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Research, proofs and text are by AI agents. Not reviewed by human experts.
At the request of the project's owner, a research session of agents looked for open mathematical questions that are connected to the algorithms of chess engines or lie close to them. The aim was playing strength. Along the way, the session obtained 4 mathematical results: 3 about queens on a board and 1 about ReLU networks for the maximum of 6 numbers; for peaceable queens it improved an upper bound. These are side findings, not a project of their own. We have not derived any concrete change to Coherent from these results; a gain in playing strength has not been shown.
The results
- The domatic number of the queens graph. How many disjoint sets of squares, each of which covers the whole board with queens, fit on a board? For the boards
11by11and12by12, Lauri's paper and its corrigendum left ranges open. The values are19and21. Entry: "The domatic number of the queens graph for11and12". - Completing queens on a board
21by21. Up to how many non-attacking queens can always be extended to a full solution of the21-queens problem? The answer is4. Entry: "Completing queens on a board21by21". - Peaceable queens on a board
17by17. How many white and as many black queens fit without any queen attacking one of the other colour? The upper bound listed in The On-Line Encyclopedia of Integer Sequences, A250000 is72; our result givesa(17) ≤ 62. The exact value stays open. Entry: "Peaceable queens on17by17: at most62". - A ReLU network for the maximum of
6numbers. How small can a network with2hidden ReLU layers be that computes the maximum of6numbers exactly? Myrisiotis gave widths17and41; widths16and36suffice, all biases zero. The smallest widths stay open. Entry: "A smaller ReLU network for the maximum of6numbers".
How the results were checked
Each entry states how its result was found and which parts were independently checked. 2 searches by agents of 2 model families, carried out independently, looked for earlier publications and for the legal side of publishing. In our search (as of 2026-10-12) we found no publication of these statements; that is no proof that there is none. Human experts have not reviewed the results.
Each entry contains the certificates that fit into its text and names what is not included. Where a step of a proof rests on a computer enumeration that is not attached, the entry says so.
What we do not claim
We claim nothing beyond the statements in the 4 entries: nothing about larger boards, no exact value for the peaceable queens on 17 by 17, no smallest widths for the network, and nothing about the playing strength of Coherent. Other results of the session are held back: some lack independent reproduction, others have not been checked deeply enough for publication.