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Statement

Setting (p. 4, Variant 3). The chromatic number $\chi(\mathbb{R}^n)$ is the least number of colors for $\mathbb{R}^n$ with no monochromatic pair at distance $1$; the paper recalls $6\le\chi(\mathbb{R}^3)\le15$ (Nechushtan 2002, Coulson 2002).

Result (p. 4, Variant 3; Section 4.3, p. 9; Appendix D, p. 18). A three-dimensional modification of Algorithm 1 yields a formal coloring of all of $\mathbb{R}^3$ except a part covering $3.4622\%$ of it with $14$ colors, no color containing two points at unit distance. Pages 2 and 18 round the figure to $3.46\%$, and the paper says a finer discretization might improve it.

Numerical findings (p. 9). The networks found near conflict-free $15$-colorings, consistent with Coulson's bound, and no conflict-free $14$-coloring; the almost-coloring networks reached a conflict rate of about $2.3\%$ before formalization.

Status of the construction. The paper shows four of the fourteen colors of a found coloring (Figure 15, p. 18) and prints no explicit description of the formal coloring; it says (p. 2) that a paper describing this result is in preparation.

Source. Konrad Mundinger, Max Zimmer, Aldo Kiem, Christoph Spiegel and Sebastian Pokutta, Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?, Proceedings of the 42nd International Conference on Machine Learning, PMLR 267 (2025), arXiv:2501.18527, read in arXiv:2501.18527v3: Contribution 2 on p. 2, Variant 3 on p. 4, Section 4.3 on p. 9, Appendix D on p. 18. The edition read is identified on the source card.

Read depth. Claims checked: the stated value was read on the print. The construction was not checked here. Nothing here is independently reviewed.

Proof pointer

Pages 6, 9 and 18: the pipeline of Variant 1 adapted to three dimensions; the adaptation is not written out.

Dependencies

Variant 1 (Algorithm 1) of the same paper.