{"paper":{"title":"Hallucination is a Consequence of Space-Optimality: A Rate-Distortion Theorem for Membership Testing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Even with perfect data and training, limited capacity forces LLMs to assign high confidence to some non-facts","cross_cats":["cs.AI","cs.CL","cs.DS","cs.IT","math.IT"],"primary_cat":"cs.LG","authors_text":"Anxin Guo, Jingwei Li","submitted_at":"2026-01-31T21:18:28Z","abstract_excerpt":"Large language models often hallucinate with high confidence on \"random facts\" that lack inferable patterns. We formalize the memorization of such facts as a membership testing problem, unifying the discrete error metrics of Bloom filters with the continuous log-loss of LLMs. By analyzing this problem in the regime where facts are sparse in the universe of plausible claims, we establish a rate-distortion theorem: the optimal memory efficiency is characterized by the minimum KL divergence between score distributions on facts and non-facts. This theoretical framework provides a distinctive expla"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"even with optimal training, perfect data, and a simplified closed world setting, the information-theoretically optimal strategy under limited capacity is not to abstain or forget, but to assign high confidence to some non-facts, resulting in hallucination.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The regime in which facts are sparse in the universe of plausible claims, together with the modeling choice that unifies discrete Bloom-filter error with continuous log-loss under a single rate-distortion objective.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Hallucinations are the space-optimal behavior for limited-capacity models performing membership testing on sparse facts, as shown by a rate-distortion theorem that equates optimal memory use to minimum KL divergence between fact and non-fact score distributions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Even with perfect data and training, limited capacity forces LLMs to assign high confidence to some non-facts","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"38afcfcfca17f0f53a74524565a3b575ad1a7c2b5c079ea23961b78a198b077c"},"source":{"id":"2602.00906","kind":"arxiv","version":7},"verdict":{"id":"d95c20da-8cad-440e-93ee-c14efa3fcfd6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T08:33:35.435664Z","strongest_claim":"even with optimal training, perfect data, and a simplified closed world setting, the information-theoretically optimal strategy under limited capacity is not to abstain or forget, but to assign high confidence to some non-facts, resulting in hallucination.","one_line_summary":"Hallucinations are the space-optimal behavior for limited-capacity models performing membership testing on sparse facts, as shown by a rate-distortion theorem that equates optimal memory use to minimum KL divergence between fact and non-fact score distributions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The regime in which facts are sparse in the universe of plausible claims, together with the modeling choice that unifies discrete Bloom-filter error with continuous log-loss under a single rate-distortion objective.","pith_extraction_headline":"Even with perfect data and training, limited capacity forces LLMs to assign high confidence to some non-facts"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.00906/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"076041c912f00d4f04be6d3068f4d74b681878e687840297e73be32065210925"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}