While the big labs are flexing with 10,000-agent swarms solving Navier-Stokes, one user just proved a new lower bound in combinatorics using OpenAI’s new Dots assistant for literally zero dollars. The agent, running on a free VM with 9 AMD Epyc cores and 10GB RAM, orchestrated a swarm of 7 GPT-6 Astra agents to improve the covering number C(24,14,4) from β‰₯ 19 to β‰₯ 20. This isn't just a toy example; the result has been formalized in Lean and passed mechanical checks by the independent Palomar registry, marking a significant milestone for accessible AI-driven mathematical research.

The Architecture of Zero-Cost Research

The user, posting as 'unexcitedneurons', leveraged the free tier of OpenAI Dots, which provides unlimited access to GPT-6 Astra agents with configurable reasoning effort (low to ultra). Unlike expensive API calls, this setup ran on a VM with 32GB storage, completely separate from the user's Codex usage limits. The swarm was organized into specialized roles: mathematical research, Lean formalization, computational search, adversarial review, and coordination. This role-based division allowed agents to check each other's work, treating single-agent outputs as merely 'plausible arguments' until verified by peers and formalized in Lean.

From Elegant Proofs to Brute Force

Initially, the swarm attempted to find elegant mathematical proofs, but the user realized a critical optimization: many subsets could be exhaustively checked by a modern computer in under a second. The agent's tendency to 'grind' for hours on trivial cases was inefficient. The user introduced a brute-forcing role, defining searches under 5 minutes as autonomous tasks. This shift from pure mathematics to hybrid computational verification was key. The swarm architecture evolved from fixed assignments to a shared task pool, where agents with warm KV caches could claim related tasks, minimizing context loss and idle time.

Verification Is the Whole Game

The most critical insight from this experiment is that AI agents are confident but often wrong. The user explicitly stated, 'I didn’t bother to trust any agent result, but only the stack of checks.' The proof required independent review and a Lean formalization with no 'sorry' placeholders. The final Lean proof spans 80 files and approximately 8,800 lines, far beyond human-readable length but mechanically verifiable. The agents even survived a catastrophic failure where their shared workspace was deleted, rebuilding their state from message logs and surviving files, demonstrating robustness in long-running agent swarms.

Key Takeaways

  • OpenAI Dots provides free, unlimited access to GPT-6 Astra agents, enabling zero-cost experimentation with agent swarms.
  • The swarm improved the covering number C(24,14,4) to β‰₯ 20, verified in Lean and registered with Palomar.
  • Role-based specialization and adversarial review are essential for filtering out confident but incorrect AI outputs.
  • Hybrid approaches combining mathematical reasoning with brute-force computational checks outperform pure LLM reasoning for certain combinatorial problems.
  • Agent swarms can recover from infrastructure failures, such as deleted workspaces, by leveraging message history.

The Bottom Line

We are entering an era where individual hackers can orchestrate mathematically significant results using free, high-reasoning AI agents. The barrier to entry for verified mathematical discovery has collapsed, provided you have the discipline to implement rigorous verification layers over the swarm's output.