How should we assess whether large language models can perform mathematical invention? I argue that this question is currently underspecified: mathematical creativity is not one capacity but several mechanistically distinct modes of meaning-making - reflexive introspection on mathematical practice, analogical import from the sciences, problem-driven construction, and the bridging of distant domains - together with a further, cross-cutting distinction between meaning pursued because a pattern was observed and meaning pursued because it is strategically wanted, a distinction I develop through the case of conjecture-formation. These mechanisms are likely non-substitutable, so that competence in one does not transfer to the others. Grounding each in a historical case study and in an architecture-level account of current transformer-based systems, I suggest that today's models concentrate their competence in modes shaped by recombination and search over existing building blocks; if that description holds, the remaining modes are out of reach in principle, not just slower - though whether it holds is itself the open, empirical part. Because proof is getting cheaper as AI improves at generating it - a shift the field's own leading voices are now diagnosing - mathematical value is migrating toward the modes current systems cannot yet perform, and evaluations of AI mathematical ability should be organized around this taxonomy rather than around aggregate benchmarks that conflate it.
Large language models produce outputs presented as discoveries - new proofs, conjectures, or molecules. Whether such an output that appears creative is truly original and effective is hard to establish: open-ended outputs require subjective judgment, the output may replicate something seen in training, or the task may be too simple to need creativity. We present ALPS (Austin-Law Proof-Synthesis), a benchmark that designs a task to measure valid creativity: producing a solution that is original and can be proven correct. Each instance is a single equational law, certified to require either the construction of an infinite mathematical structure satisfying the law, or a proof that no such structure exists. Submissions are verified by automated proof checking with no human involvement, and a public generator produces new instances without limit, so LLMs are never evaluated on problems they may have seen. A portfolio of eight configurations of leading automated provers resolves 2.2% of the 4,141-law evaluation pool, and a twentyfold budget increase adds 0.6%: the obstacle is not compute, but the absence of any method that produces the tailored structure each law requires. Under a fixed protocol, the strongest reasoning model we test succeeds in 14% of instances on the proof side, but none on the construction side. The remaining 97.2% of the pool is unresolved at every configuration and budget we test. We release ALPS in full: the corpus, the generator, and the automated judge.
"Integrative" solutions are widely praised but rarely defined: we lack an operational way to tell a genuine integration -- one that makes the world cheaper to describe -- from a tidy re-description. Building on the lineage that treats creativity and intelligence as compression, we give such a criterion for creative integration (CI): the resolution of a real conflict between A and B is CI if and only if, under a fixed description language, the description length strictly shrinks (C = L_pre/L_post > 1), with the reduction located in the conflict itself. We make the judgment decidable through four binary, conjunctive gates, and we fix its extension through a taxonomy of pseudo-integration that names and rejects the look-alikes. We back the criterion with a curated, multi-domain corpus and -- crucially -- validate it not by human inter-rater agreement but by four falsifiable tests it could fail: an independent computational check, discrimination against hard negatives, out-of-sample prediction, and description-language robustness; all pass with margin. The contribution is not "creativity is compression" but its decidability, discrimination, and corpus: on this account, what makes a move genuinely creative -- rather than merely novel -- is that it compresses a conflict, with novelty and value as downstream symptoms; whether all creativity is so constituted we state as an explicit conjecture. We claim only the sign of C-1; we judge, not generate. The result is a citable primitive for a broader program.