Three-way equivalence for LP-sharp sphere packing dimensions

Let $d$ be an integer satisfying

$$d \equiv 0 \pmod{8},\qquad d\geq 8.$$

The Cohn--Elkies linear programming (LP) bound concerns sphere packing in dimension $d$. Let $S_{d/2}(\mathrm{SL}_2(\mathbb{Z}))$ denote the space of cusp forms of weight $d/2$. An even unimodular lattice $\Lambda$ in $\mathbb{R}^d$ is called distinguished if it is either the unique even unimodular lattice of rank $d$, or the unique rootless even unimodular lattice of rank $d$, where rootless means having no vectors of squared norm $2$. An extremal Narain CFT here means a Narain conformal field theory with $U(1)^{d/2}\times U(1)^{d/2}$ symmetry and central charges $c=\bar c=d/2$ that saturates the modular bootstrap bound of Hartman--Mazáč--Rastelli.

Three-way equivalence. The following conditions are equivalent:

1. The Cohn--Elkies LP bound is sharp in dimension $d$.
2. Both

$$\dim S_{d/2}(\mathrm{SL}_2(\mathbb{Z}))\leq 1$$

and there exists a distinguished even unimodular lattice $\Lambda$ in $\mathbb{R}^d$. 3. There exists an extremal Narain CFT with $U(1)^{d/2}\times U(1)^{d/2}$ symmetry and central charge $c=\bar c=d/2$ that saturates the Hartman--Mazáč--Rastelli modular bootstrap bound.

The conjecture seeks a unified characterization of the exceptional dimensions in which the LP sphere-packing bound is sharp, linking cusp-form dimensions, uniqueness properties of even unimodular lattices, and extremal Narain conformal field theories. The paper presents the equivalence as a conjecture; no resolution is supplied in the given text.