A conditional transfer for colourings of R^n, with the conditional consequence χ(R^3) ≥ 7

Assume Theorem 2.3 (rigidity of wild character laws) of the OpenAI preprint The Euclidean plane is not five-colorable. We prove that, for every n ≥ 2 and every set Λ of positive real algebraic numbers, a Λ-avoiding k-colouring of Rn (with no regularity assumption) yields a conull Borel partition of Rn into k sets, each containing no two points at a distance in Λ. Consequently, conditionally on that theorem, χ(Rn) ≥ ⌈1/m1(Rn)⌉, where m1(Rn) is the supremum of the upper densities of measurable sets avoiding distance 1. With the published bound on m1(R3) (DeCorte, de Oliveira Filho and Vallentin) this gives χ(R3) ≥ 7 (previously known lower bound: 6, Nechushtan 2002). We also derive conditional consequences for unbounded and m-element sets of forbidden distances and for finitary fractional chromatic parameters, whose conditional novelty lies in dimensions 2 and 3. For n ≥ 4 and distances with rational squares, James Davies (arXiv:2610.12301) has independently proved, unconditionally and by a different method, that the chromatic and measurable chromatic numbers coincide; his theorems do not cover dimensions 2 and 3. The transfer in the plane is due to OpenAI; our contribution is its extension to higher dimensions, which is new for n = 3. Status. All main results are conditional on Theorem 2.3 of OpenAI's preprint (github.com/openai/math, item 158), which has not been refereed. The note was written with AI assistance (Claude, Anthropic), directed by the author, who is not a professional mathematician, and was checked by independent AI referee passes (OpenAI's ChatGPT and Google's Gemini). It has not yet been reviewed by a human expert. Corrections are very welcome.