The Cohn-Elkies implication in dimensions 8 and 24

Definitions and mathematical statement

This record is a context sentence, not an independent question. The supplied aid restores its wording as "A positive answer to Question 1 implies the complete solution to the sphere packing problem in dimensions 8 and 24." The actual adjacent Question 1 is not quoted in this record. The following states the supplied Cohn--Elkies implication and deficit identity without silently substituting for that missing question.

Let $n\geq1$, let $R>0$ be a required centre separation, and let $f$ be a real even Schwartz function on ${\mathbb{R}}^n$, with \[ \widehat f(y)=\int_{{\mathbb{R}}^n}f(x)e^{-2\pi i x\cdot y}\,{\,\mathrm{d}} x, \quad \widehat f(0)>0,\quad \widehat f(y)\geq0, \quad f(x)\leq0\text{ if }|x|\geq R. \] Such an auxiliary function gives the centre-density upper bound \[ \rho\leq f(0)/\widehat f(0) \] for all sphere packings with centre separation at least $R$, not only lattice packings. Multiplication by $\operatorname{Vol}(B_n(R/2))$ converts this to the occupied-volume density bound.

For the exact periodic identity, let the centres be \[ P=\bigcup_{i=1}^N(t_i+\Lambda),\qquad V=\operatorname{covol}(\Lambda),\qquad \rho=N/V, \] with $t_i$ distinct modulo the full-rank lattice $\Lambda$ and all distinct centre distances at least $R$. Write $\Lambda^*$ for the dual lattice and $S(y)=\sum_{i=1}^Ne^{2\pi i y\cdot t_i}$. Define nonnegative slacks \[ D_f=-\sum_{\substack{1\leq i,j\leq N,\ \lambda\in\Lambda\\ (i,j,\lambda)\ne(i,i,0)}} f(t_i-t_j+\lambda), \] where the exclusion means omit precisely the $N$ diagonal self-terms $i=j,\lambda=0$, and \[ D_{\widehat f}=\frac1V\sum_{\substack{y\in\Lambda^*\\y\ne0}} \widehat f(y)|S(y)|^2. \] Schwartz decay ensures convergence. Poisson summation gives \[ Nf(0)-D_f=\frac{N^2}{V}\widehat f(0)+D_{\widehat f}, \] and hence the exact normalized centre-density deficit \[ \frac{f(0)}{\widehat f(0)}-\rho =\frac{D_f+D_{\widehat f}}{N\widehat f(0)}. \] Equality is equivalent to both slacks vanishing; it requires primal zeros at the nonzero centre differences and $\widehat f(y)|S(y)|^2=0$ at every nonzero dual lattice point. This is complementary slackness for this certificate, not a general quantitative stability or uniqueness theorem.

The aid reports that Viazovska's dimension-eight and the CKMRV dimension-twenty-four magic functions provide sharp certificates. With the usual unimodular normalizations, $E_8$ has minimum separation $\sqrt2$ and centre density one, and the Leech lattice has minimum separation two and centre density one; the respective auxiliary ratios are one. Their attained packing densities are \[ \Delta_8=\frac{\pi^4}{384},\qquad \Delta_{24}=\frac{\pi^{12}}{12!}. \] The all-packing upper bound matches these constructions, explaining the source's optimal-density implication. The phrase "complete solution" is not expanded here to a classification or uniqueness statement absent from the supplied record.

Short English statement

The entry explains an implication rather than posing a question. A sharp Cohn--Elkies function bounds every packing, so matching the $E_8$ and Leech constructions proves their optimal densities. The periodic deficit splits exactly into nonnegative primal and dual slacks.

Sources

• Permanent UnsolvedMath ID $20001941$, exact public code catalogue entry; local registry aim-geometry-notes.json, notes[278], canonical tag question.
• AIM workshop "Sphere packings, lattices, and infinite dimensional algebra," extracted source item 1: https://aimath.org/WWN/spherepacking/spherepacking.pdf. No PDF was accessed. The actual Question 1 is described as adjacent by the aid but is not included here.
• Supplied attributions: the Cohn--Elkies auxiliary-function bound, Viazovska's dimension-eight magic function, and CKMRV magic functions for dimension twenty-four. Full bibliographic references, proposer/year metadata and original literature entries are absent.
• Research attempt 1, "The Cohn-Elkies implication in dimensions 8 and 24," labelled context_only, not_a_problem, proved and low novelty confidence; registry open and OPEN-TRIAGE. The exact field preserves the abbreviated canonical sentence, while the aid supplies its missing "A positive answer" wording.