cs.LGApr 27, 2026

Incompressible Knowledge Probes: Estimating Black-Box LLM Parameter Counts via Factual Capacity

Authors: Bojie Li

Organizations: Pine AI

Abstract

Closed-source frontier labs do not disclose parameter counts. Storing F facts requires at least F/(bits per parameter) weights, so factual recall lower-bounds parameter count--an intrinsic, serving-independent signal, though (as we show) a coarse one. We introduce Incompressible Knowledge Probes (IKPs), a benchmark of 1,400 factual questions spanning 7 tiers of obscurity, designed to isolate knowledge that cannot be derived by reasoning or compressed by architectural improvements. We score with no hallucination penalty (lambda = 0: IKP accuracy is simply the fraction of probed facts answered correctly), which removes both the penalty hyperparameter and the per-tier flooring choice; a full lambda x flooring ablation shows the calibration is robust across scoring choices while individual estimates are not, motivating the no-penalty default. We calibrate a log-linear mapping from IKP accuracy to parameter count on 93 open-weight models (135M-1,600B) spanning 19 vendors, achieving R^2 = 0.910; leave-one-out cross-validation confirms generalization (median fold error 1.48x, 72% within 2x, 86% within 3x). The instrument is deliberately coarse--its 90% prediction interval spans ~3x in either direction, wider than inference economics--so IKP recovers order-of-magnitude effective capacity and relative rankings, not precise parameter counts. For Mixture-of-Experts models, total parameters predict knowledge (R^2 = 0.67) better than active parameters (R^2 = 0.41). We evaluate 201 models from 27 vendors on a curated probe set (1,311 of 1,400 probes surviving name-collision and label-ambiguity filters) and report effective knowledge capacity for all major proprietary frontier models as prediction bands rather than point estimates; for heavily safety-tuned models these are lower bounds, since refusal policy can suppress tens of percentage points of otherwise-answerable capacity.

Explore similar work

CardsList
May 26, 2026cs.CL

Beyond Questions: Evaluating What Large Language Models (Actually) Know

Parametric knowledge in large language models (LLMs) is a cornerstone of their success, yet remains poorly understood. Existing knowledge benchmarks typically rely on predefined questions (e.g., "What is the birth date of M.L. King?"), evaluating only knowledge that benchmark designers explicitly choose to query, a problematic availability bias. In this paper, we introduce open knowledge evaluation, a new paradigm for LLM knowledge benchmarking. Instead of asking narrow questions, it evaluates models on the knowledge they choose to surface in response to open-ended elicitation prompts (e.g., "Tell me everything you know about M.L. King"). This shifts the focus from predefined answer retrieval toward characterizing the knowledge models naturally express. We instantiate this paradigm with BeQu (Beyond Questions), a benchmark of 10,000 entities paired with reference corpora for statement verification. Using BeQu, we evaluate a broad range of language models and analyze the effects of reasoning effort, model scale, prompt format, and knowledge domain. Data and leaderboard are available on this work's GitHub repository and at the benchmark's website.
Luca Giordano, Simon Razniewski
Multi-Hop Question Answering Benchmarks
AskPDF
May 28, 2026cs.LG

Inferring the Size of Large Language Models From Popular Text Memorization

The parameter counts of the most widely used large language models (LLMs) are often withheld by their developers, leaving model size -- a primary reference point for interpreting capabilities and costs -- largely undisclosed. We propose a black-box method to infer conservative lower bounds on LLM size from generated text outputs alone, requiring nothing beyond the ability to submit text fragments and observe next-token predictions. Our approach is grounded in a key observation: popular, widely-circulated texts -- such as classical literature, religious texts, and foundational documents -- are present in virtually every large-scale pretraining corpus, and how accurately a model predicts the next word across text fragments of varying length is a reliable signal of how much it has memorized them, which in turn is fundamentally limited by its total parameter count. We aggregate this memorization signal across a diverse corpus of texts and fragment lengths into a single accuracy profile vector per model, and build two complementary inference methods on top of it: a pairwise statistical test that determines which of two models is larger, and a scaling-law estimator that extracts a one-dimensional latent index from these vectors via Principal Component Analysis (PCA) to map the aggregated signal to a parameter count. Validated on a broad set of open-weight models, both methods produce accurate and reliable lower bounds. When applied to popular closed-weight models, our framework recovers internal product hierarchies and reveals a clear divergence in industry scaling strategies: while some developers yield significantly higher bounds indicative of large generational parameter growth, others operate under strict parameter ceilings, demonstrating that hidden design choices can be systematically probed even under strict API limitations.
Ivica Nikolic
MemorizationInferringScaling LawsCountingGrowth
AskPDF
Jun 3, 2026cs.AI

Knowledge Index of Noah's Ark

Knowledge benchmarks for LLMs face three issues: scaling-driven designs that do not operationalize disciplinary representativeness; flat-payment annotation that permits lazy consensus; and unaudited ranking instability under bounded test budgets. We introduce KINA, an 899-item benchmark across 261 fine-grained disciplines, with two formal results. First, we cast representativeness as a coverage-style objective over expert-elicited anchors and operationalize disciplinary representativeness through a proxy, yielding a (1-1/e) greedy approximation (Proposition 1); the guarantee applies to the proxy, not to population representativeness. Second, we prove a bonus-on-bar tournament weakly FOSD-dominates flat payment in released-review quality, with incentive-compatibility threshold B > Delta C / Delta p_min (Theorem 1). Evaluating 42 models from 13 labs, the top model, Gemini-3.1-Pro-Preview, reaches 53.17%, followed by Claude-Opus-4.6 at 49.92% and GPT-5.4 at 48.55%, leaving substantial headroom below saturation. The full leaderboard shows a tiered structure rather than a smooth total order: a small frontier tier lies above 48%, a dense strong-model tier spans roughly 38-45%, and low-performing models remain only modestly above the 10% chance baseline. Tool augmentation adds up to 5.17 points across the five tool-use evaluations, with gains varying substantially across models. We report bootstrap ranking-stability statistics to make bounded-budget variance explicit and to discourage over-interpretation of adjacent ranks.
Sheng Jin, Minghao Liu, Yunze Xiao +24
RankingRepresentation QualityPayment Networks
AskPDF