Jason Z Wang

Jason Z Wang

The most cited calibration result in deep learning -- post-temperature-scaling ECE of 0.012 on CIFAR-100 (Guo et al., 2017) -- is below the statistical noise floor. We prove this is not a failure of the experiment but a law: the minimax rate for estimating calibration error with model error rate epsilon is Theta((Lepsilon/m)^{1/3}), and no estimator can beat it. This "verification tax" implies that as AI models improve, verifying their calibration becomes fundamentally harder -- with the same exponent in opposite directions. We establish four results that contradict standard evaluation practice: (1) self-evaluation without labels provides exactly zero information about calibration, bounded by a constant independent of compute; (2) a sharp phase transition at mepsilon approx 1 below which miscalibration is undetectable; (3) active querying eliminates the Lipschitz constant, collapsing estimation to detection; (4) verification cost grows exponentially with pipeline depth at rate L^K. We validate across five benchmarks (MMLU, TruthfulQA, ARC-Challenge, HellaSwag, WinoGrande; ~27,000 items) with 6 LLMs from 5 families (8B-405B parameters, 27 benchmark-model pairs with logprob-based confidence), 95% bootstrap CIs, and permutation tests. Self-evaluation non-significance holds in 80% of pairs. Across frontier models, 23% of pairwise comparisons are indistinguishable from noise, implying that credible calibration claims must report verification floors and prioritize active querying once gains approach benchmark resolution.

Jason Z Wang

At matched accuracy, open-weight LLMs differ substantially in the shape of their error severity distribution -- a difference invisible to the scalar error rate. Hallucination benchmarks report a single error count and treat all errors as equivalent, yet a wrong date and a fabricated court ruling differ by orders of magnitude. We introduce Errorquake-10k, a 10,000-query benchmark scoring each response on a continuous 0-4 severity scale across 8 domains and 5 difficulty tiers, and we fit per-model severity distributions for 21 open-weight models. For each model we estimate a severity distribution index (b, the Gutenberg-Richter upper-tail slope) with 95% bootstrap confidence intervals. Headline: across the 210 model pairs, 85 have disjoint 95% b confidence intervals at matched accuracy (|Delta epsilon| < 0.05) on human-consensus scoring, e.g. deepseek-v3.2 vs. ministral-14b at epsilon = 0.586 and Delta b = 0.47. A 519-item three-rater human validation study confirms measurement reliability (ICC(2,k=3) = 0.85), validates the LLM-judge ranking (rho = 0.89), and confirms the dense-model scaling correlation on human data (rho_s = -0.86). We prove a Non-Reducibility Theorem showing that severity profile and error rate are informationally non-redundant (I(b; model | epsilon) = 1.56 bits; 64.5% of cross-model b variance is unexplained by epsilon). A severity mechanism taxonomy (kappa = 0.83) reveals that error type shifts categorically with severity: low-severity errors are retrievals (71%); high-severity errors are fabrications (39%) -- and this composition differs by model size (p < 0.0001). Severity distribution should be reported alongside accuracy; it carries discriminative information that the error rate cannot.

Jason Z Wang

We give a stereological theory of LLM benchmark coverage. For any suite with effective dimensionality d_eff, the visible Hausdorff distance between two convex capability profiles consistent with the same scores is bounded by epsilon + C R m^(-1/(d_eff-1)), with matching Lipschitz lower bound. Empirically, three independent leaderboards (Open LLM v2, an extended 12-benchmark suite, LiveBench) all have d_eff in [2.86, 4.80] on their competitive frontier; the structural blind spot exceeds the observed runner-up score gap by two orders of magnitude and dominates statistical noise by 52-127x. Under a chi-squared projection model, the isotropic prior is the optimistic case; across six hidden-capability priors and four ambient dimensions the simulated half-split swap rate of the top two models stays in [0.38, 0.49], and a 500-trial random visible/held-out split shows that 92% of trials swap the top-1 ranking with on average 2.83 of 5 top-5 models changing. A submodular greedy algorithm with the Nemhauser (1 - 1/e) guarantee finds a stable core of 4 benchmarks; 7 of 12 suffice for 90% coverage, and the trained subset transfers across temporal quarters with 93-97% retention. A counterfactual validation across 12 internal benchmarks and 27 Chatbot Arena categories confirms that the eigenstructure predicts which evaluations are irreplaceable (rho = -0.69, p = 0.013 for removal disruption) and which external evaluations bring new information (rho = +0.38). As a second, independent theoretical contribution, we resolve Gardner's Problem 1.5 (1995) for C^2 support functions, establishing the minimax rate Theta(R/(kappa m^(2/(D-1)))) in general dimension via optimal recovery theory on S^(D-1).

Jason Z Wang

We introduce MIRROR, a benchmark comprising eight experiments across four metacognitive levels that evaluates whether large language models can use self-knowledge to make better decisions. We evaluate 16 models from 8 labs across approximately 250,000 evaluation instances using five independent behavioral measurement channels. Core experiments are run across the full model roster; experiments with specialized infrastructure requirements report explicitly marked model subsets. We find two phenomena with direct implications for agentic deployment: (1) compositional self-prediction fails universally -- the Compositional Calibration Error ranges from 0.500 to 0.943 on the original 15-model Exp3-v1 set (and 0.434 to 0.758 on the balanced 16-model Exp3-v2 expansion), indicating that models cannot predict their own performance on multi-domain tasks, and (2) models exhibit above-chance but imperfect domain-specific self-knowledge yet systematically fail to translate even this partial awareness into appropriate agentic action-selection -- external metacognitive control reduces the Confident Failure Rate from 0.600 to 0.143 (76% reduction at temperature 0; mean 70% at temperature 0.7 across 5 models from 4 labs). Providing models with their own calibration scores produces no significant improvement (p > 0.05); only architectural constraint is effective. This suggests that external metacognitive scaffolding -- not improved self-knowledge -- is the path to safer autonomous AI systems. Code, data, and Croissant metadata will be released publicly with the benchmark.