Cohn-Elkies LP Bound

• Cohn-Elkies Linear Programming Bound is a variational formulation using radial Schwartz functions and duality to set upper bounds for sphere packing densities.
• It produces sharp bounds in dimensions 8 and 24 via modular forms while discrete reductions reveal its limitations in other higher dimensions.
• Computational techniques like symmetry reduction and high-precision rational conversion rigorously certify these bounds, guiding further research in spectral optimization.

The Cohn-Elkies Linear Programming Bound is a foundational result in discrete geometry, providing the strongest known general upper bounds for sphere packing densities in Euclidean spaces. It converts the sphere packing problem into a variational problem on radial Schwartz functions with prescribed vanishing and positivity properties, leveraging harmonic analysis and duality theory. The method was pivotal in resolving the sphere packing problem optimally in dimensions 8 and 24, where explicit modular form constructions yield sharp bounds. However, in all other dimensions greater than two, the Cohn-Elkies bound is not sharp, and recent work rigorously quantifies its limitations using discrete reductions and dual linear programs.

1. Formulation of the Cohn-Elkies Linear Programming Bound

Let $d \geq 1$ and consider the space $\mathscr{S}(\mathbb{R}^d)$ of Schwartz-class, radial functions $f:\mathbb{R}^d \to \mathbb{R}$ with integrable, real-valued Fourier transforms

$$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$

The Cohn-Elkies (CE) linear program for upper bounding the maximal center density $\delta_d$ of sphere packings in $\mathbb{R}^d$ is

$\text{Primal (CE%%%%5%%%%):} \qquad \begin{cases} \text{minimize}\;\; (r/2)^d f(0) \ \text{subject to:} \ \quad \widehat{f}(0) \geq 1, \ \quad f(x) \leq 0 \;\text{for all}\; |x| \geq r, \ \quad \widehat{f}(y) \geq 0 \;\text{for all}\; y \in \mathbb{R}^d, \ \quad f(0) > 0, \; f \in \mathscr{S}(\mathbb{R}^d) \;\text{(radial)} . \end{cases}$

Any feasible $f$ satisfies $$\delta_d \leq (r/2)^d f(0)/\widehat{f}(0) \leq (r/2)^d f(0)$$, and optimizing over feasible $f$ yields the bound.

The dual program seeks a finite, nonnegative Borel measure $\mathscr{S}(\mathbb{R}^d)$0 supported on $\mathscr{S}(\mathbb{R}^d)$1 (possibly on discrete radii) with

$\mathscr{S}(\mathbb{R}^d)$2

This duality underlines the analytic-geometric correspondence central to the Cohn-Elkies methodology (Li, 2022).

2. Sharpness and Non-Sharpness: Dimensional Dependence

The Cohn-Elkies bound is known to be sharp only in dimensions $\mathscr{S}(\mathbb{R}^d)$3. In these cases, explicit auxiliary functions (so-called "magic functions") constructed via modular forms yield extremal solutions where all inequalities in the linear program become equalities at the shell radii of optimal lattices (e.g., $\mathscr{S}(\mathbb{R}^d)$4 in $\mathscr{S}(\mathbb{R}^d)$5, Leech lattice in $\mathscr{S}(\mathbb{R}^d)$6) (Rolen et al., 2019).

For $\mathscr{S}(\mathbb{R}^d)$7, except $\mathscr{S}(\mathbb{R}^d)$8, discrete reduction methods map the infinite-dimensional CE program to computationally tractable finite linear programs. Solving these dual programs and verifying feasibility with interval arithmetic produces explicit lower bounds that strictly exceed known or best conjectured packing densities:

Dimension $\mathscr{S}(\mathbb{R}^d)$9	CE Bound $f:\mathbb{R}^d \to \mathbb{R}$0	Best Known Density
3	0.18398089	$f:\mathbb{R}^d \to \mathbb{R}$1
4	0.12914461	$f:\mathbb{R}^d \to \mathbb{R}$2
5	0.09826308	$f:\mathbb{R}^d \to \mathbb{R}$3
6	0.07632412	$f:\mathbb{R}^d \to \mathbb{R}$4
7	0.06374745	$f:\mathbb{R}^d \to \mathbb{R}$5
9–13 (odd)	$f:\mathbb{R}^d \to \mathbb{R}$6	$f:\mathbb{R}^d \to \mathbb{R}$7 value

This certifies the non-sharpness of the CE bound in all these dimensions, demonstrating its inability to resolve the sphere packing problem except in $f:\mathbb{R}^d \to \mathbb{R}$8 (Li, 2022).

3. Discrete Reductions and Finite Duality

To bridge from infinite- to finite-dimensional formulation, one applies two key restrictions:

• Step 1: Periodization onto $f:\mathbb{R}^d \to \mathbb{R}$9—Periodizing $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$0 on the torus $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$1 via the dual lattice, leading to $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$2 on $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$3 with similar bounds.
• Step 2: Modulo $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$4 reduction to $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$5—Replacing $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$6 by the finite group $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$7, forming $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$8 by periodization and working with functions and constraints indexed by group orbits under $$\widehat{f}(y) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, y \rangle} dx .$$9.

Finite-dimensional LP dual variables (nonnegative, $\delta_d$0-invariant vectors $\delta_d$1) satisfy a discrete analogue of the Poisson summation formula, subject to algebraic orbit constraints. Computer-verified solutions using commercial or open-source LP solvers, rational rounding, and interval arithmetic, yield rigorous certificates of non-sharpness by demonstrating dual feasible points with objective larger than known packing densities (Li, 2022).

4. Implications, Classification, and Limitations

These finite dual bounds show that, outside the miracle dimensions, the extremal functions that saturate the bound in $\delta_d$2 cannot exist in $\delta_d$3. Furthermore, the explicit families constructed for odd $\delta_d$4 give

$\delta_d$5

again larger than known densities. This suggests that sharpness is a rare phenomenon, possibly only occurring in the special cases with suitable rich modular form structure.

As the reduction parameter $\delta_d$6, discrete bounds approach their continuous counterparts, yet quantifying this convergence rate remains an open analytic problem (Li, 2022). Moreover, attempts to "lift" discrete feasible solutions to continuous measures may enable new directions for direct construction or improved lower bounds in the original infinite-dimensional program.

5. Computational and Structural Techniques

Key elements in practical dual CE bound computation include:

• Symmetry Reduction: Exploiting $\delta_d$7-invariance drastically reduces variable count ($\delta_d$8 real variables).
• High-Precision Float to Rational Conversion: Floating-point LP solutions are rationally rounded in $\delta_d$9.
• Certificate Verification: Strict nonnegativity constraints $\mathbb{R}^d$0 ($\mathbb{R}^d$1) buffer rounding errors. Interval arithmetic and explicit recomputation of dual variables ensure rigorous correctness.
• Dimension-Dependent Parameter Choices: The minimal $\mathbb{R}^d$2 to exceed best known density in each $\mathbb{R}^d$3 is recorded; explicit coefficients provided for closed-form families in odd $\mathbb{R}^d$4; all processes are fully algorithmic (Li, 2022).

6. Extensions and Prospects

These results delimit the power of CE-type LP bounds in Euclidean spaces. Extensions to periodic LP bounds (Mo et al., 2024) and connections to modular forms and spectral optimization continue to be active research directions. "Lifting" finite dual solutions to construct continuous measures offers a possible pathway for further structural insights or refined bounds, and quantifying the convergence of discrete to continuous bounds remains a major analytic challenge. The classification—sharp only in $\mathbb{R}^d$5, non-sharp elsewhere—serves as a benchmaking milestone for both classical and emerging methods in sphere packing upper bounds.

• Dual Linear Programming Bounds for Sphere Packing via Discrete Reductions (Li, 2022)
• A New Linear Programming Method in Sphere Packing (Mo et al., 2024)
• A note on Schwartz functions and modular forms (Rolen et al., 2019)