The Signal-Coverage Matrix: Stratifying Type and Semantic Errors in Statement Autoformalization

Headline type-correctness (TC\%) of LLM autoformalization has climbed from $\sim$53\% to $\sim$76\% in two years, yet this scalar conceals which errors each method resolves. We propose a signal-coverage matrix that crosses the Lean elaborator (pass/fail) with a semantic-equivalence judgment (equivalent/not), sorting every output into one of four cells: true success (TS), type-only (TO), semantic-only (SO), or both fail (BF). On ProofNet\# and MiniF2F-test with DeepSeek V4-Pro across Vanilla, Lean-Retry, Sample-Filter, and Stratified Autoformalization (SAF): (1) the +34 to +36 TS gain across the three elab-feedback methods is $\sim$64\% type-stratum recovery, with SO flat on net (87.5\% of original semantic errors rescued, 8 newly created). (2) The TO-to-TS rate is 23/61 for each method (Wilson 95\% CI [26.6\%, 50.3\%]), and this stratum-level recovery rate predicts $Δ$TS on held-out methods to within 2/186 and renders $Δ$TC linear in the Vanilla elab-fail rate across six (model, dataset) cells ($R^2=0.96$). (3) The two judges disagree by 26 to 37 pp on elab-feedback outputs (vs. 7 pp on Vanilla), with 30 to 56\% of symbolic-judge false negatives traceable to elaborator-forced rewrites. The persistent residual reduces to two gold-formalization errors. TC\% gains should be credited by which cell moved, not by the scalar alone.

From Knowledge to Inference: Scaling Laws of Specialized Reasoning on GlobalHealthAtlas

Public health reasoning requires population level inference grounded in scientific evidence, expert consensus, and safety constraints. However, it remains underexplored as a structured machine learning problem with limited supervised signals and benchmarks. We introduce \textbf{GlobalHealthAtlas}, a large scale multilingual dataset of 280,210 instances spanning 15 public health domains and 17 languages, stratified into three difficulty levels from health literacy to epidemiological and policy reasoning. Instances are derived from openly available public health sources and labeled by language, domain, and difficulty to support supervised learning and slice based evaluation. We further propose large language model (LLM) assisted construction and quality control pipeline with retrieval, duplication, evidence grounding checks, and label validation to improve consistency at scale. Finally, we present a domain aligned evaluator distilled from high confidence judgments of diverse LLMs to assess outputs along six dimensions: Accuracy, Reasoning, Completeness, Consensus Alignment, Terminology Norms, and Insightfulness. Together, these contributions enable reproducible training and evaluation of LLMs for safety critical public health reasoning beyond conventional QA benchmarks.