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on Sphere packing density in $\mathbb{R}^4$ ($C_{36}$)

The statement

$C_{36}=\Delta_4$ is the (optimal) sphere packing density in $\mathbb{R}^4$ , i.e. the largest fraction of $\mathbb{R}^4$ that can be covered by congruent balls with disjoint interiors. [CE2003-pack-problem] [CE2003-def-density] [CE2003-greatest-density]

More precisely, for a packing $\mathcal{P}$ in $\mathbb{R}^4$ , let $P$ denote the union of all balls in the packing, and let $B(p,R)$ denote a (Euclidean) ball of radius $R$ centered at $p$ . The (upper) density of $\mathcal{P}$ is

\Delta(\mathcal{P}) := \limsup_{R\to\infty}\sup_{p\in\mathbb{R}^4}\frac{\operatorname{vol}(P\cap B(p,R))}{\operatorname{vol}(B(p,R))}.

[CE2003-upper-density]

Then the sphere packing density in $\mathbb{R}^4$ is

\Delta_4:=\sup_{\mathcal{P}\subset\mathbb{R}^4}\Delta(\mathcal{P}),

the greatest packing density in $\mathbb{R}^4$ .

It is often convenient to work with the center density $\delta_4$ , defined (for packings of unit spheres) by

\delta_4=\frac{\Delta_4}{\operatorname{vol}(B)},

where $B$ is a unit ball in $\mathbb{R}^4$ . [dLOV2014-center-density]

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.