gpt 6 astra ultra attempt on Math-508

Sphere Packing in Higher Dimensions

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A packing of unit balls in ${\mathbb{R}}^d$ is a set of centres $X$ with $\|x-y\|_2\geq2$ for distinct $x,y\in X$. Its upper density is \[\overline\delta(X)=\limsup_{R\to\infty}\frac{\operatorname{vol}_d(B(0,R)\cap\bigcup_{x\in X}B(x,1))}{\operatorname{vol}_d B(0,R)}.\] Write $\Delta_d=\sup_X\overline\delta(X)$. The target is the exact value of $\Delta_d$ and an optimal packing, without restricting $X$ to a lattice. Scaling the common radius does not change density. [S1] [S2]

Statement recorded in the supplied source.

What is the densest packing of congruent spheres in $n$ dimensions for $n \geq 4$? [S1]

Scope and interpretation.

The source asks a family of dimension-by-dimension questions. Its background records dimensions 8 and 24 as solved subcases; those are not presented as new open cases here. [S1]

Residual task reported by the archive.

The embedded review $2026-08-17$ records: Resolve the remaining dimensions. [S1]

Short English statement

Determine how much of space identical nonoverlapping balls can occupy in each higher dimension, allowing arrangements more general than lattice packings.

Research attempt

Status: UNRESOLVED. We obtain explicit lattice lower bounds and work out a one-parameter Fourier certificate for an upper bound. The certificate can be optimized exactly, but already in dimension four it leaves a substantial gap. These are established elementary constructions and bounds, derived here as an attack on the precise unrestricted-packing target; no improvement of the literature is claimed.

Context and strategy (checked 1 October 2026).

The primary abstract of Viazovska's paper [S2] confirms optimality of the $E_8$ packing. Cohn--Elkies [E1] supplies the analytic upper-bound theorem used below. We test the simplest Gaussian times a quadratic polynomial, and compare its exact bound with a lattice construction. A numerical candidate without global sign conditions would not suffice.

A lattice construction valid in every dimension.

Let $v_d=\pi^{d/2}/\Gamma(d/2+1)$ be the volume of the unit ball. The packing $2\mathbb Z^d$ has density $v_d/2^d$. For $d\ge2$ improve this by taking \[ D_d=\{z\in\mathbb Z^d:\textstyle\sum_jz_j\equiv0\pmod2\}. \] The parity homomorphism is surjective, so this sublattice has index and covolume two. A nonzero vector cannot have squared norm one: such a vector has a single coordinate $\pm1$ and odd coordinate sum. The vector $(1,1,0,\ldots,0)$ shows that its least squared norm is two. Consequently $\sqrt2D_d$ has minimum distance two and covolume $2^{d/2+1}$, giving \[ \Delta_d\ge \frac{v_d}{2^{d/2+1}}. \] The density formula follows by averaging the disjoint ball interiors over a fundamental domain; boundary spheres have zero volume. For expanding balls the cells meeting the boundary contribute a relative error tending to zero, because a fundamental domain has bounded diameter. Thus this construction fits the catalogue's ball-window definition of density.

The precise analytic certificate.

Use $\widehat f(\xi)=\int_{\mathbb R^d}f(x)e^{-2\pi i x\cdot\xi}\,dx$. The Cohn--Elkies theorem implies \[ \Delta_d\le v_d\frac{f(0)}{\widehat f(0)} \] when $f$ is a real radial Schwartz function, $\widehat f\ge0$, $\widehat f(0)>0$, and $f(x)\le0$ for $|x|\ge2$. This statement applies to unrestricted packings; the theorem's passage from periodic packings to the general upper-density definition is an external input.

The periodic calculation explains the certificate. For a union of $M$ distinct translates of a lattice $L$, Poisson summation turns the double sum over pairs of translates into \[ \frac1{\det L}\sum_{y\in L^*}\widehat f(y) \left|\sum_{j=1}^M e^{2\pi i y\cdot x_j}\right|^2. \] It is at least $M^2\widehat f(0)/\det L$. On the original side, only the $M$ diagonal terms can be positive, so it is at most $Mf(0)$. Dividing and multiplying by $v_d$ gives the claimed bound for that packing. The nonnegative Fourier transform, not merely decay of $f$, justifies discarding nonzero frequencies.

An explicit certificate and its optimization.

For $a>0$ set $f_a(x)=(1-|x|^2/4)e^{-\pi a|x|^2}$. Its spatial sign condition is automatic. Differentiating the Gaussian transform, using $\widehat{|x|^2g}=-(4\pi^2)^{-1}\Delta\widehat g$, gives \[ \widehat f_a(\xi)=a^{-d/2}e^{-\pi|\xi|^2/a} \left(1-\frac{d}{8\pi a}+\frac{|\xi|^2}{4a^2}\right). \] Thus $a>d/(8\pi)$ makes it nonnegative and makes its value at zero positive. This yields \[ \Delta_d\le v_d\frac{a^{d/2}}{1-d/(8\pi a)}. \] Writing $b=d/(8\pi)$, the logarithmic derivative of $a^{d/2+1}/(a-b)$ is $(d/2+1)/a-1/(a-b)$. It vanishes precisely at $a=(d+2)/(8\pi)$. The bound diverges both as $a\downarrow b$ and as $a\to\infty$, and the derivative changes sign once, so this is the global minimum within the family. Therefore \[ \frac{v_d}{2^{d/2+1}}\le\Delta_d \le\min\left\{1,\frac{d+2}{2\Gamma(d/2+1)} \left(\frac{d+2}{8}\right)^{d/2}\right\}. \] The extra upper bound one follows directly from nonoverlap, and is useful if a displayed analytic bound exceeds one.

Adversarial checks and failure of sharpness.

In dimension four the lower bound is $\pi^2/16$, whereas the optimized upper bound is $27/32$. These are incompatible with a claim of exact determination. At $a=b$, the transform is still nonnegative but its value at zero vanishes, so that endpoint is not an admissible density certificate. The scaling of the lattice and the sign-change radius both use centres at distance two; confusing radius-one and diameter-one conventions would change the density by $2^d$.

There is also a structural obstruction to equality in this family for a lattice: $f_a$ is strictly negative at every distance greater than two, while the lattice has arbitrarily long nonzero vectors. Its spatial Poisson inequality is therefore strict. Adding numerical precision to this one-parameter optimization cannot remove the obstruction. A sharp certificate for a lattice must instead have zeros at its nonzero vector lengths, together with suitable Fourier zeros.

Remaining gap and next step.

The exact density and an unrestricted optimal packing in the remaining dimensions are unresolved here. The concrete missing step is a matching universal upper certificate (or a denser construction) beyond this Gaussian-quadratic family. A useful continuation is to impose selected lattice-shell zeros in a higher-degree radial polynomial and then prove its two sign conditions on whole half-lines. Checking signs only on a sampled mesh would leave an unjustified step. No finite numerical search or formal proof checking is claimed in this notebook.

Sources

• E1 H. Cohn and N. Elkies, New upper bounds on sphere packings I, Annals of Mathematics 157 $2003$, 689--714. Primary paper consulted 1 October 2026. https://arxiv.org/abs/math/0110009.

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 19 (GEO-002), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 M. Viazovska, The sphere packing problem in dimension 8, Ann. Math. 185 $2017$. https://arxiv.org/abs/1603.04246. Bibliographic information from the supplied record.