Close and evaluate the AutoGraphForge conjecture-to-proof loop

Complete the AutoGraphForge pipeline by running proof search over the surviving graph-theoretic conjectures translated into Lean 4 and evaluating the formalization-and-proving stage, including development of a preamble-specific lemma library for custom invariants such as zero-forcing numbers.

Background

AutoGraphForge currently integrates automated conjecture generation, counterexample search, Lean 4 formalization, and neural theorem provers, but the full conjecture-to-proof loop has not been completed. The generate–refute stages have produced thousands of surviving conjectures, and the formalization-and-proving infrastructure has passed sanity checks on trivial inequalities; however, proof search has not yet been run systematically on the surviving candidates.

The unresolved task is especially significant because many candidate conjectures involve custom graph invariants, including zero-forcing variants, that are defined in the project’s own formalization preamble and are absent from the training data of the integrated provers. The authors therefore identify both the execution and controlled evaluation of proof search, and the creation of supporting lemmas for these invariants, as the principal outstanding step toward closing the loop.

References

The main open task---and our current work---is closing the loop. The decisive gap is proving: the formalization-and-proving stage is implemented and integrated (\S\ref{sec:formalize-prove}) but has not yet been run on the candidate conjectures. Our immediate next step is to run proof search over the surviving conjectures translated to Lean---together with a preamble-specific lemma library to ground the provers on the custom invariants, which we anticipate to be the main obstacle to closing the conjecture-proof loop because the provers we use were not trained on these invariants but on other parts of mathematics.

— AutoGraphForge: Towards Automated Graph Theory Discovery  (2609.03478 - Pastorek, 3 Sep 2026) in Section Discussion and future work