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

Bojie Li

arXiv:2604.24827v2cs.LGcs.AI

TL;DR

Closed-source labs do not disclose parameter counts, motivating an intrinsic estimate based on the factual knowledge models have stored. The paper introduces IKPs, calibrates factual accuracy against open-weight model size, and finds a useful but coarse estimator whose uncertainty and refusal sensitivity require prediction bands and cautious interpretation.

  • Problem

    Closed-source frontier labs do not disclose parameter counts, while inference-economics estimates carry acknowledged 2×+ uncertainty from serving factors external to the model.

  • Method

    The paper introduces IKPs to isolate incompressible factual capacity, scores accuracy with λ = 0, and calibrates it against 93 open-weight models across 19 vendors.

  • Results

    R2 = 0.910, with 72% of models predicted within 2× and 86% within 3×; total parameters predict MoE knowledge better than active parameters, R2 = 0.67 versus 0.41.

  • Takeaways & Limitations

    IKP provides order-of-magnitude effective-capacity sizing and relative ranking, while showing that factual scaling continues despite reasoning-benchmark saturation.

  • Takeaways & Limitations

    The 90% prediction interval is ∼3.20× in either direction, and heavily safety-tuned models can read smaller than their underlying weights because refusals score as ignorance.

Abstract

from arXiv · show

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.

1 Introduction

The paper introduces IKPs to measure incompressible factual capacity and invert that measurement into coarse parameter estimates for black-box models. It argues that factual capacity continues scaling even when procedural benchmarks saturate, while providing calibration, MoE findings, fingerprinting, and an open evaluation toolkit.

  • Frontier estimates: The estimate has a ∼3× prediction interval, supporting order-of-magnitude sizing and relative ranking rather than precise parameter counts for proprietary models.Proprietary frontier scores are projected onto the open-weight calibration curve as effective-size bands.
  • Incompressibility framework: IKPs isolate incompressible factual capacity from procedural capability using tiered probes accessible through black-box APIs.The framework treats factual associations as information that cannot generally be derived or compressed through procedural improvements.
  • Calibration and validation: R2 = 0.910 across 93 open-weight models from 19 vendors, with leave-one-out validation placing 72% within 2× and 86% within 3× of known size.The calibration is deliberately coarse rather than a precise parameter-count estimator.
  • Densing Law falsification: Benchmark saturation does not imply scaling has ended: the IKP time coefficient is +0.0013/month, indistinguishable from zero, while rejecting the Densing prediction of +0.0129/month at p < 10−15.Factual capacity continues to scale log-linearly with parameters across generations and vendors.
  • Frontier estimates and MoE: For MoE models, total parameters predict knowledge capacity better than active parameters, with R2 = 0.67 versus 0.41.The comparison is consistent with factual knowledge being stored across all expert weights.
  • Knowledge fingerprinting: Knowledge fingerprints combine rare-fact Jaccard overlap with hallucination similarity to distinguish weight-sharing siblings, post-training lineages, and full retrains without model weights.The paper also releases the evaluation toolkit, probe set, and results for API-based estimation.

2 Background and Related Work

Related work links IKPs to scaling laws, factual memorization, API-based model measurement, and knowledge-distillation detection. These lines of work motivate measuring rare factual knowledge while distinguishing IKP’s complementary parameter-estimation and fingerprinting goals.

  • Factual storage: Transformer factual memorization research estimates storage around 2–4 bits per parameter and distinguishes memorization from generalization.These results provide the theoretical link between parameter count and factual storage capacity.
  • Scaling and factual memorization: Prior work establishes power-law relationships among model size, compute, data, and loss, while rare-fact accuracy scales log-linearly with model size.Larger models extend further down the power-law frequency tail of web knowledge.
  • Long-tail knowledge: Long-tail knowledge loss can arise from gradient dilution, representation limitations, and knowledge overshadowing, making obscure facts useful for discrimination.Parametric memory can fail on less popular facts even when retrieval succeeds.
  • Black-box model measurement: Inference economics estimates parameter counts from throughput and pricing but carries acknowledged 2×+ uncertainty, while API fingerprinting identifies model versions without estimating unknown size.IKP addresses a complementary measurement problem using factual knowledge.
  • MoE analysis: Prior MoE work studies memory efficiency and the effects of total size, active size, expert count, and shared-expert ratios on performance.These factors inform the paper’s effective-parameter analysis.

3 Theoretical Framework

The theoretical framework defines factual knowledge as information-bearing associations that cannot be derived from other known facts, then connects their storage requirements to model size and aggregate probe accuracy. It separates factual, procedural, and linguistic parameter roles and models tier accuracy with shifted sigmoids.

  • Factual knowledge as incompressible information: An incompressible fact is an association (e, a, v) whose value cannot be derived, computed, or inferred from other known facts or language regularities.The definition targets entity, attribute, and value combinations that require explicit storage.
  • Limits of incompressibility: Factual knowledge is not perfectly incompressible because prior distributions narrow possibilities, but IKPs minimize such compressibility through probe design.An exact founding year still requires substantial information after broad temporal narrowing.
  • From facts to parameter counts: Observed accuracy constrains parameter count because factual storage has an information lower bound and empirical fact capacity scales approximately as acc ≈a·log(N)+b.The framework combines storage-capacity arguments with observed scaling of rare-fact accuracy.
  • Functional parameter roles: The model decomposes parameters as N = Nfact + Nproc + Nling, with IKPs designed primarily to probe Nfact while still requiring basic linguistic and procedural abilities.Aggregate accuracy therefore provides a lower bound on total parameter count to the extent the probes isolate factual storage.
  • Long-tail scaling: Web knowledge follows a power-law frequency distribution, so larger models memorize farther down the long tail and aggregate accuracy scales logarithmically with model size.The framework interprets this as a natural model-specific frequency cutoff.
  • Tiered probe model: Per-tier accuracy follows a logistic sigmoid, while the mean of seven shifted sigmoids approximates a straight line over a wide range of log N.This supplies the theoretical basis for the aggregate log-linear calibration.

4 Methodology

The methodology constructs empirically tiered factual probes, filters ambiguity and reasoning-solvable items, scores responses with λ = 0, and maps aggregate accuracy to parameter count through log-linear calibration.

  • Probe construction: 1,400 probes span seven difficulty tiers, with tiers assigned using landmark-model performance rather than proxy metrics alone.A probe belongs to tier k when the Tk landmark answers correctly but Tk−1 does not; non-monotonic probes are dropped.
  • Probe construction: 401 probes come from LLM-generated candidates, while corpus-grounded sampling supplies harder T3–T7 probes and Wikidata contributes 557 verified items.The remaining 97 probes are manual or supplementary additions balancing T1–T4 coverage.
  • Probe quality: Quality filters exclude computable and ambiguous questions, while discriminating attributes and collision audits reduce title and researcher-name ambiguity.The audit specifically flags same-name researchers and same-label Wikidata entities lacking sufficient disambiguation.
  • Scoring: At λ = 0, IKP accuracy is the fraction of probed facts answered correctly, with no penalty hyperparameter or per-tier flooring decision.Researcher responses retain the STRONG > WEAK distinction, while REFUSAL and WRONG both score zero.
  • Calibration: The calibration fits A = α · log10(N) + β by ordinary least squares and estimates target size by inverting that regression.A denotes aggregate accuracy and N denotes parameter count in billions.

5 Experimental Setup

The experiment evaluates a 1,400-probe benchmark across 16 domains and 201 models from 27 vendors, using open-weight models for calibration and a temperature-zero automated judge.

  • Data and models: 1,400 probes cover seven tiers and 16 domains, including 345 researcher-subfield questions.The probe set includes founding years, geography, history, science, culture, and other factual domains.
  • Data and models: 201 models from 27 vendors are accessed through the OpenRouter API and a local Ollama server.The calibration subset contains 93 open-weight models with known sizes from 135M to 1,600B parameters.
  • Evaluation: Gemini 3 Flash Preview judges responses at temperature = 0 with low reasoning effort, with an estimated 0.1–0.2% error rate.The estimate was verified by manual audit.
  • Researcher probes: Researcher probes require a computer-science subfield plus a verifiable artifact, using evidence bundles to distinguish strong from weak correctness.Evidence bundles include subfields, venues, systems, affiliations, and co-authors.

6 Results

IKP calibration links factual accuracy to effective parameter capacity, while tier analyses show where models differ and how reasoning mode and refusal policy affect scores. The results support coarse frontier estimates and reject time-based per-parameter factual improvement predicted by the Densing Law.

  • Calibration: R^2 = 0.910 links IKP accuracy to parameter count, with total MoE parameters outperforming active parameters (R^2 = 0.67 vs. 0.41).The fit uses 93 open-weight models under no-penalty scoring; total parameters are consistent with knowledge being stored across expert weights.
  • Calibration: 1.48× median leave-one-out error places 72% of models within 2× and 86% within 3× of their known size.The 90% prediction interval factor is 3.20×, so the estimator is coarse rather than precise.
  • Per-tier discrimination: T3 is the strongest single proxy for overall ranking, with slope 0.275 per decade, Spearman ρ = 0.965, and accuracy ranging from near-zero to 97%.T1 and T2 saturate earlier, while T3 provides the widest within-set discrimination.
  • Per-tier discrimination: T6 separates the frontier: GPT-5.5 reaches 67.4% and GPT-5.5-Pro 72.2%, while the leading open-weight cluster remains lower.Meaningful T6 accuracy indicates effective capacity beyond the calibration range, although T6 is a softer separator under no-penalty scoring.
  • Per-tier discrimination: T7 remains nonzero under λ = 0, reaching approximately 22–28% at the frontier, but lucky or partially informed guesses and roughly nine miscategorized probes limit interpretation.The paper treats T7 as the least-saturated tier and future headroom, not as a clean knowledge measurement or permanent ceiling.
  • Thinking mode: Thinking improves accuracy in 18 of 30 base/think pairs by a mean 2.3 percentage points, with benefits concentrated at T3–T4 and absent at T7.The largest regression is Claude 3.7 Sonnet at −11.0 percentage points; the paper attributes regressions largely to increased refusal conservatism.
  • Densing Law falsification: The IKP time coefficient is +0.0013/month and indistinguishable from zero, rejecting the Densing prediction of +0.0129/month at p < 10^-15.Controls for thinking mode and MoE architecture do not change the conclusion; newer models’ factual gains track larger size rather than improved factual storage efficiency.
  • Benchmark comparison: IKP explains more parameter-count variance than standard benchmarks on every matched subset, with the largest gap for GPQA Diamond: R^2 = 0.52 versus 0.90.Reasoning-heavy benchmarks drift over time at fixed size, whereas IKP and SimpleQA drift near zero.

7 Discussion

The discussion identifies safety tuning, calibration uncertainty, probe-quality issues, and benchmark-construction trade-offs that constrain IKP interpretation. It also reports model-generation regressions and extensive auditing needed to make long-tail probes reliable.

  • Safety tuning: 21.5 percentage points separate Sonnet 4 and Claude 3.7 Sonnet, but refusal rates suggest much of the gap reflects policy rather than factual capacity.Sonnet 4 refuses 85% of probes versus 15% for Sonnet 3.7.
  • Safety tuning: Refusal-heavy models can be systematically underestimated because no-penalty scoring treats refusal like genuine ignorance.The paper therefore interprets such estimates as lower bounds when refusal may mask retained knowledge.
  • Model behavior: 89–99% hallucination rates in some smaller or older models contrast with 3–35% rates in several newer frontier variants, revealing strong vendor and generation differences.The wrong-answer-to-non-correct-response ratio is used as a vendor fingerprint on difficult probes.
  • Model behavior: Newer models sometimes score below predecessors, with RLHF conservatism and size differences offered as competing explanations for apparent knowledge regression.For example, GPT-3.5 Turbo scores 62.2% versus 55.4% for GPT-4o-mini.
  • Calibration limits: The 90% prediction interval is approximately 3.20× in either direction, limiting IKP to coarse capacity estimation rather than precise parameter recovery.Leave-one-out results place 72% of models within 2× and 86% within 3×.
  • Calibration limits: Only two calibration anchors exceed 1T parameters, so estimates above that range face wider uncertainty than the global interval suggests.The high-end slope is effectively extrapolated from very few points.
  • Probe construction: The benchmark pipeline mitigates contamination and allocation concerns, but private probes limit direct reproducibility and uniform tier allocation leaves discriminative power concentrated in T3–T5.The paper releases methodology and scoring code while withholding the probe set.
  • Probe construction: Probe quality is constrained by miscategorized T7 items, unreliable long-tail source fields, and name or label ambiguity requiring manual auditing and correction.Nine T7 probes exceeded 25% accuracy, while some audited Wikidata properties had pass rates as low as 25%.

4. Do knowledge fingerprints persist through continued pretraining?

The paper proposes testing whether knowledge fingerprints remain useful after substantial continued pretraining. If they do, the fingerprints could support training-free provenance checks for open-weight models.

  • Persistence and use: Knowledge fingerprints survive fine-tuning and distillation, and continued-pretraining survival would make them useful for training-free provenance enforcement.The stated application is open-weight licensing enforcement.

8 Conclusion

The conclusion presents IKP as a black-box instrument for measuring effective factual knowledge capacity and characterizing which facts, people, and artifacts models internalize. It combines calibrated probing, validation, and reproducible probe-generation procedures while retaining explicit scoring and data-quality safeguards.

  • Contribution: IKP measures effective factual knowledge capacity from black-box API access using facts intended to resist reconstruction from language alone.The framework targets stored factual associations rather than compressible procedural capability.
  • Validation: R^2 = 0.910 across 93 open models from 19 vendors spanning 135M to 1,600B, with 72% predicted within 2× and 86% within 3×.The calibration uses no-penalty scoring and leave-one-out validation.
  • Results: Total parameters predict MoE knowledge better than active parameters, with R^2 = 0.67 versus 0.41, while within-family variation is explained by training and post-training differences.The paper also reports effective-capacity estimates for 97 proprietary models.
  • Interpretation: IKP suggests factual storage continues scaling even when procedural benchmarks saturate, separating compressible capability from incompressible factual capacity.The conclusion frames this as a conceptual reinterpretation of benchmark saturation.
  • Reproducibility: The toolkit, responses, scripts, and supporting data are released to reproduce the paper’s figures and tables.The paper also provides an interactive companion site.
  • Probe design: Seven tiers are calibrated to model-size ranges and checked against a landmark ladder, with contradictory probes reassigned or dropped.T7 is defined beyond current models as a ceiling probe.
  • Probe construction: The probe pipeline combines LLM-generated candidates with externally grounded Wikidata and researcher corpora, using regional, domain, citation, and landmark calibration controls.Phase A supplied 401 of 1,400 final probes, while Phase B used independently verifiable corpora.
  • Probe construction: Researcher probes require a subfield plus an associated paper, system, institution, or co-author, raising the verification bar beyond plausible subfield guessing.The format also scopes questions to computer science and stores structured researcher evidence bundles.

C Dense vs. MoE Calibration

The dense-versus-MoE calibration analysis finds that total MoE parameters are more informative than active parameters, while combined calibration remains the headline choice because MoE-only fitting does not improve validation. The appendix also documents substantial probe-audit and source-quality constraints.

  • Calibration: R^2 = 0.67 for MoE total parameters versus 0.41 for active parameters, supporting total weights as the better predictor of factual knowledge.Dense and MoE-total slopes are similar, while MoE-total fits more loosely than dense models.
  • Calibration: 1.58× versus 1.52× median fold error shows MoE-only calibration does not outperform the combined curve in leave-one-out validation.For models at least 100B, both approaches have 1.55× median fold error.
  • Frontier estimates: 18% higher top estimates result from switching to the MoE-total curve, but the shift remains within the 3.2× prediction interval.Examples include Claude Fable 5 rising from approximately 3.5T to 4.0T and GPT-5.5 from approximately 4.7T to 5.5T.
  • Probe audit: The audit taxonomy is intended as a reusable diagnostic for future Wikidata-sourced evaluation sets.The appendix assigns every problematic probe to exactly one of five buckets.
  • Probe audit: The audit identifies stale, semantically mismatched, wrong, ambiguous, disputed, and multiply valid ground-truth cases in long-tail Wikidata probes.Examples include stale headquarters, foundry-versus-sculptor attribution, generic titles, and disputed sovereignty.
  • Probe audit: Probe repairs correct answer fields, add discriminating qualifiers, drop problematic items, or accept multiple documented answers according to failure class.Replacement probes are sourced through repeated SPARQL rounds and landmark calibration.

D.3 Per-Fact-Type Reliability Ranking

Reliability varies sharply by Wikidata fact type: river mouths and inception dates are strongest, while headquarters, screenwriter, and creator fields are substantially less reliable. The evaluation therefore exposes field-specific limitations behind aggregate probe quality.

  • 100% pass rate makes river-mouth facts the most reliable Wikidata type in the audited ranking.The paper describes geography as well-curated.
  • 75–100% reliability for country fields is qualified by systematic failures on politically contested entities because Wikidata selects one claimant.
  • ∼95% reliability makes inception dates the best-curated date field, with most failures attributed to pipeline-side date selection bugs.
  • 75% reliability for founder fields is weakened when Wikidata returns a founding organization instead of a person.
  • ∼70% sculpture-creator reliability is limited because Wikidata often records the bronze foundry rather than the sculptor.
  • 25% headquarters reliability is the lowest listed rate because company moves are not propagated quickly, leaving stale locations.
  • Probe-tier behavior: T4 has the widest accuracy spread and is the best population discriminator, while T1–T2 are compressed above 98% and T6–T7 distinguish only the strongest models.

F Densing Law Falsification: Full Regression Results

The regression analysis finds no detectable time trend in IKP accuracy after accounting for parameter count, contradicting the Densing Law’s predicted monthly gain. Robustness checks support total parameters as the preferred explanatory variable, while date uncertainty does not alter the estimate in expectation.

  • The time coefficient is statistically indistinguishable from zero in every parameter-controlled specification, while the time-only model’s explanatory power collapses after log-parameters are included.Table 17 reports R2 = 0.22 for the time-only model.
  • The Densing Law predicts approximately +0.01285 IKP-accuracy points per month from the observed parameter slope.
  • The 95% bootstrap confidence interval for the time coefficient is [−0.00037, +0.00326], spanning zero and remaining well below the Densing target.
  • Robustness: Excluding distilled models or sub-1B anchors preserves near-zero time estimates, while active-parameter regression produces a misleading larger effect because MoE knowledge is stored across all experts.
  • Date uncertainty: Systematic backdating of post-2025 releases by more than three months each would be required to hide a Densing-sized effect.

G Cross-Generation and Model Family Analysis

IKP scores track changing factual knowledge across model generations and reveal internal family structure. Improvements are common but regressions appear after efficiency or safety optimization, while GPT-5 variants form distinct size tiers.

  • Most tracked model families improve across generations, but efficiency or safety optimization can produce factual-recall regressions.
  • Across six generations, GLM shows the steadiest improvement among the highlighted families.
  • GPT-5 family: GPT-5 T5 accuracy separates nano at 5–6%, mini at 25–53%, base at 60–95%, and pro at 60–94%.
  • GPT-5 family: The GPT-5, GPT-5-pro, and GPT-5-think variants score nearly identically, suggesting a shared base with different inference configurations.

H Knowledge Fingerprinting: Detailed Tables

Knowledge fingerprints compare models through correct-answer overlap, expected-overlap lift, and shared hallucinations, distinguishing shared bases, lineages, and retrains. The analysis finds strong positive controls and recurring release patterns, but sparse joint-wrong probes limit causal interpretation of cross-vendor similarities.

  • Metrics: HSS measures the rate at which two models produce the same normalized wrong answer on probes both models miss, complementing Jaccard and lift.
  • Release patterns: Lineage releases preserve many base-model rare-fact mistakes, while full retrains show statistically indistinguishable same-wrong counts from cross-vendor models.
  • Cross-vendor outliers: Baidu ERNIE 4.5-300B-A47B exceeds the cross-vendor threshold against four independent models, a pattern expected from mixtures of distilled frontier outputs.
  • Cross-vendor outliers: Llama 3.1 70B has HSS ≥0.30 against several models, consistent with broad use as a synthetic-data base but not proving direct distillation.
  • Limitations: Individual high-HSS frontier pairs remain weak evidence when joint-wrong counts are too small to reject innocent explanations such as shared web snapshots.
  • Limitations: The method cannot distinguish direct distillation from shared training data, and HSS is noisy when frontier models have few joint-wrong rare-fact probes.
  • Positive controls: Known-provenance pairs show elevated lift of 2.1–7.9 and J of 0.15–0.81 when at least 10 joint-wrong probes are available.
  • Release patterns: Shared-base variants such as GPT-5/GPT-5-pro/GPT-5-think concentrate at HSS ≥0.30, whereas full retrains can resemble different-vendor models in same-wrong counts.

I.1 USTC Hackergame — Watching a Fact Arrive Over Three Years

The Hackergame case study shows long-tail factual knowledge arriving sharply across model generations, while revealing that meta-knowledge, detailed content, refusal behavior, and reasoning access can diverge. The broader IKP benchmark turns this kind of observation into quantitative evaluation across 1,400 facts and 201 models.

  • Sharp, dateable transition: ∼19 verified 2023 challenge titles marked Claude 3.7 Sonnet as the first tested model to list real per-year Hackergame challenges.GPT-4o hallucinated a deterministic fake list in May 2024; later models inherited and extended the knowledge.
  • Meta-fact versus content: 2014 as the contest’s start year did not imply knowledge of its per-year challenges: DeepSeek V4 Pro stated the meta-fact but fabricated the detailed content.The case study treats structural knowledge and long-tail content as independently stored.
  • Measurement interactions: Refusal, hallucination, and empty answers varied by vendor, affecting how factual knowledge appears in measurement.Claude Opus 4.7 declined to fabricate, whereas GPT-4o and DeepSeek V4 Pro confidently fabricated and GPT-5/5.5 produced empty content after extended reasoning.
  • From case study to benchmark: 1,400 facts and 201 models extend the Hackergame observation into a systematic IKP evaluation with quantitative scoring.The motivating case was verified against official Hackergame writeup repositories across 12 representative models.
  • Obscurity tiers: T3 probes typically separate small models below 10B parameters from models above ∼30B, which more often answer moderately obscure scientific facts correctly.The liquid-helium probe names Heike Kamerlingh Onnes as the gold answer, while examples include refusal and hallucination.
  • Obscurity tiers: T6 probes are generally answered correctly only by the largest frontier models, while even strong models may refuse obscure researcher questions.The Janick Edinger example contrasts refusals with answers identifying distributed systems and related areas.
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