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machine reasoning

The more informative test of machine mathematical reasoning will come from problems that require sustained theory development rather than a single construction, and from fields with less pre-existing partial progress to lean on. Watch whether labs eventually publish results in areas without an equivalent curated database, since that would be a stronger signal that the underlying capability generalises beyond the specific structure that has made Erdős problems such convenient targets this summer.

• Deep theory-building fields such as algebraic geometry, arithmetic geometry or large parts of analysis require constructing new conceptual frameworks, not finding a single object matching a specification.
• Many resolved Erdős problems were already known to be 'probably true' from partial results or computational evidence, narrowing the search considerably before any model was involved.
• Difficulty within the Erdős catalogue varies enormously; some entries are closer to exercises than to genuinely hard open questions, and announcements rarely specify which end of that range was targeted.
• A model finding one clever construction says little about its ability to sustain a multi-year, multi-paper research programme of the kind most genuine mathematical advances require.