Korean neologisms arise through diverse mechanisms, a demanding test of language models beyond lexical memorization. To enable systematic evaluation, we introduce NeoTrace, an automated, rerunnable framework that collects and dates neologisms across genres. Its 4,271 entries support KoAGE, a cutoff-aware benchmark with three acquisition and two generation tasks---the latter, to our knowledge, the first to compare model coinages with attested neologisms under their original materials and community context. Across 21 models, lexical recall diverges from contextual inference: in a five-model temporal analysis, models define only 13-30% of cutoff-year words, yet can infer meaning from one usage. However, structural understanding remains limited: even correct definitions barely improve formation analysis of pre-cutoff neologisms. Given word-building materials, single models often follow human formation mechanisms, whereas multi-agent coinage favors abbreviation and depends on interaction framing. KoAGE thus separates what models know, learn from use, and coin.

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