The Steiner ratio is at least 0.8564

The Steiner ratio is the infimum, over finite sets of points in the Euclidean plane, of the length of a Steiner minimal tree divided by the length of a minimum spanning tree. Gilbert and Pollak conjectured in 1968 that it equals √3/2 ≈ 0.8660, and the conjecture is open. We prove that it is at least 2141/2500 = 0.8564, improving the bound 0.8559 of Ke, Huang, Shu, He, Gai and Wang. Their method reduces the induction step of Du and Hwang’s 1983 proof of the bound 0.8 to finitely many inequalities on boxes in a space of local configurations of a Steiner minimal tree, which are checked by computer. These inequalities use estimates for the positions of terminals outside the local configuration. We prove all the geometric lemmas that our certificate uses, filling gaps in two proofs of Ke et al., and we add two such estimates that hold for more parameter values than the published ones. Within this method some new estimate is necessary: with the published estimates, no certificate of this kind reaches 0.856, whatever the boxes. The computer-assisted part is also verified in Lean 4, by a proof whose checks all run in Lean’s kernel; the rest of the argument enters the formal proof as hypotheses.

The certificate has 555,102,905 boxes; the programs check it in ball arithmetic in about 200 core-hours. It is archived, with these programs and the Lean proofs, at doi:10.5281/zenodo.23716011. The kernel-checked Lean proof covers every box of the certificate; its hypotheses, listed in Section 7 of the paper, state the rest of the argument, which the paper proves or, for the tables of Ke et al., checks by program.