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We model the generation of valuable mathematics as nested language generation in the limit: a verifiable formal language $F$, accessed through a membership oracle (the proof checker), contains an unknown valuable language $H \\in \\mathcal{H}$ revealed only through an adversarial enumeration of a core $C \\subseteq H$ of exact density $\\alpha$ (the literature). Every output is valuable ($\\in H$), trivial ($\\in F \\setminus H$), or a hallucination ($\\notin F$). We settle four questions. First, the verifier is not taste: the collections admitting generation with breadth are exactly those of the oracle-free model, characterized fiber-wise by Angluin's condition. Second, the verifier does buy sound coverage, covering all unseen valuable statements ","title":"Flood and Harvest: The Provable Necessity of Trivia for Generating Valuable Mathematics via the Lens of Language Generation in the Limit","url":"https://arxiv.org/abs/2606.14688","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.14688v1 Announce Type: cross \nAbstract: AI systems coupled to proof assistants now generate formal mathematics at scale, and the gap between what a checker can verify and what a mathematician would value has become the binding constraint. We model the generation of valuable mathematics as nested language generation in the limit: a verifiable formal language $F$, accessed through a membership oracle (the proof checker), contains an unknown valuable language $H \\in \\mathcal{H}$ revealed only through an adversarial enumeration of a core $C \\subseteq H$ of exact density $\\alpha$ (the literature). Every output is valuable ($\\in H$), trivial ($\\in F \\setminus H$), or a hallucination ($\\notin F$). We settle four questions. First, the verifier is not taste: the collections admitting generation with breadth are exactly those of the oracle-free model, characterized fiber-wise by Angluin's condition. 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