Kissing number in dimension 5

Upper bound, last-listed row

| $44$ | [[MV2009](#MV2009)] | [MV2009-range-40-44] |

exact pinned row

| $44$ | [[MV2009](#MV2009)] | <a href="#MV2009-range-40-44">[MV2009-range-40-44]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $40$ | [[KZ1873](#KZ1873)] | Achieved by the $D_5$ root system (40 points), giving a kissing configuration of size $40$ in $\mathbb{R}^5$. [CR2024-lb-40-D5] |

exact pinned row

| $40$ | [[KZ1873](#KZ1873)] | Achieved by the $D_5$ root system (40 points), giving a kissing configuration of size $40$ in $\mathbb{R}^5$. <a href="#CR2024-lb-40-D5">[CR2024-lb-40-D5]</a> |

tracked since 2026-07-24 — no movement seen yet

Checked against material for its cited source

• The row shown above: bound row $40$ line 49 citing [KZ1873] — the claim we checked: that Korkine and Zolotareff's 1873 paper contains the D5 lattice, as a quadratic form whose 40 minimal vectors give a kissing configuration in dimension 5. What we read supports it as an implicit construction: the paper's general form and its count of minimal vectors give 40 at five variables, but it prints neither the five-variable form nor the number 40. This verdict does not cover naming the form D5 or reading its minimal vectors as a kissing configuration; neither is printed in the 1873 paper, and the row's other citation, [CR2024], calls the construction implicit there. source link (read in the cited edition, as the Göttingen GDZ scan of Mathematische Annalen vol. 6: pp. 367–369, read from page images. The paper gives a quadratic form in n variables and states that its minimum has n(n−1) representations, a vector and its negative counted once. It does not write that form out for five variables, and it prints neither the number 40 nor anything about spheres. Its stated count gives 20 such pairs at n = 5, hence 40 minimal vectors; that step, and reading them as a kissing configuration, are not in this paper). 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. A row not listed here has NOT been checked.