Inevitable Hallucination in LLMs

Updated 31 December 2025

Inevitable hallucination in LLMs is defined as a mathematically proven error mechanism arising from diagonalization and computability theory, ensuring infinite misgeneration cases.
Statistical approaches can reduce hallucination probability on practical inputs, yet empirical benchmarks show error rates ranging from 59% to 82% in major LLMs.
Architectural biases in transformer models and adversarial vulnerabilities stress the need for hybrid mitigation strategies and continuous oversight.

LLMs are subject to the phenomenon of hallucination: the generation of nonfactual, unfounded, or contextually inappropriate content. This limitation is not merely an artifact of engineering or data selection but is grounded in deep results from computability theory, statistical learning, causal analysis, architectural design, and empirical benchmarking. Hallucination is both an inevitable consequence of the generalization problem under the open world assumption and a persistent challenge in applied NLP, demanding both theoretical understanding and practical mitigation.

1. Computability and Mathematical Foundations

The inevitability of hallucination in LLMs has been rigorously formalized using diagonalization arguments and computability theory. Any LLM $h: \Sigma^* \to \Sigma^*$ is a computable function; for every such model, there exists an acceptable-output-set map $F_0: \Sigma^* \to 2^{\Sigma^*}$ (with $F_0(s) \neq \emptyset$ for all $s$ ) such that the set $\{s \in \Sigma^* | h(s) \notin F_0(s)\}$ is infinite. The proof constructs, via enumeration and diagonalization, an infinite sequence of inputs on which any computable $h$ inevitably misgenerates, regardless of architecture, training algorithm, or data quality (Suzuki et al., 15 Feb 2025, Xu et al., 2024). This computability-theoretic “no-algorithm” barrier establishes that there is no model free from hallucination across all conceivable inputs.

Further, drawing on Gödel’s incompleteness theorem and undecidability results (halting, acceptance, emptiness problems), it has been shown that every LLM component (training, retrieval, classification, generation) possesses a strictly nonzero probability of contributing to hallucination. This guarantee persists under any scaling of data or architecture and is termed “structural hallucination” (Banerjee et al., 2024).

2. Statistical Learning and Practical Negligibility

While computability arguments assure inevitability, in practice, hallucination probability can be made statistically negligible for real-world input distributions. Given a distribution $\mu$ over inputs and a suitable length-CDF prior, it is possible to design trainers such that, with sufficiently large data and appropriate algorithms, the hallucination-probability $\mathrm{HP}_{\mu}(h)$ for a model $h$ falls below any chosen operational threshold. The error set remains infinite, but its measure under $\mu$ can be arbitrarily shrunk. This reconciliation—inevitable but negligible—mirrors the treatment of rare codewords in information theory (source coding) (Suzuki et al., 15 Feb 2025).

For realistic language usage concentrated in moderate-length inputs, practical dataset sizes suffice to achieve low hallucination risk. Natural strings cluster at $n \lesssim 200$ tokens, and collecting $m \sim O(|\Sigma|^n / \delta)$ examples can control error rates. Algorithmic checks and data-centric engineering are required for operational safety.

3. Empirical Evidence and Benchmarking

Large-scale evaluation on the DefAn “Definitive Answer” dataset ( $>$ 75,000 prompts, 8 domains) demonstrates the empirical inevitability of hallucination. Across six major LLMs (GPT-3.5, Gemini, LLaMA, Mixtral, Zephyr), overall factual hallucination rates range from 59% to 82%, while consistency across paraphrases rarely exceeds 60%. Numeric recall is especially poor—near-zero reliability—whereas person, location, and date domains fare only moderately better. Even instruction-tuned and scaled models show persistent high error (Rahman et al., 2024).

Three key metrics frame this empirical challenge:

Factual Hallucination (FCH): Fraction of responses contradicting ground truth.
Prompt-Misalignment (PMH): Fraction of responses deviating from required format.
Response Consistency (RC): Fraction of matched claims across multiple paraphrases.

These benchmarks confirm that next-token prediction architectures lack explicit grounding, statistical pattern completion overrides factuality, and exposure bias undermines repeatability. Hallucinations persist despite scaling and tuning—factual engines remain elusive.

4. Architectural and Causal Drivers

Transformer architectures and their self-attention modules are structurally incentivized toward hallucination. As statistical coherence engines, they maximize fluency in continuation but lack existential grounding—temporality, affordance, social context, and disclosure. Ackermann & Emanuilov distinguish ontological hallucination (violating world constraints) and residual reasoning hallucination (approximate mimicry of human inference without true causal modeling) (Ackermann et al., 19 Sep 2025). Case studies aligned with Heideggerian existential structures reveal pervasive failure modes: anachronism, mood inconsistency, mis-affordance, referential overwriting, and cultural misattribution.

Causal interventions on self-attention (zeroing selected layers) modestly alter factuality, suggesting that confounders propagate through multiple redundant paths. Middle layers encode core knowledge; disabling front or tail layers yields minor reductions in hallucination but does not extinguish it. Only external knowledge mediators or dynamic confounder monitors can begin to address the systemic bias (Li et al., 2024).

5. Mechanisms and Predictors: Knowledge Overshadowing

LLMs experience “knowledge overshadowing,” wherein dominant facts obscure rarer ones during text generation. This leads to predictable hallucination rates governed by a log-linear law:

$R(P, L, S) = \alpha \log P + \beta \log L + \gamma \log S + \delta$

where $P$ is fact popularity ratio, $L$ is relative knowledge length, $S$ is model size, and $R$ the hallucination rate. Empirical ablation across synthetic and real tasks confirms that frequency, length, and scale monotonically increase hallucination; CoDa (contrastive decoding adjustment) can mitigate, but not eliminate, this structural dynamic (Zhang et al., 22 Feb 2025). Overshadowing is pronounced in large models and long contexts, pointing to a “rich get richer” effect inherent in next-token modeling.

6. Adversarial and Robustness Perspectives

Hallucination manifests as an adversarial vulnerability: random or weakly perturbed prompts reliably elicit controlled, spurious outputs. Gradient-guided token swaps induce targeted hallucinations, even absent semantic coherence. Success rates for “hallucination attacks” are substantial; Vicuna-7B reaches 92% under weak- and 81% under out-of-distribution attacks. Entropy thresholding defenses filter some attacks but cannot guarantee correctness (Yao et al., 2023). Robustness demands adversarial evaluation and defenses grounded in statistical signal detection, but such mechanisms remain partial.

7. Mitigation Strategies and Open Challenges

Prompting-level interventions such as CounterFactual Multi-Agent Debate (CFMAD) reduce, but do not eradicate, hallucination. By forcing debate, critique, and third-party judgment on candidate answers, bias-driven fabrication is made conspicuous and more easily eliminated via collective scrutiny. CFMAD outperforms self-reflection and chain-of-thought baselines, yet computational overhead, stance recall, and open-ended generation remain unresolved (Fang et al., 2024). Empirical studies reveal that LLM hidden states are internally sensitive to hallucination; activation engineering can steer generation towards truthfulness in some cases, although architectural limits persist (Duan et al., 2024).

Ultimate elimination of hallucination is unattainable without external retrieval, symbolic reasoning modules, or fundamentally new architectural principles. Hybrid systems—retriever, verifier, generator pipelines—offer greater reliability but still admit nonzero error under open-world generalization (Xu, 29 Sep 2025).

Table 1: Theoretical Drivers of Hallucination

Principle	Consequence	Key Reference
Diagonalization/Computability	Infinite hallucinations on countable input	(Suzuki et al., 15 Feb 2025, Xu et al., 2024)
No Free Lunch/Open World	Nonzero generalization error on unseen inputs	(Xu, 29 Sep 2025)
Knowledge Overshadowing	Log-linear law of hallucination rate	(Zhang et al., 22 Feb 2025)
Structural Hallucination	Gödel/Undecidability keeps $P_h > 0$	(Banerjee et al., 2024)
Causal Propagation	Redundant paths sustain error	(Li et al., 2024)
Statistical Fluency Bias	Pattern completion overrides factuality	(Fang et al., 2024, Ackermann et al., 19 Sep 2025)

These theoretical foundations, validated by empirical benchmarks and architectural analysis, establish hallucination as a persistent, structurally inevitable phenomenon in LLMs.

Conclusion

Hallucination in LLMs is a mathematically inevitable consequence of computability barriers, open-world generalization, knowledge overshadowing mechanisms, and transformer design. While practical rates can be reduced by data scaling, algorithmic advances, and prompt engineering, persistent error—especially on out-of-distribution, long-tail, or adversarial inputs—cannot be eradicated. The coexistence of mathematical inevitability and practical negligibility mandates a paradigm shift: from elimination to management, detection, and toleration of hallucination through hybrid system design, uncertainty quantification, and continuous oversight.