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REVIEW 3 major objections 5 minor 65 references

Evaluating the Unseen Capabilities: How Many Theorems Do LLMs Know?

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By estimating the knowledge an LLM does not output, KnowSum shows observed performance captures only about 20-50% of internal knowledge and can misrank models.

desk verdict A genuinely useful evaluation lens with real held-out validation at small extrapolation factors, but the headline 20–50% SKR and ranking shifts depend on an unvalidated t=100 extrapolation that the paper's own saturation argument cannot certify. read the letter →

arxiv 2506.02058 v1 pith:KJFWHFVI submitted 2025-06-01 cs.CL cs.IRcs.LGstat.APstat.ME

Xiang Li , Jiayi Xin , Qi Long , Weijie J. Su This is my paper · ORCID
classification cs.CLcs.IRcs.LGstat.APstat.ME
keywords LLMevaluationunseenknowledgeestimationGood-Turingestimatorspeciescountingcrisisseenratiodiversitymeasurement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that standard LLM evaluations systematically undercount what a model knows, because they only score what the model happens to output. It introduces KnowSum, a five-step pipeline—generate, verify, cluster, count frequencies, extrapolate—that estimates how many additional knowledge items would appear if the same model were queried many more times. Applying the pipeline to named theorems, human diseases, biomedical retrieval, and open-ended generation, the authors find that nine models typically express only 20-50% of their estimated internal knowledge. They also show that ranking models by observed knowledge alone can disagree with ranking them by estimated total knowledge, with some pairs of models swapping order once unseen knowledge is counted.

What carries the argument

The load-bearing object is the smoothed Good-Turing estimator, which predicts unseen knowledge as $\hat{N}_{\mathrm{unseen}}(t) = \sum_{s=1}^{k} h_s n_s$, where $n_s$ is the number of knowledge items observed exactly $s$ times and $h_s = -(-t)^s \, P\big(\mathrm{Bin}(k, 1/(t+1)) \ge s\big)$. Rare items—those seen once or twice—act as statistical fingerprints of the hidden mass, and truncating at a small $k$ (typically 6 to 10, chosen by cross-validation) stabilizes the estimate. The paper pairs this estimator with a modular pipeline in which generation, external verification, semantic clustering, and prevalence counting turn raw LLM answers into a frequency histogram. A proven bound, $\hat{N}_{\mathrm{unseen}}(t) \le e^{kt/(t+1)} N_{\mathrm{seen}}$, relates the unseen estimate to the observed count and motivates the seen-knowledge ratio as a dimensionless summary of knowledge exposure.

What would settle it

Take a domain with a complete, known list of items (e.g., all 24,762 mathematical concepts in the paper's reference databases), run the KnowSum pipeline on a model, then actually perform about 100 times more queries; if the number of new items that surface in those extra queries differs substantially from $\hat{N}_{\mathrm{unseen}}(100)$, the total-knowledge claim fails.

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Extended reading notes

Core claim

The central claim is that an LLM's observable output is a small sample of a much larger internal knowledge store, and that the size of the hidden part can be inferred statistically. KnowSum models repeated queries as draws from a prevalence distribution over knowledge items, builds a histogram of how many items were seen once, twice, and so on, and applies the smoothed Good-Turing estimator to predict how many new items would surface in $t \cdot n$ further queries. For all nine evaluated models, the estimate of total knowledge substantially exceeds the observed count, with seen-knowledge ratios (observed divided by estimated total) mostly between 0.2 and 0.5. Because the estimate depends on the full frequency histogram—not just the observed count—a model that repeats a few common items can end up with a lower estimated total than a model that emits a larger variety of rare items, which is why rankings change.

Load-bearing premise

That extrapolating to 100 times more queries, so that observed plus estimated unseen items equals the model's true total knowledge, is valid—an assumption validated only for the small extrapolation factors $t = 1, 2, 3$.

Editorial extensions

If this is right

  • Observed benchmark counts should not be read as capacity, because two models with identical observed knowledge can have very different estimated totals.
  • Ranking evaluations should be re-run with unseen-knowledge estimates, since several pairwise orderings reverse once the hidden mass is counted.
  • The framework transfers to other countable outputs: biomedical retrieval coverage and semantic diversity of open-ended generations both show seen-knowledge ratios near 20-25%.
  • The estimator adds no extra queries or compute beyond the samples already collected for evaluation, since it reuses the same frequency histogram.
  • Estimates saturate around $t = 80$, so the paper adopts $t = 100$ as a standard extrapolation factor for reporting total knowledge.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the prevalence distribution is heavy-tailed with unbounded support, the $t = 100$ number is an extrapolation statement about 100 times more draws, not a proven bound on true knowledge; the total-knowledge reading is the paper's interpretation, not an established fact.
  • The same pipeline could estimate unseen items in other countable LLM outputs—named entities, code identifiers, chemical reactions—provided a verifier and a semantic clustering rule exist.
  • The reported ranking flips suggest a practical evaluation recipe: sample far more than is typical, or explicitly model the unseen mass, before declaring one model more knowledgeable than another.
  • A direct check of the $t = 100$ assumption is possible in closed domains with known reference lists by generating 100 times more samples and comparing the number of new items actually found with the estimate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces KnowSum, a five-step pipeline (generation, verification, clustering, prevalence estimation, and smoothed Good–Turing extrapolation) for estimating the number of knowledge items an LLM could produce but has not yet produced under repeated sampling. The central claims are that LLMs typically express only 20–50% of their estimated internal knowledge and that accounting for unseen knowledge changes comparative rankings of models (Figure 1, Table 1). The method is validated by held-out experiments for extrapolation factors t = 1, 2, 3, and is applied to three tasks: counting theorems and diseases, BioASQ information retrieval, and open-ended diversity measurement.

Significance. If the total-knowledge interpretation of bNtot were valid, this would be a valuable contribution: it provides a principled, computationally cheap way to quantify latent knowledge, builds on a statistically well-motivated estimator (SGT) with near-optimality guarantees, and identifies a real evaluation blind spot. The held-out validation for small t is genuine and methodologically sound, and the sensitivity analyses across t, k, prompt, temperature, and clustering threshold are a strength. The ranking-reversal examples are practically consequential for model comparison. The main reservations are that the headline extrapolation factor t=100 is not validated against ground truth, and that Application 2 counts units taken from gold-standard documents rather than from model outputs.

major comments (3)
  1. [§5.1–§5.2, Eq. (1), Tables 1–2, Figure 1] The headline numbers and all SKR/ranking claims use extrapolation factor t=100, but the only ground-truth validation in §5.1 covers t=1, 2, 3. The saturation argument in §5.2 (saturation around t=80) is read off from the same estimator on the same data and cannot by itself certify that bNtot(100) equals the total knowledge encoded in the model. For a heavy-tailed prevalence distribution, bNunseen(t) estimates the number of distinct items that would appear in t·n additional draws, not the size of the support; the gap between these two quantities is exactly the zero-frequency mass that the SGT truncation extrapolates. Consequently the '20–50% expressed knowledge' statement and the ranking changes in Figure 1 are conditional on an unvalidated extrapolation. Please add validation at large t on a setting with a known support (e.g., synthetic sampling from a known prevalence distribution, or a domain with an exhaustive gold-standard list) or explicitly reframe the claims as being about a 100× extended query budget rather than total internal knowledge.
  2. [§4.2, Table 2] In both IR subtasks, the counted MeSH keywords are taken from gold-standard documents, not from the model's generated output: in document retrieval, all MeSH keywords of any retrieved gold document are credited; in question answering, all MeSH keywords of documents linked to a correctly answered question are credited. Nseen and bNunseen therefore measure how many gold-document MeSH terms are covered by the retrieved/answered questions under the chosen query distribution, which is a property of the corpus and annotation, not of the model's internal knowledge. The SGT extrapolation then estimates how many additional gold MeSH terms would be covered by more questions from the same pool. This conflates retrieval success with latent knowledge and undermines the 'latent retrieval capacity' conclusions drawn from the left two columns of Table 2.
  3. [Definition 1, §5.2, Figure 8] The claim that SKR(t) measures the proportion of 'exposed knowledge' is a direct function of the chosen extrapolation factor t, and the paper's own Figure 8 shows SKR(t) decreasing monotonically in t. The assertion that 'when both t and n are sufficiently large (e.g., t=10^2, n=10^5), the denominator ... serves as an estimate of the total knowledge encoded in the model' is not established; the 20–50% range is therefore partly an artifact of the choice t=100. The paper should either justify the total-knowledge interpretation with external evidence or present SKR(100) as a conditional measure tied to a 100× sampling budget, with the dependence on t clearly stated in the abstract and conclusion.
minor comments (5)
  1. [Appendix A, proof of Theorem 1] The proof begins by defining n as the total number of valid items, n = Σ_{s≥1} s·n_s, but then bounds bNunseen(t)/Nobs with denominator Σ_{s≥1} n_s (the number of distinct items Nseen). As written, inequality (2) does not follow unless Nobs is redefined as Nseen, and the statement that n = Σ s·n_s is then irrelevant. Please clarify the notation.
  2. [Appendix B.3.1] The subsection title 'Human Disease Oncology' should read 'Human Disease Ontology'.
  3. [Table 1] The multi-part column headers 'Theorem onlyAll math concepts' and 'Anatomical diseaseHuman diseases' appear without spacing in the rendered text; this should be fixed for readability.
  4. [Figure 7] The x-axis label 'Extrapolation t' is used for three subplots with different ranges; labeling each subplot with the corresponding robs value (or the effective t) would make the validation design easier to follow.
  5. [§4.3] The clustering threshold for the diversity task is set as the q-quantile of 10-nearest-neighbor distances aggregated across all models; this global choice can systematically affect models with different output-density profiles. The sensitivity analysis over q is appreciated, but a per-model threshold calibrated on a held-out set would be a more neutral alternative.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the SGT estimator is independently validated on held-out data, and the SKR/total-knowledge relation is an operational definition, not a fitted prediction.

full rationale

The paper's derivation chain is self-contained and non-circular. The core estimate bNtot = Nseen + bNunseen(t) is computed from Eq. (1), the smoothed Good–Turing estimator cited to external statistical literature [19, 41], not fitted to the ranking claims. Section 5.1 validates the estimator against ground truth on held-out fractions for t = 1, 2, 3, independently of the headline '20–50% SKR' numbers. The SKR is defined as Nseen / (Nseen + bNunseen(t)), so it is by construction the ratio of observed to estimated total knowledge; this is an operational definition rather than a circular derivation, and the estimator itself is not derived from SKR or from the ranking results. The only co-authored citation [59] is a related-work mention about creativity testing and is not load-bearing. The concern that t = 100 extrapolates far beyond the validated t = 3 regime is a correctness/validity limitation, not circularity, and the paper discloses the saturation argument as an internal property of the estimator. No step reduces to its own input by construction, so the circularity score is 0.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The framework relies on no new physical or conceptual entities. Its free parameters are experimental and estimation hyperparameters: t, k, clustering thresholds, verification criteria, and sampling settings. The key modeling axioms are the i.i.d. sampling assumption and the saturation/finite-support assumption that lets the authors call the t=100 estimate 'total knowledge'.

free parameters (6)
  • Extrapolation factor t = 100
    Set to 100 as default because Ntot saturates around t=80 in Figure 8; all headline estimates and SKR values depend on this choice.
  • Truncation level k = 6, 8, or 10 per model and application (Table 5)
    Selected by cross-validation from {6,8,10}; controls how many frequency counts enter the estimator.
  • Clustering threshold q for diversity = 0.5 main; 0.2, 0.3, 0.7 for robustness
    For diversity measurement, responses are merged when embedding distance is below the q-quantile of 10-nearest-neighbor distances; directly determines Nseen and Ntot.
  • Sampling temperature and nucleus sampling = 1, disabled
    Chosen in Section 5.2 because they maximize Nseen and Ntot; affects the exposure of rare knowledge.
  • Verification criteria (strict vs relaxed) = Strict: 'theorem' word or anatomical disease; relaxed: any of 12 math terms or any DOID disease
    These choices change observed counts and therefore the estimated unseen totals and rankings.
  • Fuzzy matching threshold for disease verification = 0.9 (rapidfuzz similarity)
    Determines which generated disease names pass verification and affects the observed frequency histogram.
assumptions (4)
  • domain assumption LLM outputs are i.i.d. draws from a fixed output distribution
    Section 3.1/3.2 treats repeated prompting as a sampling process analogous to drawing from a Poisson or multinomial distribution; this is required for Good-Turing estimation.
  • standard math Each knowledge item emerges according to its own Poisson process
    Section 3.2 states 'if each knowledge item emerges according to its own Poisson process'; this is the classical model underlying the smoothed Good-Turing estimator.
  • domain assumption External databases (Wikipedia, MathSciNet, ProofWiki, Disease Ontology, MeSH) provide a complete and correct ground truth for verification
    Section 4.1 and 4.2 use these databases to validate generated items; if they are incomplete or contain errors, observed counts and unseen estimates are biased.
  • ad hoc to paper The total knowledge is finite or saturates by t=100
    Definition 1 and the default choice of t=100 assume that Nseen + Nunseen(100) approximates the total knowledge encoded in the model; this is not proven and is questionable if the prevalence distribution is heavy-tailed with unbounded support.

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Pith. "Pith review of Evaluating the Unseen Capabilities: How Many Theorems Do LLMs Know?." pith.science (2026). https://pith.science/paper/KJFWHFVI

@misc{pith2026250602058,
  author       = {Pith},
  title        = {Pith review of: Evaluating the Unseen Capabilities: How Many Theorems Do LLMs Know?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJFWHFVI}},
  note         = {Machine review of arXiv:2506.02058}
}
read the original abstract

Accurate evaluation of large language models (LLMs) is crucial for understanding their capabilities and guiding their development. However, current evaluations often inconsistently reflect the actual capacities of these models. In this paper, we demonstrate that one of many contributing factors to this \textit{evaluation crisis} is the oversight of unseen knowledge -- information encoded by LLMs but not directly observed or not yet observed during evaluations. We introduce KnowSum, a statistical framework designed to provide a more comprehensive assessment by quantifying the unseen knowledge for a class of evaluation tasks. KnowSum estimates the unobserved portion by extrapolating from the appearance frequencies of observed knowledge instances. We demonstrate the effectiveness and utility of KnowSum across three critical applications: estimating total knowledge, evaluating information retrieval effectiveness, and measuring output diversity. Our experiments reveal that a substantial volume of knowledge is omitted when relying solely on observed LLM performance. Importantly, KnowSum yields significantly different comparative rankings for several common LLMs based on their internal knowledge.

Figures

Figures reproduced from arXiv: 2506.02058 by the authors.

Figure 1
Figure 1. Estimating the number of unseen theorems changes LLM rankings based on [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) Examples of common and rare theorems from an LLM. (b) Obtain theorem names by repeatedly prompting the LLM. (c) Comparison between observed results and our proposed approach. quantifying retrieval coverage in biomedical document retrieval and question answering, and (3) measuring semantic diversity in open-ended generation tasks. Our findings reveal two key insights: (i) LLMs typically express only 20-50% of the… view at source ↗
Figure 3
Figure 3. Schematic illustration of the unseen knowledge estimation framework. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Frequency histogram of theorems (top) and human diseases (bottom) found in LLM [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Top three most frequently generated theorems (left) and human diseases (right). The black [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Performance on selected traditional IR metrics. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: SGT estimates (colored curves) versus ground truth (black dotted lines) for theorem [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Effects of the extrapolation factor t (left two plots) and prompts (right two plots) for theorem estimation. Counterpart results for human diseases are shown in [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Effects of the truncation level k (left two plots) and sampling strategies (right two plots) for theorem estimation. Counterpart results for human diseases are shown in [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Distribution of math concepts (left) and human disease subcategories (right). These distributions are used as filtering criteria to exclude invalid knowledge items under the strict criteria setting in Application 1. B.5 Used Truncation Levels All the truncation levels…
Figure 11
Figure 11. Figure 11: SGT estimates (colored curves) versus ground truth (black dotted lines) for disease [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: Effects of the extrapolation factor t (left two plots) and prompts (right two plots) for disease estimation. B.8 Additional Results on Truncation Levels and Sampling Strategies We present additional results for human diseases in [PITH_FULL_IMAGE:figures/full_fig_p027…
Figure 13
Figure 13. Figure 13: Effects of the truncation level k (left two plots) and sampling strategies (right two plots) for disease estimation. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.