The session behind Quantum Advantage for Distributed Symmetry Breaking started with three prompts. About an hour after “Keep pushing”, Codex reported a complete proof candidate. The fourth prompt asked it to write up and check the result.
9 September 2026 · Helsinki time (EEST, UTC+3)
The human prompts are verbatim; Codex replies are excerpted.
Check these two papers: https://arxiv.org/abs/2608.11720 and https://arxiv.org/abs/2609.09091 - download them and any related work needed to fully understand them (download arXiv source if it's easier to read for you than pdf-to-text results), and understand the techniques used there so that you are ready for follow-up research. Feel free to use this directory /workspace to organize your notes and files as you see best.
Done—I’ve worked through both papers’ proofs and appendices and am ready for follow-up research.
See if you can combine ideas from both papers to tackle the real question: can you 3-color cycles in O(1) rounds in quantum-LOCAL w.h.p.? Do the two techniques "fit together" in some meaningful manner? Some concrete steps forward starting with the more recent paper 2609.09091 could be: 1. Does this generalize from 4-coloring to e.g. 10-coloring? 2. If yes, does it generalize from coloring to local symmetry breaking? 3. If yes, can we do teleportation stuff? 4. If teleportation requires a 2-sided bound instead of a 1-sided bound as a seed, can this be extended to 2-sided communication? Feel free to make good use of multiagent setup to study this. Avoid wasting time running low-value computational experiments.
Yes—there is a meaningful connection, but I have not resolved the -round question.
Keep pushing, as far as possible, beyond the current frontier of what is known
A concrete positive construction has emerged for a large, fixed number of colors. If it survives the remaining checks, standard color reduction would give constant-round 3-coloring.
I reached a complete positive proof candidate for the main question:
Cycles admit quantum-LOCAL 3-coloring with probability at least , for any fixed , in 8 rounds with an orientation and 21 rounds without one.
This is awesome if true. Write this up properly. Do very careful adversarial audit with multiagent setup of your claims. The claim is extremely strong and surprising and unexpected, so be sceptical, try to check it in all possible ways.