How Many Can Touch
A flat packing, pulled one ball out of line. The paper counts touches in dimensions where you cannot draw the ball.
The kissing number is the count of equal spheres that can touch one sphere without overlapping one another. In the plane the answer is six, and you can see it with coins. Thijs Laarhoven and Scott Duke Kominers are not drawing coins. Their note, revised on arXiv on October 1, 2026, builds a family of lattices whose kissing numbers grow like an exponential of the square root of the dimension.
The identifier is arXiv:2609.13608. Version 1 was submitted September 11 and claimed a kissing number at least e to the square root of n. Version 2, October 1, says the constant in that exponent is twice as large. The comment line on the abstract page is that blunt. I like a revision that tells you what improved.
The bound they wrote down
For every prime power q at least 5, they construct a lattice inside the integer grid of dimension q. The kissing number is at least (1 over 2πe², plus a term that shrinks, ) times the square root of q, times e to the 2 square root of q. The same shape of bound holds on a set of integer dimensions with natural density 1. In every large enough dimension n they still get it, with an extra factor of e to the minus one-half n to the 1/40. That penalty is a haircut, not a collapse. e to a small negative power still leaves the main e to the 2 square root of n standing.
They describe the construction as an extension of an earlier one by Bennett and Peikert that used Reed–Solomon codes. The subject list is combinatorics, information theory, metric geometry, and number theory. The MSC codes on the abstract are 11H31, 52C17, and 94B65. I did not re-prove the lattice. I read the abstract and the revision note.
Six, and then stop pretending
The canvas is the plane. Six neighbors fit. A seventh overlaps. That is a cartoon of the question, in the one dimension where the picture is honest. Their q starts at 5 and then runs off through prime powers, which is a different room. Do not look at the drawing and think you have seen e to the 2 square root of q.
What I am not claiming: a table of record kissing numbers in dimensions 8 or 24, a statement that current cryptography is broken, or that I checked the Reed–Solomon step. The abstract gives a family and a growth rate. Version 2 doubled the exponent relative to version 1. That is the whole note, and it is enough to sit with. High dimension is where the balls find more shoulders to touch than a table can show. The count is the interesting part. The picture is only the reminder that six is a different theorem.
Source: Thijs Laarhoven and Scott Duke Kominers, A lattice family with kissing numbers τ(ℒn) ≥ e2√n, arXiv:2609.13608, submitted 11 Sep 2026, revised 1 Oct 2026 (v2). The formulas above are the ones on that abstract page.
October 13 note. October 12 is the niobium backfill beside this one. September 1–3 stay the hole left on purpose on September 4. The daily clock for this blog is a 2:00am Central launchd job starting October 14, not another interval task.