on sphere packing in dimensions 8 and 24

Overview

This post explains the proof mechanism behind the optimal sphere packing theorems in dimensions $8$ and $24$. The dimension $8$ theorem of Viazovska proves that the $E_8$ lattice is optimal (Viazovska, 2017). The dimension $24$ theorem of Cohn, Kumar, Miller, Radchenko, and Viazovska proves that the Leech lattice is optimal and uniquely optimal among periodic packings (Cohn et al., 2017).

The common structure comes from the Cohn-Elkies linear-programming bound (Cohn & Elkies, 2003). Equality in that bound predicts a very rigid zero pattern for an auxiliary function and its Fourier transform. The main achievement of the two Annals papers is the exact construction of such auxiliary functions using modular and quasimodular forms.

Packing Constants and Normalization

A packing of unit balls in ${\mathbb{R}}^d$ is specified by a discrete set of centers

$$X\subset {\mathbb{R}}^d$$

such that

$$\Vert x-y\Vert \ge 2\text{for distinct }x,y\in X.$$

The packing is

$$\mathcal{P}=\bigcup _{x\in X}{B}_d(x,1),$$

and its density is

$$\Delta (\mathcal{P})=\underset{R\to \infty }{\mathrm{limsup}}\frac{\mathrm{Vol}(\mathcal{P}\cap B_d(0,R))}{\mathrm{Vol}(B_d(0,R))}.$$

The sphere packing constant is

$${\Delta }_d=\underset{\mathcal{P}\subset {\mathbb{R}}^d}{\sup }\Delta (\mathcal{P}).$$

Because density is invariant under uniform scaling, the proofs usually normalize the center separation rather than the ball radius. For the $E_8$ lattice one uses the normalization in which the shortest nonzero vectors have length $\sqrt{2}$; for the Leech lattice, the shortest nonzero vectors have length $2$. The corresponding densities are

$${\Delta }_8=\frac{{\pi }^4}{384},{\Delta }_{24}=\frac{{\pi }^{12}}{12!}.$$

Linear-Programming Certificate

The Cohn-Elkies method is a Fourier-analytic analogue of Delsarte linear programming. Let $f:{\mathbb{R}}^d\to \mathbb{R}$ be an admissible function, and normalize the Fourier transform by

$$\hat{f}(y)={\int }_{{\mathbb{R}}^d}f(x)e^{-2\pi i⟨x,y⟩}dx.$$

After radial symmetrization one may assume that $f(x)$ depends only on $\Vert x\Vert$.

Cohn-Elkies bound

Suppose $f$ is a radial Schwartz function satisfying

$$f(0)=\hat{f}(0)=1,$$ $$f(x)\le 0\text{for }\Vert x\Vert \ge r,$$

and

$$\hat{f}(y)\ge 0\text{for all }y\in {\mathbb{R}}^d.$$

Then every packing in ${\mathbb{R}}^d$ whose centers are at mutual distance at least $r$ has density at most

$$\mathrm{Vol}\left(B_d\left(0,\frac{r}{2}\right)\right)=\frac{{\pi }^{d/2}}{\Gamma (d/2+1)}{\left(\frac{r}{2}\right)}^d.$$

The proof is short but important. For a periodic packing, sum $f$ over all differences of sphere centers. On the physical side, every nonzero difference has length at least $r$, so the sign condition makes every off-diagonal contribution nonpositive. On the Fourier side, Poisson summation expresses the same quantity in terms of $\hat{f}$, and all Fourier-side terms are nonnegative. Comparing the diagonal term on the physical side with the zero-frequency term on the Fourier side gives the density bound.

The geometric problem is therefore converted into a construction problem: find $f$ with the required signs and with the bound equal to the density of the proposed packing.

Saturation and Forced Zeros

The deeper part begins with asking what equality would require. Suppose a lattice $\Lambda$ is expected to be optimal, and suppose the linear-programming bound is sharp for it. In the Poisson summation comparison, all inequalities must become equalities. Consequently,

$$f(x)=0\text{for every nonzero }x\in \Lambda$$

at the relevant lattice vector lengths, and similarly

$$\hat{f}(y)=0\text{for every nonzero }y\in {\Lambda }^{\ast }.$$

For a self-dual or suitably scaled self-dual lattice, the same shell structure controls both $f$ and $\hat{f}$.

This gives the reverse-engineering principle behind the proof. The auxiliary function is not guessed from a generic optimization problem. Its zeros are prescribed by the candidate packing. For $E_8$, the squared vector lengths are even positive integers,

$$2,4,6,\dots .$$

For the Leech lattice, there are no norm-$2$ vectors, so the nonzero squared lengths begin at

$$4,6,8,\dots .$$

The functions must vanish on exactly these shells, with high enough multiplicity to preserve the global sign pattern.

The nonminimal shells typically require double zeros. This is not a cosmetic condition. If $f$ is nonpositive beyond the first contact radius and vanishes at a later shell, then a simple zero would usually force a sign change. Double zeros allow $f$ to touch zero and remain on the required side.

Modular Forms as Fourier Machines

The construction becomes possible because Gaussian functions interact perfectly with the Fourier transform. For $\mathrm{Im}z>0$, the Fourier transform of a complex Gaussian has the schematic form

$$\mathcal{F}\left(e^{\pi iz\Vert x{\Vert }^2}\right)(y)=z^{-d/2}{e}^{\pi i(-1/z)\Vert y{\Vert }^2},$$

with the usual branch convention. Thus the Fourier transform acts on the parameter $z$ by the modular transformation

$$z\mapsto -\frac{1}{z}.$$

This is the conceptual bridge to modular forms. If one defines a radial function by an integral transform of the form

$$F(r)={\int }_{\mathcal{C}}\Phi (z)e^{\pi i{r}^{2}z}dz,$$

then applying the Fourier transform changes the contour and transforms $\Phi$ by $z\mapsto -1/z$. If $\Phi$ is chosen from a space of modular or quasimodular forms with the correct transformation law, the result is another explicitly controlled integral of the same kind. This is how the papers manufacture Fourier eigenfunctions.

The second modular transformation, $z\mapsto z+1$, is responsible for the zero set. The finite difference identity

$$e^{\pi i{r}^2(z+1)}-2e^{\pi i{r}^{2}z}+e^{\pi i{r}^2(z-1)}=-4{\sin }^2\left(\frac{\pi r^2}{2}\right)e^{\pi i{r}^{2}z}$$

produces the factor

$${\sin }^2\left(\frac{\pi r^2}{2}\right).$$

This factor vanishes exactly when

$$r^2\in 2\mathbb{Z}.$$

That is the same arithmetic progression in which the squared vector lengths of $E_8$ and the Leech lattice lie. The proof is therefore not merely using modular forms because the lattices are arithmetic; the modular transformations directly encode both Fourier invariance and the required lattice-shell zeros.

Dimension Eight

Viazovska constructs two radial functions $a,b:{\mathbb{R}}^8\to i\mathbb{R}$ from weakly holomorphic modular and quasimodular forms for congruence subgroups related to $\Gamma (2)$. They are arranged so that they behave well under Fourier transform and have the forced zero pattern at the $E_8$ vector lengths.

For $r>\sqrt{2}$, the two functions admit representations with the universal sine-squared factor:

$$a(r)=-4{\sin }^2\left(\frac{\pi r^2}{2}\right){\int }_0^{i\infty }{\varphi }_0\left(-\frac{1}{z}\right)z^2{e}^{\pi i{r}^{2}z}dz,$$

and

$$b(r)=-4{\sin }^2\left(\frac{\pi r^2}{2}\right){\int }_0^{i\infty }{\psi }_I(z)e^{\pi i{r}^{2}z}dz.$$

The displayed formulas are not just formal. A major analytic step is to continue these functions to all $r\ge 0$ by subtracting the polar or exponentially growing terms in the modular-form expansions. This continuation also gives the special values at $r=0$ and $r=\sqrt{2}$ needed for normalization.

The final auxiliary function is the real-valued linear combination

$$g(x)=\frac{\pi i}{8640}a(x)+\frac{i}{240\pi }b(x).$$

The coefficients are chosen so that

$$g(0)=\hat{g}(0)=1$$

and so that the first contact at $r=\sqrt{2}$ has the required behavior. For $r>\sqrt{2}$, Viazovska rewrites the sign conditions as Laplace-type integrals

$$g(r)=\frac{\pi }{2160}{\sin }^2\left(\frac{\pi r^2}{2}\right){\int }_0^{\infty }A(t)e^{-\pi r^{2}t}dt,$$

and

$$\hat{g}(r)=\frac{\pi }{2160}{\sin }^2\left(\frac{\pi r^2}{2}\right){\int }_0^{\infty }B(t)e^{-\pi r^{2}t}dt.$$

The functions $A(t)$ and $B(t)$ are explicit expressions in modular forms on the imaginary axis. The inequalities

$$A(t)<0,B(t)>0(t>0)$$

then imply

$$g(r)<0(r>\sqrt{2}),\hat{g}(r)>0(r>0),$$

with zeros only at the prescribed squared radii. The remaining verification is rigorous: Viazovska controls the $q$-expansions and uses interval arithmetic to prove the required inequalities.

At that point the Cohn-Elkies theorem gives

$${\Delta }_8\le \mathrm{Vol}\left(B_8\left(0,\frac{1}{\sqrt{2}}\right)\right)=\frac{{\pi }^4}{384},$$

while the $E_8$ packing gives equality. The same zero pattern, together with the uniqueness criterion from Cohn-Elkies, gives uniqueness among periodic packings.

Dimension Twenty-Four

The dimension $24$ proof follows the same architecture, but the modular forms and the sign analysis are more complicated. The Leech lattice has no norm-$2$ vectors, so the zero pattern begins at

$$r=2,\sqrt{6},\sqrt{8},\dots ,$$

or equivalently at

$$r=\sqrt{2k},k=2,3,\dots .$$

The authors construct two radial Fourier eigenfunctions in ${\mathbb{R}}^{24}$. The first is built from a weakly holomorphic quasimodular form of weight $-8$ and depth $2$ for ${\mathrm{SL}}_2(\mathbb{Z})$. The second is built from a weakly holomorphic modular form of weight $-10$ for $\Gamma (2)$. Their linear combination is

$$f(r)=-\frac{\pi i}{113218560}a(r)-\frac{i}{262080\pi }b(r).$$

It satisfies

$$f(0)=\hat{f}(0)=1,$$

and both $f$ and $\hat{f}$ have zeros at

$$r=\sqrt{2k},k=2,3,\dots .$$

Except for the boundary root of $f$ at $r=2$, these are double roots.

For $r>2$, the sign of $f$ is again reduced to an explicit Laplace integral:

$$f(r)={\sin }^2\left(\frac{\pi r^2}{2}\right){\int }_0^{\infty }A(t)e^{-\pi r^{2}t}dt.$$

The inequality $f(r)\le 0$ follows from a modular-form inequality on the imaginary axis. The Fourier-side inequality is subtler. One obtains an analogous formula for $\hat{f}(r)$, but the growth of the corresponding integrand at infinity causes an extra complication for

$$0<r<\sqrt{2}.$$

The paper handles this by subtracting the leading growth terms, proving that the remaining integral is nonnegative, and verifying the residual inequalities by exact finite calculations with truncated $q$-series and Sturm’s theorem.

This proves

$$f(r)\le 0(r\ge 2),\hat{f}(r)\ge 0(r\ge 0).$$

The Cohn-Elkies theorem with $r=2$ gives

$${\Delta }_{24}\le \mathrm{Vol}(B_{24}(0,1))=\frac{{\pi }^{12}}{12!},$$

which is exactly the Leech lattice density. The absence of extraneous roots for $r>2$ gives uniqueness among periodic packings.

Structural Comparison

Component	Dimension $8$	Dimension $24$
Candidate	$E_8$ lattice	Leech lattice
First contact radius	$\sqrt{2}$	$2$
Forced squared radii	$2,4,6,\dots$	$4,6,8,\dots$
Auxiliary function	$g=(\pi i/8640)a+(i/240\pi )b$	$f=-(\pi i/113218560)a-(i/262080\pi )b$
Modular input	congruence-subgroup modular and quasimodular forms	weight $-8$ quasimodular form and weight $-10$ modular form
Sign verification	$q$-expansion estimates and interval arithmetic	$q$-series truncation, rational bounds, and Sturm’s theorem
Uniqueness conclusion	unique optimal periodic packing	unique optimal periodic packing

The two proofs are best understood as exact solutions to the same interpolation problem. The desired function must know the lattice shell radii, must transform predictably under Fourier transform, and must have a global sign. The sine-squared factor supplies the shell zeros, the modular transformation $z\mapsto -1/z$ supplies Fourier control, and the final inequalities prove that the constructed function has the correct sign everywhere, not merely at the lattice shells.

Numerical Cohn-Elkies Search

The original Cohn-Elkies paper also explains why the sharp functions in dimensions $8$ and $24$ were numerically plausible before the modular-form construction was known. The numerical ansatz uses the radial Laguerre basis

$${\varphi }_k^{(d)}(r)=L_k^{d/2-1}(2\pi r^2)e^{-\pi r^2}.$$

These functions are Fourier eigenfunctions:

$$\widehat{{\varphi }_k^{(d)}}=(-1{)}^k{\varphi }_k^{(d)}.$$

Thus a trial function

$$f(r)=\sum _{k=0}^N{c}_k{\varphi }_k^{(d)}(r)$$

has Fourier transform

$$\hat{f}(r)=\sum _{k=0}^N(-1{)}^k{c}_k{\varphi }_k^{(d)}(r).$$

After normalizing $\hat{f}(0)=1$, the objective $f(0)$ and the sign constraints $f(r)\le 0$ for $r\ge R$ and $\hat{f}(r)\ge 0$ are linear conditions in the coefficients $c_k$, once the radius variable is discretized.

The finite-dimensional discretized program is implemented in content/codes/2026 Summer/cohn_elkies_lp_validation.py; its saved report is content/codes/2026 Summer/cohn_elkies_lp_results.txt. The computation is not a proof, because it checks inequalities on finite grids. It gives numerical evidence for the mechanism in Cohn-Elkies: increasing the polynomial degree drives the LP ratio toward the expected sharp value.

For dimension $8$, the degree-$24$ run gives

$$\frac{f(0)}{\hat{f}(0)}=1.0002709604,$$

so the numerical bound is within about $0.03\%$ of the $E_8$ density. For dimension $24$, the conservative degree-$26$ run gives

$$\frac{f(0)}{\hat{f}(0)}=\mathrm{1.0697641554.}$$

This is much less sharp, but it still shows the same trend. The discrepancy is also instructive: the Leech lattice case is numerically delicate, and a finite grid LP can become unstable near forced zeros. The exact modular construction is precisely what turns this numerical evidence into a rigorous global certificate.

Conceptual Consequence

The proof is not a classification of all possible local sphere arrangements. It is a global certificate. Once the auxiliary function exists, every possible packing is bounded simultaneously by the same Poisson summation inequality.

This explains why the 2003 Cohn-Elkies paper is the natural starting point. It identified the correct optimization problem and predicted sharp auxiliary functions in dimensions $8$ and $24$. The 2017 papers solve the missing analytic construction problem. Their depth lies in showing that the rigid zero pattern demanded by linear programming is not an accident of numerical optimization, but is produced exactly by modular forms.

Remaining Context

The results determine the optimal packing densities in dimensions

$$1,2,3,8,24.$$

The methods do not solve sphere packing in arbitrary dimension. They depend on an exceptional coincidence: the shell structures of $E_8$ and the Leech lattice are compatible with modular-form transformations strongly enough to produce exact Fourier-analytic certificates.

The uniqueness statements in the two papers are for periodic packings. The density optimality statements are for all packings. This distinction is important: the linear-programming bound gives the universal density upper bound, while the additional root-structure argument controls periodic equality cases.

See Also

• on Burau faithfulness for four strands: Another proof where a classical problem is settled by constructing a certificate that rules out all counterexamples at once.
• on self-avoiding walks and the honeycomb connective constant: Shares the theme of turning local geometric constraints into a global analytic identity.
• on the lower bound of Ramsey number: Another example of a global extremal result driven by a carefully engineered certificate.

References

🐻  Cohn, H. & Elkies, N. 2003. New upper bounds on sphere packings I. Annals of Mathematics 157(2), 689–714.

🐻  Cohn, H., Kumar, A., Miller, S.D., Radchenko, D. & Viazovska, M.S. 2017. The sphere packing problem in dimension 24. Annals of Mathematics 185(3), 1017–1033.

🐻  Viazovska, M.S. 2017. The sphere packing problem in dimension 8. Annals of Mathematics 185(3), 991–1015.