Hard or Just Unreached? Diagnosing the Sampling Blind Spot in Math-Reasoning Difficulty Estimation

Math and science reasoning benchmarks rely on pass@k, the fraction of sampled chains that reach gold, as the canonical per-example difficulty signal. The same signal drives RL with verifiable rewards, math data curation, synthetic curricula, and verifier training. We show this proxy has a persistent blind spot on its hardest stratum: on the eight free-form math cells we test (GSM8K and MATH across four open-weight models), 10.3-22.9% of the examples that no sampling seed solves in six tries are instead solved at matched compute by a six-chain deterministic regime. These are greedy decoding plus five cheap residual-stream perturbations applied via activation grafting, while greedy alone solves at most 6% on these math cells. Recovery scales with the additional budget, across perturbations whose mechanistic distinctness we verify across all twelve cells (cross-kind fix-set Jaccard <= 0.47 in every setup). Activation grafting is used as an intervention on internal representations, not a decoding method; we use it purely as a diagnostic and diversification tool, and our recovered items show that the pass@k= 0 % stratum is structurally identifiable in the residual stream rather than that the unmodified model reaches them under ordinary inference.

The Loss Is Not Enough: Sampling Conditions and Inductive Bias in Contrastive Representation Learning

Contrastive learning has become a leading paradigm for self-supervised representation learning, yet the conditions under which it recovers meaningful latent geometry remain incompletely understood. We develop a measure-theoretic framework formalizing the diversity condition, a support requirement on positive-pair sampling that is necessary for isometric latent recovery. We show that the standard full-support von Mises-Fisher setting implies the satisfaction of the diversity condition and as a consequence global contrastive loss minimizers recover latent geometry up to orthogonal transformation, while restricted conditionals can make non-orthogonal maps attain strictly lower asymptotic contrastive loss. We introduce a support-corrected Information Noise Contrastive Estimation (InfoNCE) variant as a theoretical fix: this correction makes orthogonal latent space recovery achievable but does not uniquely select it. Experiments on synthetic benchmarks validate the identifiability predictions, and CIFAR-10 experiments are consistent with the qualitative prediction that architectural inductive bias becomes more important when sampling diversity is limited. Together, our results clarify how sampling mechanisms and encoder inductive bias interact in contrastive representation learning.