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on Kissing number in dimension $5$ ($C_{29}$)

The statement

In geometry, the kissing number problem asks for the maximum number $\tau_n$ of unit spheres that can simultaneously touch the unit sphere in $\mathbb{R}^n$ without pairwise overlapping. The value of $\tau_n$ is only known for $n=1,2,3,4,8,24$ . [BV2008-def-known-dims]

We define

C_{29}\ :=\ \tau_5,

the kissing number in dimension $5$ .

Dimension $5$ is the first dimension in which the kissing number is not known; currently the rigorous range is

40\ \le\ \tau_5\ \le\ 44.

[MV2009-range-40-44]

One standard reformulation is in terms of spherical codes. Let $A(n,\theta)$ be the maximal size of a code on the unit sphere $S^{n-1}$ with minimal angular distance at least $\theta$ . Then the kissing number problem is equivalent to computing $A(n,\pi/3)$ . [BV2008-A-pi3]

Equivalently (inner-product form), if $A(n,s)$ is the maximal size of a spherical code $C\subset S^{n-1}$ with $\langle x,y\rangle\le s$ for all distinct $x,y\in C$ , then

A(n,1/2)\ =\ \tau_n.

[BDM2012-A-n-s-and-tau]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.