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

arXiv:2604.24827v1 Announce Type: cross
Abstract: Closed-source frontier labs do not disclose parameter counts, and the standard alternative — inference economics — carries $2times$+ uncertainty from hardware, batching, and serving-stack assumptions external to the model. We exploit a tighter intrinsic bound: storing $F$ facts requires at least $F/$(bits per parameter) weights, so measuring how much a model emphknows lower-bounds how many parameters it emphhas. We introduce textbfIncompressible 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 calibrate a log-linear mapping from IKP accuracy to parameter count on 89 open-weight models (135M–1,600B) spanning 19 vendors, achieving $R^2 = 0.917$; leave-one-out cross-validation confirms generalization (median fold error $1.59times$, $68.5%$ within $2times$ and $87.6%$ within $3times$). For Mixture-of-Experts models, total parameters predict knowledge ($R^2 = 0.79$) far better than active parameters ($R^2 = 0.51$). We evaluate 188 models from 27 vendors and estimate effective knowledge capacity for all major proprietary frontier models; for heavily safety-tuned models the estimates are lower bounds, since refusal policy can hide tens of percentage points of “refused but known” capacity.
The widely-reported saturation of reasoning benchmarks does not imply the end of scaling. Procedural capability compresses under the “Densing Law,” but across 96 dated open-weight models the IKP time coefficient is $-0.0010$/month (95% CI $[-0.0031, +0.0008]$) — indistinguishable from zero, and rejecting the Densing prediction of $+0.0117$/month at $p < 10^-15$. Factual capacity continues to scale log-linearly with parameters across generations and across vendors.