Grokability in five inequalities

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in $\mathbb{R}^n$, sharper $L_2$-$L_1$ moment comparison inequalities on the Hamming cube $\{-1,1\}^n$, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest $g$-Sidon sets in $\{1,\dots,n\}$, and an optimal balanced Szarek's inequality.

EvA: An Evidence-First Audio Understanding Paradigm for LALMs

Large Audio Language Models (LALMs) still struggle in complex acoustic scenes because they often fail to preserve task-relevant acoustic evidence before reasoning begins. We call this failure the evidence bottleneck: state-of-the-art systems show larger deficits in evidence extraction than in downstream reasoning, suggesting that the main limitation lies in upstream perception rather than reasoning policy. To address this problem, we propose EvA (Evidence-First Audio), a dual-path architecture that combines Whisper and CED-Base through non-compressive, time-aligned fusion. EvA first aggregates intermediate CED layers to preserve multi-scale acoustic cues, then aligns the aggregated CED features to the Whisper timeline and adds the two streams without changing sequence length. We also build EvA-Perception, a large-scale open-source training set with about 54K event-ordered captions (150 h) and about 500K QA pairs. Under a unified zero-shot protocol, EvA achieves the best open-source Perception scores on MMAU, MMAR, and MMSU, and improves over Kimi-Audio-7B on all reported metrics, with the largest gains on perception-heavy splits. These results support the evidence-first hypothesis: stronger audio understanding depends on preserving acoustic evidence before reasoning.