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Benefit Ledger · provisional · Mathematics

Rybin, using GPT-5.6 Pro, counters the Dinitz–Garg–Goemans unsplittable-flow cost conjecture

On 22 July 2026 Dmitry Rybin announced an explicit 7-vertex graph in which a feasible fractional unsplittable-flow costs 58 while every congestion-legal unsplittable flow costs at least 60, disproving Goemans’ cost-preserving rounding conjecture. He attributes the construction to GPT-5.6 Pro. An Isabelle Archive of Formal Proofs entry later checks the finite instance.

22 Jul 2026Tier 2 NotableMethodology 0.1

Current score

+0.35

3 base · Notable (tier 2 of 5, 3 pts)
× 0.7500 attribution · Critical contribution
× 0.4000 evidence · External expert evaluation
× 0.5500 realization · Experimentally validated
× 0.7000 durability
Event-level product before credit split: 0.35

An explicit counterexample to a named 1999 cost conjecture is notable (tier 2). GPT-5.6 Pro found the instance under Rybin’s prompting (0.75). Evidence is a public announcement plus an Isabelle check of the finite graph (0.40). Realization is a reproducible construction (0.55).

What happened

Dinitz, Garg, and Goemans proved that a fractional single-source flow can be rounded to an unsplittable flow with congestion at most the largest demand. Goemans conjectured that rounding can also preserve cost. Rybin posted a 7-vertex, 9-arc instance with demands 15, 10, and 15: fractional cost 58 versus unsplittable cost ≥60 under capacity violation ≤15. The 1999 congestion theorem is unaffected. A later AFP development verifies the finite counterexample in Isabelle; that formalization used AI for proof engineering, but the mathematical discovery credit remains GPT-5.6 Pro with Rybin.

Model attribution

+0.35

GPT-5.6

Produced the explicit 7-vertex counterexample in a GPT-5.6 Pro session posted by Dmitry Rybin.

Rybin’s announcement and later writeups name GPT-5.6 Pro as the system that found the graph.

Attribution 0.7500 · Credit share 100% · OpenAI

Claims

  • An explicit graph has fractional unsplittable-flow cost 58 versus unsplittable cost at least 60, contradicting the Dinitz–Garg–Goemans cost conjecture.

    outcome · supported

Sources

primary sources

independent sources

Secondary domains: Computer Science

Revision history

  • 13 Sep 2026 · 0.00 → 0.35

    Imported events under methodology 0.1.