On the Fundamental Limits of LLMs at Scale (2511.12869v1)

Abstract: LLMs have benefited enormously from scaling, yet these gains are bounded by five fundamental limitations: (1) hallucination, (2) context compression, (3) reasoning degradation, (4) retrieval fragility, and (5) multimodal misalignment. While existing surveys describe these phenomena empirically, they lack a rigorous theoretical synthesis connecting them to the foundational limits of computation, information, and learning. This work closes that gap by presenting a unified, proof-informed framework that formalizes the innate theoretical ceilings of LLM scaling. First, computability and uncomputability imply an irreducible residue of error: for any computably enumerable model family, diagonalization guarantees inputs on which some model must fail, and undecidable queries (e.g., halting-style tasks) induce infinite failure sets for all computable predictors. Second, information-theoretic and statistical constraints bound attainable accuracy even on decidable tasks, finite description length enforces compression error, and long-tail factual knowledge requires prohibitive sample complexity. Third, geometric and computational effects compress long contexts far below their nominal size due to positional under-training, encoding attenuation, and softmax crowding. We further show how likelihood-based training favors pattern completion over inference, how retrieval under token limits suffers from semantic drift and coupling noise, and how multimodal scaling inherits shallow cross-modal alignment. Across sections, we pair theorems and empirical evidence to outline where scaling helps, where it saturates, and where it cannot progress, providing both theoretical foundations and practical mitigation paths like bounded-oracle retrieval, positional curricula, and sparse or hierarchical attention.

• The paper rigorously demonstrates that intrinsic computational and statistical limits cause unavoidable failure modes in large language models.
• It employs techniques from computability theory and information theory to mathematically derive phenomena like hallucination, context compression, and reasoning degradation.
• The findings imply a need for confidence-aware benchmarks, neurosymbolic integration, and calibrated system design to manage LLM fallibility.

Formalizing the Fundamental Limits of LLMs at Scale

Overview and Problem Framing

This work presents a rigorous, unified theoretical framework identifying and formalizing the fundamental limits of LLMs as they scale in size, data, context length, and modality (2511.12869). In contrast to prior empirical and descriptive surveys, the authors draw on proofs from computability theory, information theory, and statistical learning to establish that many key LLM failure modes are direct, necessary consequences of mathematical constraints—not transient engineering artifacts. The paper proceeds to synthesize these impossibility results across five principal axes: hallucination, context compression, reasoning degradation, retrieval fragility, and multimodal misalignment.

Figure 1: Five principal and interacting fronts—long context window, reasoning, hallucination, retrieval quality, and multimodality—that fundamentally limit the reliability of scaled LLMs.

Hallucination: Proof of Inevitability

The work establishes that hallucination is an intrinsic property of scalable LLMs and cannot be eliminated by architecture, data, or algorithmic improvements alone. Hallucination is shown to arise from three distinct but interlocking sources:

1. Computability Theory: Using diagonalization arguments, the paper proves that no computably enumerable set of models can answer all computable queries without failure—there always exist adversarially constructed inputs on which any model, regardless of training, must hallucinate. Moreover, The Halting Problem makes it inevitable that there are infinite queries on which a model must fail, independent of scale or architecture.
2. Statistical and Information-Theoretic Limits: The authors derive risk and sample complexity bounds showing that finite-capacity models must commit irreducible error on incompressible or long-tail facts, and that exhaustive memorization of arbitrary facts is infeasible, even with astronomical data.
3. Data-Induced Pathways: Real-world data exacerbates hallucination via incompleteness, noise and misinformation, memorization failures for rare facts, temporal decay, internal contradictions, and exposure bias.

   Figure 2: Taxonomy of hallucination sources in LLMs, spanning formal computability limits, data flaws, and evaluation misalignment.

The paper further critiques contemporary evaluation protocols that reinforce confident fabrication—binary grading and RLHF policies incentivize guessing, and reward overconfidence and plausible fabrication, amplifying hallucination beyond theoretical baselines. The tradeoff between creativity and factuality is formalized via entropy and sampling-temperature arguments, showing that increased generative diversity unavoidably raises hallucination rates.

Compression of Effective Context: Distributional and Architectural Barriers

While LLMs support massive nominal context windows, empirical and formal results reveal that the effective usable context is strictly sublinear in window size due to three factors:

1. Training Distribution Skew: Pretraining data is left-skewed toward short spans, resulting in severe undertraining on positions and dependencies beyond the early sequence. The cumulative gradient updates and attention weights for distant positions shrink multiplicatively, leading to marginal improvements at extreme window lengths.
2. Encoding Attenuation: The paper mathematically demonstrates that standard positional encodings (sinusoidal, RoPE) suffer from cancellation and near orthogonality at long distances, causing decaying overlap and sharply attenuating the effect of long-range context on attention.
3. Softmax Crowding: Formal analysis establishes that in quadratic softmax attention, the required logit margin to isolate a relevant token among $N$ distractors scales as $\log N$, and thus, maintaining consistent attention on any specific distant token is practically impossible as context grows.

   Figure 3: Three main mechanisms—distributional skew, encoding attenuation, and attention softmax crowding—effectively compress the usable long-context window in practical LLMs.

Taken together, these factors explain the empirical finding that, despite 128k-token context support, actual reasoning and memory reliability falls off rapidly at much shorter effective ranges.

Reasoning Degradation: Objective Mismatch and Architectural Constraints

Despite the adoption of chain-of-thought-style prompting and scaling, the work formalizes the persistent brittleness of LLMs in systematic, multi-step, and compositional reasoning tasks. The authors show that next-token likelihood objectives inherently reward localizing plausible recitation rather than enforcing the consistent, symbolic, or causal relationships required for reliable algorithmic reasoning.

Figure 4: Mathematical framework adaptations for LLM reasoning: standard likelihood-based approaches versus architectures that seek verification, constraint satisfaction, or neuro-symbolic integration.

Key technical insights include:

• Disposable mediators in chain-of-thought traces: Intermediate reasoning steps $Z$ are only weakly constrained by causal influence on answer $Y$, so models often learn to treat them as ancillary, optimizing final answer likelihood.
• Spurious knowledge injection: External evidence is easily cited in rationales without being deployed as an instrumental, causal mediator in actual inference.
• Search and compute pathologies: Greedy and locally sampled decoding myopically misallocates compute, while multi-path expansion rapidly becomes intractable.
• Interface and metrics fragility: Evaluation metrics that reward only final answer accuracy fail to penalize inconsistency or irrationality in intermediate steps; process-aware validation is required for reliable reasoning.

The remedy, per the authors, is to design reasoning objectives that explicitly enforce correctness and constraint at every step (e.g., program-aided execution, neuro-symbolic integration, chain-of-verification loops), and to optimize for reasoning efficiency (quality per compute).

Figure 5: Reasoning strategies: (a) chain-of-verification, (b) program-aided execution, (c) neuro-symbolic integration, and (d) multimodal LLM grounding.

Retrieval Fragility: Budgeted, Positional, and Adversarial Failures

In retrieval-augmented LLMs, failure modes originate both from retrieval optimization under finite token budgets and from the model's over-reliance on retrieved context, which can be easily corrupted or diluted. Fundamental challenges include:

• Relevance–Coverage Dilemma: Under hard context size limits, improving coverage sacrifices relevance and vice versa. Increased recall leads to prompt dilution and generation drift, while strict precision excludes essential peripheral evidence.
• Ranking and Chunking Failures: Marginally sub-threshold or mid-prompt evidence is frequently omitted or disregarded due to LLM positional attention bias, ("lost-in-the-middle" effect).
• Adversarial Vulnerability: Retrieval systems are highly susceptible to knowledge poisoning—maliciously engineered documents can disproportionately occupy top ranks in the context window, dominating generation and leading to confident but incorrect outputs.
• Context Fusion Limitations: Query ambiguity and context merging result in the LLM performing unprincipled mixtures over paraphrased or conflicting evidence without control, amplifying hallucination and calibration errors.

  Figure 6: Performance landscape over relevance–coverage space with token budget constraints; budgeted RAG optimizes along tradeoff curves, not global maxima.

Multimodal Misalignment: Architectural, Representational, and Scaling Obstacles

The extension to multimodal LLMs is shown to amplify, rather than eliminate, existing text-only failures. The incorporation of vision and audio encoders into LLM backbones leads to multiple new forms of bottleneck, including:

• Textual Dominance: Projected visual tokens are structurally and statistically overwhelmed by pretrained language representations. Backpropagated gradients with respect to visual features are orders of magnitude smaller than for text, yielding text-centric optimization and representational hegemony.
• Alignment Noise and Semantic Drift: Vision encoders retain co-occurrence statistics from their contrastive pretraining, not true perceptual structure, introducing semantic drift and anchoring errors at scale.
• Fusion Bottlenecks and Quantization Artifacts: Cross-modal attention's quadratic cost leads to pooling, patching, or vector quantization, introducing lossy compression and collapsed codebooks. The resulting information bottleneck sharply delimits the mutual information between input modality and downstream reasoning.
• Scaling Anti-Synergy: Multimodal scaling law exponents are shown to be bounded by, and often inferior to, unimodal exponents; increasing model or data size can hurt aggregate performance if inter-modality scaling is imbalanced.

Collectively, these limitations expose the illusion of grounding in modern MLLMs and argue for architectures that provide external, neurosymbolic reasoning components, or embodied data, rather than larger fusion models.

Benchmarking Limitations and Implications for Progress

The work emphasizes that current leaderboards and evaluation frameworks systematically obscure these ceilings via data contamination, judge bias, compute inflation, and protocol idiosyncrasy. Improved scores can result from memorization, verbosity, or favorable prompt framing, rather than fundamental advances. Consequently, headline gains may reflect evaluation artifacts rather than genuine progress towards overcoming theoretical limits.

Practical Implications and Future Directions

The authors advocate a paradigm shift: reliable language modeling is fundamentally an exercise in bounding and allocating inevitable failure. This calls for selective prediction (calibrated abstention), confidence-aware evaluation, benchmark design that rewards uncertainty over confident error, and architectures that integrate external verification, constraint satisfaction, or selective modality fusion. The formal ceilings derived here render certain error rates intrinsic and invite work to move from existence proofs and pessimistic impossibility to practical bounds, system-level mitigations, and quantification of where and how models are likely to fail.

Conclusion

"On the Fundamental Limits of LLMs at Scale" provides a rigorous, proof-informed synthesis of why key reliability failures persist under scale. By unifying impossibility results from computability, information theory, and statistical learning, it reframes scaling as an enterprise necessarily bounded by intrinsic computational and epistemic ceilings. The implications are broad: future progress lies in quantifying, predicting, and managing fallibility through theoretically minded architectural innovation, confidence-aware benchmarks, and selective, calibrated system design—not in pursuing a mirage of infallibility via unbounded scale (2511.12869).