💬 NLP

On the Fundamental Impossibility of Hallucination Control in Large Language Models

This paper argues that hallucination in large language models is fundamentally unavoidable due to inherent trade-offs in aggregating internal knowledge, proving that while external evidence can verify support for an answer, no internal mechanism can guarantee factual truth or resolve the semantic imbalances that lead to fabrication.

Original authors: Michał P. Karpowicz

Published 2026-08-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Michał P. Karpowicz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a giant jigsaw puzzle, but instead of one person doing it, you have a team of experts sitting around a table. Each expert has only a few pieces of the picture in their hands, and they can't see the whole image. To solve the puzzle, they have to shout out their best guesses about what the final picture should look like and then vote on the answer. This is basically how modern Artificial Intelligence, specifically Large Language Models (LLMs), works. These models are like massive teams of digital "experts" (called components or attention heads) that each hold a tiny fragment of knowledge. When you ask the AI a question, all these fragments compete to shape the answer.

For a long time, scientists and engineers have been trying to fix a annoying problem: these AI teams sometimes make things up. They might tell you a fake fact with total confidence, a glitch known as "hallucination." People have tried to fix this by giving the AI better training, more data, or by making it check its work. But what if the problem isn't just a bug that can be patched? What if the very way the AI is built makes it impossible to be perfectly truthful, perfectly confident, and perfectly creative all at the same time? This is the big question Michal P. Karpowicz explores in a new paper. The paper suggests that hallucination isn't just a mistake; it's a fundamental trade-off, like trying to have your cake and eat it too.

The Great Idea Auction

The paper imagines the AI's brain as a bustling marketplace called an "Auction of Ideas." In this auction, every little part of the AI (like a specific attention head or a circuit) is a bidder. Each bidder holds a secret piece of knowledge and tries to convince the group that their version of the answer is the best. They "bid" by sending signals to influence the final output.

The author proves that sometimes, this auction hits a wall. Imagine two bidders are arguing about the same fact. One says, "The sky is blue," and the other says, "The sky is green." If the AI tries to combine these two opposing views into one single, confident answer, it runs into a mathematical impossibility. The paper shows that you cannot have a system that does four things at once:

Tells the truth (honestly reports what it knows).
Conserves information (doesn't make up new facts out of thin air).
Lets everyone participate (uses every relevant piece of knowledge it has).
Gives the best answer (optimizes for the user's satisfaction).

The paper proves that if the bidders are fighting over the same fact, the system must fail at one of these goals. It's like a magic trick where you can't pull three rabbits out of a hat if you only have two rabbits to begin with. You have to choose: either the AI lies (hallucinates a detail to make the story complete), or it becomes overly cautious and admits it doesn't know, or it ignores some of the knowledge it actually has. There is no "free lunch" where the AI is perfectly honest, perfectly confident, and perfectly helpful all at once.

The Confidence Trap

The paper digs deeper to show why this happens using some clever math. It looks at how the AI combines the "votes" from its different parts. When the AI adds up these votes to decide on an answer, it often ends up sounding more confident than any single part of it actually is.

Think of it like a group of weather forecasters. If Forecaster A says there's a 50% chance of rain, and Forecaster B says there's a 50% chance of rain, but they are looking at different clouds, the group might end up saying, "It's definitely going to rain!" The math shows that the group's confidence is "manufactured." It's an extra layer of certainty that didn't exist in the individual parts. This is the "excess confidence" that leads to hallucinations. The AI isn't necessarily lying on purpose; it's just that the way it combines information naturally creates a gap between what it knows and how sure it sounds.

Can We Fix It?

The paper is very clear about what it doesn't say. It doesn't claim that AI will always lie, or that we can never build a reliable AI. Instead, it says that hallucination is a structural limit, like the speed of light or the uncertainty of a spinning coin. You can't just "train" it away if the conditions of the auction are met.

However, the paper does offer a way to manage the problem. It suggests that instead of trying to make the AI perfectly truthful (which is impossible under these conditions), we should focus on certifiable support. Imagine the AI is a lawyer. We can't always prove if the lawyer's story is the absolute truth, but we can check if the lawyer's story is supported by the evidence they were given.

The paper shows that if we give the AI a specific set of "authorized evidence" (like a trusted database or a set of documents), we can mathematically prove whether the AI's answer stays within the bounds of that evidence. If the AI steps outside those bounds, we know it's hallucinating relative to the evidence, even if we don't know the absolute truth. This turns the problem from "Is this true?" (which is hard) to "Is this supported?" (which is checkable).

The Bottom Line

This paper is a wake-up call for the AI world. It tells us that the "hallucination" we see in AI isn't just a glitch to be fixed with more data or better code. It's a fundamental feature of how these systems aggregate information. When different parts of the AI disagree or compete over the same facts, the system has to make a choice: be honest and weak, or be confident and potentially wrong.

The author suggests that we should stop trying to eliminate this trade-off and start managing it. By understanding that the AI's confidence is sometimes "manufactured" by the math of combining ideas, we can build better systems that know their limits. We can design AI that is honest about what it knows and what it is just guessing, or we can build systems that are strictly bound to the evidence we give them. The paper doesn't promise a perfect AI, but it gives us a clear map of where the walls are, so we can stop banging our heads against them and start building better doors.

Technical Summary: On the Fundamental Impossibility of Hallucination Control in Large Language Models

Problem Statement
Large Language Models (LLMs) frequently produce outputs that are factually incorrect, inconsistent, or unsupported by their training data, a phenomenon known as hallucination. Despite extensive research into architectural innovations (e.g., Mixture-of-Experts, Retrieval-Augmented Generation) and training techniques (e.g., Chain-of-Thought, RLHF), hallucination persists. This paper posits that hallucination is not merely a bug to be patched but reflects a fundamental, conditional limit of the inference process. The core problem is whether a model can simultaneously satisfy four desirable properties of an ideal response: (1) Truthfulness (components report knowledge accurately), (2) Semantic Information Conservation (no knowledge is created ex nihilo), (3) Relevant-Source Participation (all relevant knowledge sources contribute), and (4) Knowledge-Constrained Optimality (the response is the best possible given available knowledge).

Methodology and Theoretical Framework
The author develops a rigorous mathematical framework to analyze inference as a mechanism-design problem, specifically an "auction of ideas."

Knowledge Representation: Knowledge is modeled as a Polish space (a complete, separable metric space) to accommodate the continuous nature of neural embeddings and transformations. Reasoning is formalized via an "emergence operator" ( $E_C$ ) that makes latent knowledge accessible within a computational budget $C$ , without creating new information.
Semantic Information Measure: A measure $\mu_C$ is defined to quantify the information available within a context and budget. The framework distinguishes between ground truth (external reality) and support (derivation from authorized evidence).
The Auction of Ideas: Inference is modeled as a game where internal components (e.g., attention heads, circuits) act as agents with private knowledge. They "bid" to influence the final output. The paper analyzes three complementary settings to prove impossibility results:
- Idealized Private Knowledge: Components hold independent private knowledge.
- Probabilistic Predictions: Components output probability distributions scored by strictly proper, convex loss functions.
- Transformer Architecture: A concrete analysis of additive-logit models and the log-sum-exp operation in transformers.

Key Contributions and Results

The paper establishes three conditional Impossibility Theorems, demonstrating that the four desirable properties cannot coexist when specific "switches" are activated.

Theorem 6 (Independent Private Knowledge): In an idealized setting where components hold independent private knowledge, if the query forces components to contest the same relevant facts (i.e., multiple sources hold and compete to assert the same fact), no mechanism can simultaneously achieve truthfulness, semantic information conservation, participation, and optimality. This result relies on the Green-Laffont theorem from mechanism design. The obstruction arises because the "credit" owed to sources for their joint influence cannot be balanced by report-independent transfers when knowledge overlaps.
Theorem 8 (Probabilistic Aggregation): In settings where components report probability forecasts scored by a strictly proper, convex loss (e.g., cross-entropy), if the forecasts disagree about the realized outcome, the aggregation process creates a "Jensen gap." The aggregate distribution exhibits strictly higher confidence (lower loss) than the weighted average of the individual components' losses. This creates a positive balance defect in the accounting of information contributions, making conservation impossible.
Theorem 9 (Transformer Architecture): Applied to a concrete transformer model, the theorem shows that summing channel logits (from attention heads and FFNs) and applying a softmax normalization inherently leaves a positive "log-sum-exp gap." This gap represents an excess confidence in the aggregate output relative to the declared contributions of the channels. The gap admits a decomposition into two parts: a floor that is strictly positive for any finite channelization (independent of whether components disagree) and an excess-confidence term that measures the degree of disagreement among the channels. Thus, the gap is always present due to the floor, but its magnitude is relative to the declared channelization and accounting rule; the excess-confidence term vanishes only when channel profiles coincide, yet the floor remains.

Certifiable Grounding vs. Truth
A crucial distinction is made between truth (alignment with ground truth) and support (alignment with authorized evidence). The paper proves (Theorem 11) that while truth cannot be certified internally without an oracle, support can be effectively certified. If an external body of authorized evidence is provided, one can mechanically check if a response stays within the deductive closure of that evidence. However, a supported answer can be false, and a true answer can be unsupported.

Conservation-Reasoning Dichotomy
Theorem 12 establishes a dichotomy: Strict semantic information conservation (zero net information gain) and meaningful reasoning (generating new insights or making latent knowledge accessible) are mutually exclusive. Enforcing strict conservation renders reasoning vacuous. Therefore, strict conservation is an inappropriate lever for safety; instead, safety should be defined as "bounded creativity" within an authorized envelope.

Significance and Claims
The paper claims to offer a constructive rather than purely negative result. It does not prove that hallucination is inevitable in all cases, but rather maps the precise structural conditions under which it becomes unavoidable.

Structural Limit: The impossibility is structural, residing in the aggregation step of inference, rather than a limitation of data, compute, or specific architectural choices.
Trade-off Management: The theorems suggest that hallucination control is not about elimination but about managing trade-offs. Different applications (e.g., medical diagnosis vs. creative writing) may prioritize different properties (truthfulness vs. optimality).
Accounting vs. Verdict: The "excess confidence" identified in the theorems is a bookkeeping gap relative to a specific accounting rule (e.g., channelization). It is a diagnostic of the aggregation mechanism, not a direct verdict on factual incorrectness.
Analogy to Fundamental Limits: The paper draws analogies to Gödel's incompleteness, Heisenberg's uncertainty, and Arrow's impossibility theorem, suggesting that LLMs, like formal systems and physical systems, operate under fundamental constraints on what can be simultaneously known, aggregated, and reported.

The paper concludes that while hallucination cannot be universally eliminated, understanding these structural obstructions allows for principled system design, such as using external evidence to certify support, rather than attempting to force a model to satisfy mutually exclusive properties.

The Great Idea Auction

The Confidence Trap

Can We Fix It?

The Bottom Line

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