arxiv: 2509.04664 · v1 · submitted 2025-09-04 · 💻 cs.CL

Recognition: 3 theorem links

· Lean Theorem

Why Language Models Hallucinate

Adam Tauman Kalai , Ofir Nachum , Santosh S. Vempala , Edwin Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-13 12:28 UTC · model grok-4.3

classification 💻 cs.CL

keywords language modelshallucinationsuncertaintybenchmarksevaluationtrainingstatistical causes

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The pith

Language models hallucinate because training and evaluations reward guessing when uncertain instead of admitting limits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Language models produce plausible but incorrect statements rather than expressing uncertainty on hard questions. The paper shows that this stems directly from statistical pressures in pretraining where facts cannot be separated from errors, turning hallucinations into ordinary classification mistakes. Current benchmarks make the problem worse by scoring only final answers and penalizing any admission of doubt, so models learn to always respond to maximize scores. The authors conclude that the fix lies in changing how existing leaderboards grade responses to reward honest uncertainty rather than adding new tests.

Core claim

Hallucinations need not be mysterious; they originate simply as errors in binary classification. If incorrect statements cannot be distinguished from facts, then hallucinations in pretrained language models will arise through natural statistical pressures. They persist due to the way most evaluations are graded: language models are optimized to be good test-takers, and guessing when uncertain improves test performance.

What carries the argument

The combination of binary classification errors during pretraining and misaligned benchmark scoring that penalizes uncertain answers.

If this is right

Changing how current benchmarks award points will steer models toward acknowledging uncertainty.
Hallucination rates will fall as a direct result of removing the incentive to guess on uncertain questions.
Leaderboards will better track trustworthy behavior once uncertain responses are no longer penalized.
The same statistical pressures that produce hallucinations will continue unless evaluation incentives change.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This view suggests similar guessing incentives may affect other model behaviors such as overconfident predictions in new domains.
Modifying dominant benchmarks could shift training practices across the field without requiring entirely new evaluation suites.
Real-world deployments might benefit if the same scoring principle is applied to user-facing interactions that currently favor complete answers.

Load-bearing premise

That hallucinations are ordinary binary classification errors and that simply changing benchmark scoring will reduce them without creating new problems.

What would settle it

Measure whether the rate of hallucinations drops on existing benchmarks after their scoring is changed to reward expressions of uncertainty instead of penalizing them.

read the original abstract

Like students facing hard exam questions, large language models sometimes guess when uncertain, producing plausible yet incorrect statements instead of admitting uncertainty. Such "hallucinations" persist even in state-of-the-art systems and undermine trust. We argue that language models hallucinate because the training and evaluation procedures reward guessing over acknowledging uncertainty, and we analyze the statistical causes of hallucinations in the modern training pipeline. Hallucinations need not be mysterious -- they originate simply as errors in binary classification. If incorrect statements cannot be distinguished from facts, then hallucinations in pretrained language models will arise through natural statistical pressures. We then argue that hallucinations persist due to the way most evaluations are graded -- language models are optimized to be good test-takers, and guessing when uncertain improves test performance. This "epidemic" of penalizing uncertain responses can only be addressed through a socio-technical mitigation: modifying the scoring of existing benchmarks that are misaligned but dominate leaderboards, rather than introducing additional hallucination evaluations. This change may steer the field toward more trustworthy AI systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper offers a useful incentive-based view of hallucinations but leaves the connection to standard pretraining objectives underspecified.

read the letter

The one thing to take away is that this paper treats hallucinations as a predictable outcome of how we train and score language models, specifically because guessing beats admitting uncertainty on most benchmarks. They make a solid case that the problem is not mysterious but follows from statistical pressures when incorrect statements are hard to distinguish from correct ones. The emphasis on evaluation procedures as the main driver is useful, and the proposal to adjust scoring on dominant benchmarks to reward uncertainty is a direct, if simple, suggestion for change. The paper does well at keeping the argument accessible and tying it to real-world deployment issues without adding unnecessary complexity. It avoids claiming new empirical results and sticks to analyzing the pipeline conceptually. The soft spots are more noticeable in the details. The binary classification framing sits uneasily with the autoregressive next-token prediction used in pretraining, which is a multi-class problem over a large vocabulary. The paper does not provide a derivation or even a sketch showing how the cross-entropy loss induces the claimed binary errors or prevents the model from learning to hedge on uncertain cases. Without any experiments or formal steps, the statistical causes remain high-level and unverified. The stress-test concern about this mismatch holds up based on the abstract and argument presented. This is the kind of paper for researchers focused on making AI systems more reliable through better evaluation design. It could be valuable in a reading group for sparking debate on incentives, though it might not change practice on its own. I would send it to peer review. The core idea is clear and relevant enough that referees could help strengthen the connection to actual training dynamics.

Referee Report

2 major / 1 minor

Summary. The paper claims that language models hallucinate because training and evaluation procedures reward guessing over acknowledging uncertainty. It argues that hallucinations originate simply as errors in binary classification when incorrect statements cannot be distinguished from facts, arising through natural statistical pressures in the modern training pipeline, and persist due to benchmark scoring that optimizes models to be good test-takers. The proposed mitigation is modifying the scoring of existing benchmarks rather than adding new hallucination evaluations.

Significance. If the central statistical explanation holds, the work offers a non-mysterious account of hallucinations grounded in standard classification principles and training dynamics, with potential to redirect evaluation practices toward more trustworthy systems. The absence of formal derivations, empirical measurements, or controlled experiments leaves the claim plausible but untested in detail.

major comments (2)

[statistical causes analysis] The core reduction in the statistical-causes section—that hallucinations originate simply as errors in binary classification—does not explicitly map onto the autoregressive next-token prediction objective used in pretraining. Standard pretraining optimizes multi-class cross-entropy over a large vocabulary; the manuscript provides no derivation showing how this induces the claimed binary fact/non-fact error rate or why low-probability mass cannot be assigned to uncertain continuations.
[socio-technical mitigation discussion] The claim that modifying benchmark scoring will address the issue without introducing new problems (e.g., new gaming behaviors or degraded performance on other metrics) is asserted but not supported by any analysis or simulation of downstream effects on training dynamics.

minor comments (1)

[Abstract] The abstract and introduction use the phrase 'binary classification' without an early clarifying footnote or equation that distinguishes it from the actual multi-class pretraining loss.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. The comments highlight areas where the manuscript can be strengthened with greater formalization and analysis. We address each major comment below and outline planned revisions.

read point-by-point responses

Referee: The core reduction in the statistical-causes section—that hallucinations originate simply as errors in binary classification—does not explicitly map onto the autoregressive next-token prediction objective used in pretraining. Standard pretraining optimizes multi-class cross-entropy over a large vocabulary; the manuscript provides no derivation showing how this induces the claimed binary fact/non-fact error rate or why low-probability mass cannot be assigned to uncertain continuations.

Authors: We agree that an explicit mapping would strengthen the argument. The binary classification framing is intended as an abstraction: under next-token prediction, when a factual continuation must be chosen from the vocabulary, the model faces an effective binary decision between the correct token(s) and plausible incorrect alternatives if their probabilities cannot be reliably distinguished. We will add a short derivation in the revised statistical-causes section showing how the multi-class cross-entropy loss, when the correct token has low probability mass relative to incorrect but high-likelihood distractors, produces the same error pattern as binary misclassification. We will also clarify why the training dynamics do not favor assigning low probability to uncertain continuations (as this would increase loss on the observed data). revision: partial
Referee: The claim that modifying benchmark scoring will address the issue without introducing new problems (e.g., new gaming behaviors or degraded performance on other metrics) is asserted but not supported by any analysis or simulation of downstream effects on training dynamics.

Authors: We acknowledge that the socio-technical mitigation proposal is currently stated at a high level without quantitative analysis of side effects. In the revision we will expand the discussion to include a qualitative analysis of potential new gaming behaviors (e.g., models learning to hedge in ways that reduce informativeness) and impacts on other metrics, drawing on prior work on benchmark misalignment. A full simulation of training dynamics is beyond the current scope but will be noted as valuable future work. revision: partial

Circularity Check

0 steps flagged

No circularity: argument rests on general statistical classification principles applied to standard LM training

full rationale

The paper frames hallucinations as arising from binary classification errors under the standard next-token cross-entropy objective and misaligned benchmark scoring. No equations, fitted parameters, or self-citations are invoked in a load-bearing way that would make any claimed prediction or cause equivalent to its own inputs by construction. The derivation therefore remains self-contained against external benchmarks of training dynamics and evaluation practices.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that language model training reduces to next-token prediction under binary correctness signals and that evaluation is dominated by accuracy-based scoring; no free parameters or new entities are introduced.

axioms (1)

domain assumption Hallucinations originate simply as errors in binary classification when incorrect statements cannot be distinguished from facts.
This premise underpins the statistical pressure argument in the abstract.

pith-pipeline@v0.9.0 · 5479 in / 1154 out tokens · 65878 ms · 2026-05-13T12:28:06.229364+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Hallucinations need not be mysterious—they originate simply as errors in binary classification. If incorrect statements cannot be distinguished from facts, then hallucinations in pretrained language models will arise through natural statistical pressures.
IndisputableMonolith.Foundation.DAlembert.Inevitability bilinear_family_forced unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We argue that language models hallucinate because the training and evaluation procedures reward guessing over acknowledging uncertainty
IndisputableMonolith.Foundation.HierarchyEmergence hierarchy_emergence_forces_phi unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

This “epidemic” of penalizing uncertain responses can only be addressed through a socio-technical mitigation: modifying the scoring of existing benchmarks

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · cited by 28 Pith papers · 2 internal anchors

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