Gilbert-Pollak conjecture (Steiner ratio)

Upper bound, last-listed row

| $\sqrt{3}/2 \approx 0.866$ | [GP1968] | equilateral triangle |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $0.860$* | [S2026] | Certificate layer, independently verified; same lemma set as [KHSHGW2026]. Asterisked: the source is a self-archived preprint, and the bound is conditional on the [KHSHGW2026] lemma layer. |value changed 2026-09-05

Checked or tried against material for its cited source

• Another row, not one shown above: bound row $0.5$ line 22 citing [GP1968] — the cited source could not be read at all (tried 2026-09-11). the source we could not read (NOT read — the publisher returned a bot challenge). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• Another row, not one shown above: bound row $0.577$ line 23 citing [GH1976] — the claim we checked: that the cited 1976 reference gives the lower bound 0.577 (the row's comment is empty, so the value is the whole claim). What we read supports it: the paper's theorem proves that the length of a Steiner minimal tree is at least 1/√3 times the length of a minimal spanning tree, in every dimension, and 1/√3 is 0.57735…, which the row gives to three places. This verdict does not cover the correctness of the proof. source link (read in the cited journal edition (Bull. Inst. Math. Acad. Sinica 4, no. 1, pp. 177–182), all six pages as page images, from the scan on R. L. Graham's papers page). About the reference itself, not the bound: The paper's first page (p. 177) prints its title in capitals as A REMARK ON STEINER MINIMAL TREES, that is, “A Remark on Steiner Minimal Trees”; the table's reference entry for [GH1976] gives the title as “Remarks on steiner minimal trees”. Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. Also attempted, and the source could NOT be read: the bound row $0.5$ citing [GP1968] line 22. That row is not counted as read. A row not listed here has NOT been checked.