Worked solution: Viazovska's modular form solution to sphere packing in dimensions 8 and 24

Step 6 of 9:

E_8

is the unique densest packing in

\mathbb{R}^8

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In plain words

With the magic function $f$ in hand, satisfying every sign condition and achieving equality exactly at the $E_8$ vectors, the Cohn–Elkies machine from Step 2 finishes the job automatically: it converts the mere existence of $f$ into a proof that nothing beats $E_8$ 's density, and — because equality can only happen when the packing's vectors are exactly the zero set of $f$ and $\hat f$ — into a proof that $E_8$ is essentially the only periodic packing that achieves it.

\Delta_8 = \frac{\pi^4}{384}

Detailed analysis

Since $f$ and $\hat f$ constructed in Steps 4–5 vanish (to double order) at exactly the nonzero vectors of $\tfrac{1}{\sqrt2}\Lambda_8$ and $\sqrt2\Lambda_8$ respectively, and $f(0)=\hat f(0)$ , the chain of inequalities from Step 3 becomes a chain of equalities, so the Cohn–Elkies bound is attained exactly by the $E_8$ -lattice packing, giving $\Delta_8 = \pi^4/384 \approx 0.2537$ . Combining this with the general uniqueness argument for Cohn–Elkies bounds (equality forces the packing's difference set to equal the zero set of $f$ ), $E_8$ is not merely a densest packing but the unique periodic packing of maximal density, up to scaling and isometry — the theorem announced in Viazovska's 2016 preprint and published in the Annals of Mathematics in 2017.

Knowledge used in this step

Convex and discrete geometry