A 1.37^(√d) Lower Bound for the Euclidean Borsuk Number

Preprint. Let b(d) be the least number of parts of strictly smaller diameter needed to partition any bounded set of positive diameter in Euclidean d-space. We prove that for every sufficiently large integer d one has b(d) ≥ 1.37 to the power √d. More precisely, the lim inf as d → ∞ of b(d) to the power 1/√d is at least (1249327/1000000) to the power √2, hence greater than 1.37. The elementary argument uses four rational endpoint calculations and exact integer rounding. This record contains the main article, a supplement with exact finite certificates, and a one-page extended abstract. Companion manuscripts: A 1.30^d Lower Bound for the Chromatic Number of Euclidean Space (https://doi.org/10.5281/zenodo.22716236) and the coefficient-rank transfer method for geometric graphs.