .

Figure 5: Irreversibility summary under autoregressive decoding (HaluEval QA, Llama-3.2-1B, best layer). We report basin-entry, conditional irreversibility, escape-after-entry, and factual entry rates. This verifies Theorem 5.9 .

Figure 6: Layer-wise attention entropy for factual versus hallucinated generations under uninformative contexts (autoregressive extraction). Entropy trends provide a complementary signal to basin-separation metrics. Supports the uniform attention assumption.

Figure 7: Causal Intervention Paths: Llama-3.2-1B (HaluEval QA)

Figure 8: Causal Intervention Paths: Llama-3.2-3B (HaluEval QA)

Figure 9: Causal Intervention Paths: Qwen-2.5-1.5B (HaluEval QA)

Figure 10: 2D PCA Evolution: Llama-3.2 1B (QA).

Figure 11: 2D PCA Evolution: Llama-3.2 1B (Summarization)

Figure 12: 2D PCA Evolution: Qwen-2.5 1.5B (QA)

Figure 13: 2D PCA Evolution: Gemma-2 2B (Summarization)

Parameter	Setting
Random seed	42 (NumPy, PyTorch, scikit–learn)
Determinism	Seeds fixed; some GPU ops may remain non-deterministic
Batch size (train / extraction)	8 (training loaders); 4 (hidden-state extraction)
Generation batch size	Per experiment; demo runs use batch $=$ #prompts (typically 3)
Maximum sequence length	512 tokens (truncation applied)
Tokenizer padding	pad_token $=$ eos_token ; left padding for decoder-only models

Appendix

Table 4: Global data generation, model loading, and evaluation hyperparameters shared across all experiments.

Category	Parameter	Setting
Detection	Evaluation layer	Middle layer ( $\lfloor L/2\rfloor$ )
	Covariance estimate	Ledoit–Wolf shrinkage
	Detection model	Logistic regression (L2) on standardized features
Causality	Interpolation grid	$\alpha\in\{0,0.1,\ldots,1.0\}$
	Control directions	Random + orthogonalized (Gram–Schmidt)
	Effect metric	Fold change $\frac{\mathbb{P}(\mathrm{hall})_{\text{int}}}{\mathbb{P}(\mathrm{hall})_{\text{base}}}$

Appendix

Table 5: Experiment-specific hyperparameters used across detection, causality, steering, and ablation studies.

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We address hallucination detection in Large Language Models (LLMs) and Multimodal Large Language Models (MLLMs) by framing the problem through the lens of dynamical systems stability theory. Rather than treating hallucination as a straightforward classification task, we conceptualize (M)LLMs as dynamical systems, where factual knowledge is represented by stable equilibrium points within the representation space. Our main insight is that hallucinations tend to arise at the boundaries of knowledge-transition regions separating stable and unstable zones. To capture this phenomenon, we propose Lyapunov Probes: lightweight networks trained with derivative-based stability constraints that enforce a monotonic decay in confidence under input perturbations. By performing systematic perturbation analysis and applying a two-stage training process, these probes reliably distinguish between stable factual regions and unstable, hallucination-prone regions. Experiments on diverse datasets and models demonstrate consistent improvements over existing baselines.

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