Inevitability of Hallucination in LLMs

• Hallucination in LLMs is the phenomenon where models produce plausible but factually incorrect outputs due to statistical and computational constraints.
• The analysis employs mathematical proofs, adversarial testing, and statistical bias studies to demonstrate that hallucinations are inherently unavoidable in LLM architectures.
• Mitigation strategies like retrieval augmentation and watchdog frameworks can reduce hallucination rates, though complete elimination remains theoretically impossible.

LLMs exhibit a persistent propensity for producing “hallucinations,” i.e., outputs that are plausible, coherent, and linguistically fluent, but factually incorrect, contextually inconsistent, or unverifiable relative to input data, training corpus, or real-world reference. Decades of research across learning theory, computational complexity, information theory, cognitive science, and empirical evaluation converge on the finding that hallucination in LLMs is not merely an occasional artifact of poor data or suboptimal design, but an inescapable, structural property of probabilistic architectures based on next-token prediction, statistical memorization, and finite representational capacity. A vast body of recent literature rigorously formalizes, measures, and benchmarks hallucination through mathematical proof, behavioral studies, and adversarial testing, while simultaneously elucidating its sources, manifestations, and limits of mitigation.

1. Formal Foundations and Computational Theory

The inevitability of hallucination is rooted in the mathematical nature of LLMing and computability. At the most abstract level, an LLM is a computable function $h: S \to Y$ mapping strings $S$ to outputs, commonly trained to maximize $p(y|s)$, where $y$ is a completion or answer. Hallucination is rigorously defined as any instance where the model output $h(s)$ deviates from the ground truth function $f(s)$:

$\exists s \in S: h(s) \neq f(s)$

Diagonalization arguments (Cantor-Gold style) are leveraged to show that for any countable family of computable LLMs, one can construct an adversarial ground-truth function $f$ that disagrees with each model on at least one (typically infinite many) inputs (Xu et al., 22 Jan 2024, Suzuki et al., 15 Feb 2025, Cossio, 3 Aug 2025, Shi et al., 10 Aug 2025).

More specifically, for every computable model, the set $\{s \in S: h(s) \neq f(s)\}$ is infinite, regardless of data quality, architecture, or inference strategy (Suzuki et al., 15 Feb 2025). For models with time/space complexity constraints (e.g., polynomial time), there always exist computable—but intractable—functions $f$ (e.g., instances of SAT, Presburger arithmetic, or exhaustive listing problems) the model cannot emulate, enforcing unavoidable errors.

Complementary results invoke Gödel's incompleteness and Turing’s undecidability: no machine learning system, regardless of its training data or size, can decide all queries or retrieve all facts deterministically (Banerjee et al., 9 Sep 2024, Shi et al., 10 Aug 2025). Thus, at a structural level, there is always a nonzero probability of hallucination on some queries.

2. Statistical Learning, Memory, and Inductive Biases

Hallucinations persist in LLMs trained on enormous or even perfectly curated datasets because of the underlying statistical objectives and memorization-generalization tradeoff (McKenna et al., 2023, Zhang et al., 22 Feb 2025). Two principal train-time biases propagate into inference-time hallucination:

• Attestation Bias: Sentence-level memorization causes LLMs to inappropriately “entail” hypotheses merely because they match memorized fragments in pretraining data, irrespective of logical consistency with the prompt (McKenna et al., 2023).
• Relative Frequency Bias: LLMs learn frequent associations and overpredict common predicates; if a premise contains less frequent predicates than the hypothesis, models favor entailment, echoing corpus-level statistics over logical inference (McKenna et al., 2023).

The law of knowledge overshadowing extends these findings: when multiple relevant facts exist, the more frequent, longer, or dominant knowledge representations suppress less frequent ones, causing the model to hallucinate by substituting popular knowledge even when less popular ground truth is required (Zhang et al., 22 Feb 2025). This is formalized in a “log-linear law”:

$$R(P) \propto \log(\text{popularity}), \quad R(L) \propto \log(\text{length}), \quad R(S) \propto \log(\text{model size})$$

where $R(*)$ is the hallucination rate as a function of knowledge popularity, segment length, or model size, respectively.

3. Adversarial and Thermodynamic Perspectives

Hallucinations can be framed as adversarial examples: small, semantic-preserving perturbations to prompts—or even random, out-of-distribution token sequences—can reliably induce LLMs to generate specific, pre-defined, incorrect outputs (Yao et al., 2023). This derives from the model's high-dimensional non-linearity and gradient-based optimization, making them susceptible to input manipulations that trigger deterministic misfires in generation.

Thermodynamic analogies (field-theoretic modeling) further support the inevitability of hallucination. When LLM outputs are analyzed in terms of “free energy” (negative log probability) and “entropy” (uncertainty), hallucinated responses correspond to unstable regions—where small changes in sampling temperature or likelihood cause erratic shifts in output energy and entropy. Since sampling in high-dimensional probability spaces always involves such fluctuations, some degree of hallucination is inherent (Vu et al., 12 Sep 2025).

4. Taxonomy and Manifestation

Comprehensive taxonomies divide hallucinations into:

• Intrinsic hallucinations: Contradictions or inconsistencies relative to the explicit input context or prompt.
• Extrinsic hallucinations: Content that is unsupported by, or inconsistent with, the model’s training data or real-world facts (Bang et al., 24 Apr 2025, Cossio, 3 Aug 2025).

Further distinctions are made between factuality (absolute correctness) and faithfulness (adherence to source). Manifestations include fabricated facts, logical or contextual inconsistencies, temporal disorientation, ethical violations, and domain-specific errors (code, multimodal tasks).

Consistent with theoretical predictions, experiments across all major LLM families—including GPT-3.5, PaLM, LLaMA, and more recent models—show that hallucinations remain prevalent both in benchmarked tasks (e.g., precise QA, summarization) and adversarial settings (e.g., hypothetical term generation, multilingual tests in low-resource languages) (Uluoglakci et al., 25 Feb 2024, Das et al., 30 Jul 2025, Bang et al., 24 Apr 2025).

5. Detection, Mitigation, and Limits

Empirical research shows that, while hallucinations are theoretically ineliminable, their practical likelihood can be rendered statistically negligible (Suzuki et al., 15 Feb 2025). If the input distribution is known (with a CDF tending to 1 for query lengths) and training data are sufficiently high quality and comprehensive, the model’s probability of hallucination for randomly drawn queries can be made arbitrarily small.

However, worst-case (adversarial) or tail events cannot be eliminated. Certain hallucinations occur with high certainty even when the model “knows” the correct answer (CHOKE phenomenon), eluding uncertainty-based detection or abstention strategies (Simhi et al., 18 Feb 2025). Signal detection methods exploiting LLM internal states—e.g., eigenvalue-based semantic consistency metrics, feature-space probes, or field-theoretic variations—can flag, but not fully prevent, such errors (Chen et al., 6 Feb 2024, Ji et al., 3 Jul 2024, Vu et al., 12 Sep 2025).

Mitigations include:

• Black-box “watchdog” frameworks that estimate the model’s generalization boundary via probabilistic and semantic space exploration (Liu et al., 21 Jul 2025).
• Contrastive decoding and “escaping reward” schemes to enhance retrieval of non-dominant facts suppressed by knowledge overshadowing (Zhang et al., 22 Feb 2025).
• Dynamic benchmarking and continuous domain adaptation (retrieval-augmented generation, continual learning) to move ephemeral computational boundaries, albeit at external cost (Shi et al., 10 Aug 2025, Bang et al., 24 Apr 2025).

Nonetheless, these only reduce, not eliminate, hallucination—especially in low-resource domains or adversarial contexts.

6. Evaluation Benchmarks and Practical Impact

Taxonomy-aware benchmarks (HalluLens, HypoTermQA, TruthfulQA, and others) measure hallucination rates across intrinsic and extrinsic axes, including outright fabrication (nonexistent entities), misalignment with up-to-date facts, and ground-truth inconsistency with training data (Uluoglakci et al., 25 Feb 2024, Bang et al., 24 Apr 2025, Cossio, 3 Aug 2025). Dynamic test set generation mitigates data leakage, ensuring robustness and continued challenge even as models memorize previous evaluations.

Hallucinations have substantial impact in applications demanding reliability—healthcare, law, finance—where even rare high-confidence errors can erode trust and cause tangible harm. The problem is exacerbated in low-resource languages, where insufficient and lower-quality data increase extrapolation and thus hallucination rates (Das et al., 30 Jul 2025).

7. Theoretical Boundaries and “Escape Routes”

The cumulative body of research formalizes a computational necessity hierarchy for hallucination inevitability, spanning:

• Diagonalization: For any (probabilistic) LLM, adversarially constructed queries provably induce error.
• Uncomputability: Full elimination of hallucination would yield deciders for the Halting or Acceptance problems, a logical impossibility.
• Information-theoretic limits (learner pump lemma): Bounded model complexity ensures that, above a certain function complexity threshold, hallucination risk exceeds any nonzero tolerance.

Two “escape routes” are formalized: (1) External oracle augmentation (Retrieval-Enhanced Generations) achieves local, absolute escape from hallucination but only for queries within oracle range; and (2) Continual adaptation and dynamic capacity expansion via internalized memory can locally and incrementally move boundaries, but never eradicate systemic hallucination in the global sense (Shi et al., 10 Aug 2025).

---

In conclusion, hallucination in LLMs is mathematically, statistically, and empirically unavoidable due to the limits of computable, finite-capacity, probabilistic models operating under statistical learning objectives and imperfect data. While mitigation strategies can suppress hallucinations for the majority of practical inputs—and information-theoretic perspectives assure that error probability can be made negligible for most “in-distribution” queries—complete elimination is provably impossible. Future efforts focus on robust detection, dynamic mitigation, and continuous human oversight to contain, quantify, and contextualize hallucinations, thus ensuring responsible deployment in high-stakes applications where error tolerance is limited.