---
title: "Kalai & Vempala's theorem is silent on cross-model hallucination agreement, so Correlated Errors' convergent-fabrication finding is neither modeled nor anticipated by it"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
source_url: "https://cdn.openai.com/pdf/d04913be-3f6f-4d2b-b283-ff432ef4aaa5/why-language-models-hallucinate.pdf"
source_sha: "578cea662c00ead3841f3053c18ce544f0c2edae2ea9219a534774017f3be283"
source_title: "Why Language Models Hallucinate"
source_author: "Adam Tauman Kalai, Ofir Nachum, Santosh S. Vempala, Edwin Zhang"
source_date: "2025-09-04"
source_venue: "OpenAI technical report (cdn.openai.com); also arXiv preprint cs.CL 2509.04664"
source_quote: "Kalai and Vempala (2024) considered a special case of arbitrary facts where there is no learnable pattern in the data, like the earlier birthday hallucination example."
source_tier: 1
source_url_2: "https://arxiv.org/pdf/2311.14648"
source_sha_2: "a481ef538d58b39f023db1795b2b1fa29f41cb10076a9a5ef4078220988e2f91"
source_title_2: "Calibrated Language Models Must Hallucinate"
source_author_2: "Adam Tauman Kalai, Santosh S. Vempala"
source_date_2: "2023-11-24"
source_venue_2: "Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024); arXiv preprint cs.CL 2311.14648 (v3, 2024-03-20)"
source_tier_2: 1
note_on_source_2: "The universality clause ('for any algorithm A: X^n -> Delta(X)') and the W5 negative-correlation construction were verified directly against the extracted text at read time; they are mathematical notation, not quotable prose, so no source_quote_2 is recorded, per the quote-shape rule in sources.md."
independent_roots: 2
root_shared: "overlapping author team (Kalai and Vempala co-author both papers); distinct documents, not the same source reprinted"
provenance: "Promotion from 10-inbox/raw/2026-11-02-this-weeks-ai-safety-capture-found-llm-errors.md, 2026-11-03 (headless)"
origin: "batch"
derived_from: "10-inbox/raw/2026-11-02-this-weeks-ai-safety-capture-found-llm-errors.md"
date_created: "2026-11-03T00:00:00.000Z"
tags: ["ai-safety","hallucination","correlated-errors","algorithmic-monoculture","calibration","nlp-evaluation"]
audit_status: "capture-verified (quote and sha recorded at capture time, 2026-11-02)"
seek_code_commit: "d40ee39"
quote_sweep: "PASS 2026-11-04 — source_quote verbatim (normalized) in the capture-time archive (sha256 578cea662c00…), same-night model-free sweep"
verified_verbatim: "2026-11-04 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
---


Kalai & Vempala's bound in "Calibrated Language Models Must Hallucinate" is
stated as holding for any algorithm A: X^n → Δ(X) — a universal rate guarantee
across all possible learning algorithms, not a claim about the relationship
*between* two algorithms' outputs. The paper's object of study throughout is a
single language model's distribution over documents compared against the true
fact distribution; it contains no formalism, result, or discussion of whether
two different models, or the same model's two independent samples, would select
the *same* unobserved factoid when hallucinating. The paper's own "W5"
negative-correlation construction models anti-correlation *within* one person's
possible meals, not agreement *between* models' guesses. The later OpenAI paper
"Why Language Models Hallucinate," written by an overlapping author team and
explicitly building on this bound, names the 2024 result as covering "a special
case of arbitrary facts where there is no learnable pattern in the data, like
the earlier birthday hallucination example," and frames its own
hallucination-generation step the same single-model way, as drawing uniformly
random errors from a set.
[[claim-correlated-errors-in-llms-exceed-chance-and-rise-with-accuracy]] (Kim,
Garg, Peng & Garg, ICML 2025) is therefore not a test, confirmation, or
extension of this theorem's claims; it is an independent empirical finding, from
a different methodology — pairwise agreement rates across hundreds of real
models on real benchmarks — answering a question the theorem does not address.

> [!note] Seek's commentary: this is the capture's actual answer to its own
> governing question, and it's a "no, not that" rather than a clean yes or no.
> Kalai & Vempala predicts whether a task should have a high or low
> hallucination floor, by how singleton-heavy its fact distribution is — and,
> read through its own mechanics rather than its stated claims, leans toward
> divergence as the default shape of error on arbitrary facts specifically (see
> [[claim-kalai-vempala-2023-construction-models-arbitrary-fact-hallucination-as-uniform-random-draw]]).
> But "convergent fabrication rising with model accuracy" is a finding about
> agreement *between* models, a dimension this theorem never represents in its
> formalism at all. Citing it as this lineage's origin is accurate for the
> rate/arbitrary-vs-systematic framing both later papers inherit by name, and
> overclaiming if read as having anticipated or explained the convergence
> result itself. The 2026-10-31 decision to defer a Kalai/Vempala hub rather
> than build one on an unread paper's word was the right call — read directly,
> the paper answers a different question than the one that hook asked, and that
> gap is itself the finding. — Seek