How GPT‑5.6 Refuted a 30‑Year‑Old Math Conjecture with a 58‑Word Prompt

The Dinitz‑Garg‑Goemans conjecture, proposed in 1999, concerns single‑source unsplittable flow: it posits that one can enforce both a bounded congestion increase and a cost no higher than the optimal splittable solution.

Researcher Dmitry Rybin engaged GPT‑5.6 Pro in a four‑round dialogue, providing only four prompt statements totaling 58 English words, and asked the model to produce a complete, unconditional counterexample.

GPT‑5.6 built a linear‑program verification framework, explored thousands of small instances, and tried various structures (hypercubes, layered graphs, merge‑split networks). The first three rounds returned no valid counterexample, but each iteration narrowed the search space.

In the final round the model produced a directed graph with 7 nodes and 9 edges (see image). Three commodities have demands 15, 10, 15. Each commodity can choose a high‑cost route (cost 30) or a zero‑cost route that shares edges with others. Allowing split flow yields total cost 58, but the unsplittable requirement forces at least two commodities onto costly routes, raising the minimum feasible cost to 60, thereby violating the conjecture’s claim that both congestion and cost can be simultaneously bounded.

Enumerating the 2³ = 8 possible routing combinations shows four satisfy capacity limits with costs of 60, while the other four are cheaper but overload edges (loads 25, 30, 40 versus limits 24, 29, 39). This two‑unit cost gap suffices to refute the conjecture.

The experiment also revealed that checking only a few hundred preset routes is insufficient; a valid counterexample must consider every possible unsplittable routing in the network.

Rybin’s concise prompts guided the model through iterative refinement, demonstrating that simple, persistent prompting can enable AI to uncover substantive mathematical counterexamples.