Zero: Programming Long-Horizon Mathematical Research Across Agents and Tools

We introduce Zero, a new framework for programming long-horizon mathematical research. Zero represents research procedures as executable workflows. A meta-agent constructs and revises these workflows, while a runtime executes them across agents and mathematical tools. Researchers can start from a problem, provide procedural knowledge, or augment an existing research system. Zero lifts an existing system's workflow into its executable representation, where the different stages can be modified, reassigned to different resources, or extended with new capabilities. Across four studies, Zero (i) proves optimality of a previously open discrete-packing instance and finds novel constructions for more than 15 related variants; (ii) improves the state of the art for the five-color almost-coloring problem of the plane, lowering the removed fraction from the previous state-of-the-art value of 3.74% to below 3.5%, with the construction validated by exact-arithmetic geometric checks; (iii) resolves two open questions in Lagrangian relaxation; and (iv) extends existing research harnesses with formal verification and open-solver backends. Together, these results suggest a shift from programming individual mathematical agents to programming the mathematical research process itself.