GPT-5.6 Overturns 30-Year-Old Math Conjecture with Just 58 Words of Prompts

The Dinitz-Garg-Goemans conjecture originated from a 1999 seminal paper on single-source unsplittable flow. It posited that when converting a splittable flow to an unsplittable one while respecting capacity constraints, the total cost would not exceed that of the splittable solution. In simple terms, 'not too congested, nor more expensive.' However, the conjecture remained unproven or undisproven for nearly 30 years on general graph structures.

The counterexample constructed by GPT-5.6 Pro is a small graph with 7 nodes and 9 directed edges, containing a common source and three destinations, with commodity demands of 15, 10, and 15 respectively. Each commodity has two possible routes: one with cost 30, and another with cost 0 but requiring shared edges. The splittable flow total cost is 58; but under unsplittable flow, any two commodities taking the free route simultaneously would overload a certain edge by 1 unit, so at most one commodity can take the free route, forcing the other two to take the toll route, raising the minimum cost to 60.

Verification of the counterexample requires only exhaustive enumeration of 2^3 = 8 path combinations, of which 4 satisfy capacity constraints but have costs 90, 60, 60, 60, while the other 4 are cheaper but cause overloads. This gap directly refutes the conjecture's claim that the cost would not exceed 58.

Rybin's conversation with GPT-5.6 Pro is publicly available. The first prompt was simply 'Continue researching, find a complete, unconditional counterexample,' accompanied by the problem file. The AI then established a linear programming verification method, tried various structures, and screened thousands of instances, but failed to find a valid counterexample in the first and second rounds, even warning that 'packaging it as a counterexample would lead to wrong conclusions.'

Rybin did not provide new formulas or guidance, only repeated nudges: 'Continue researching,' 'Based on a deep understanding of the problem structure, first formulate a clear strategy,' and 'Let's wrap up with a complete, unconditional counterexample.' In the third round, the AI narrowed the search to routing structures with 24 states, but still did not produce a complete counterexample. Only in the fourth round did the AI deliver the final counterexample.

Throughout the process, the AI repeatedly found seemingly valid candidate counterexamples, but exhaustive checking revealed previously overlooked 'mixed paths'—paths that splice segments from different preset routes to create new paths that bypass capacity constraints. The AI concluded: 'Checking only a few hundred preset routes is far from enough; all possible unsplittable routes in the network must be accounted for.'

Wharton professor Ethan Mollick raised a question: Should the author of this work be Rybin, who wrote the 58 words, or GPT-5.6 Pro, which performed hours of continuous reasoning? This controversy highlights the blurring of AI's role in scientific research.

From a practical standpoint, Rybin's key contribution was recognizing that the AI's first three outputs were intermediate results and refusing to stop early. This suggests that the most effective prompt to push an AI to work harder can sometimes be as simple as telling it to 'keep digging.' In the past week, the speed at which AI finds mathematical counterexamples has begun to seem 'a bit ridiculous.'