Meta-Architectures for AI-Native Mathematical Research

Meta-Architectures for AI-Native Mathematical Research argues that as machine exploration, proof search, and formal verification become abundant, the unit of mathematical work should shift from the isolated theorem to an evolving, typed research state of conjectures, counterexamples, certificates, and abstractions, and it unifies eight meta-architectures for working in that state, from research copilot to human-AI collective, into a single Mathematical Research Operating System with trusted proof kernels inside the search loop rather than after it. The framework is backed by theorems rather than slogans: exact asymptotics showing that replicated AI verification fails precisely on the mass of shared blind spots while a single kernel check escapes that floor; a soundness-invariance result showing the verified fragment of the research state remains sound under arbitrary, even adversarial, agent policies; and compression bounds, sharpened through functorial and information-geometric formulations, showing that abstraction gain is capped by shared algorithmic information and degenerates without structural constraints. The unifying thesis is that the value of abstraction is a bounded-rationality phenomenon, since conservative abstractions prove nothing new yet compress proofs beyond any elementary bound, so the right ambition for AI in mathematics is not faster proof production but a research system that discovers reusable abstractions, maintains auditable provenance, and reserves judgments of significance for human mathematicians.