The Kissing Number Problem

Kissing Number Problem

Definitions and mathematical statement

After scaling the centers of touching unit balls to the unit sphere, the kissing number has the equivalent finite-dimensional definition \[ \tau_n=\max\left\{|X|:X\subset S^{n-1},\ \langle x,y\rangle\le\tfrac12 \text{ for distinct }x,y\in X\right\}. \] The inner-product condition is equivalent to angular separation at least $\pi/3$. The problem asks for the exact integer $\tau_n$ for every $n>4$, not only asymptotic bounds or the best lattice configuration. Centers may be arranged arbitrarily, subject to nonoverlap.

Formulation and concept references: [S1].

Short English statement

Exactly how many equal balls can simultaneously touch one equal central ball in each Euclidean dimension greater than four?

Research attempt

Current frontier.

The target is the entire function $n\mapsto\tau_n$ for $n>4$, not just an improvement in one dimension. The source-defined spherical-code model is unambiguous. Cohn--Rajagopal, Section 2 and Theorem 2.1, give four nonisometric $40$-point configurations in dimension five, while retaining $40\leq\tau_5\leq44$. Thus even a complete classification of perturbations of $D_5$ would not classify all plausible optimizers. [S2]

Recent lower-bound work remains genuinely constructive rather than an exact determination in all dimensions: the September 2026 version of Cohn--Li improves dimensions $17$--$21$ and acknowledges Ho's subsequent improvement in dimension $19$. A September 2026 preprint of Takhanov--Yun reports additional higher-dimensional constructions. Their newest claims are not independently certified by this notebook and are not used in any proof below. The local/global distinction is already explicit in the rigidity theory of Cohn--Jiao--Kumar--Torquato. [E1] [E2] [E3]

Attack route.

Three possibilities were tested: add points into holes of a root configuration; deform that configuration to create a new hole; or produce a global positive-definite-polynomial upper certificate. The first needs a precise covering-radius calculation, the second a nontrivial feasible infinitesimal displacement, and the third a bound below the next integer. We compute all three diagnostics. They yield exact restricted statements and a particularly concrete counterexample to the inference that "saturated and rigid" means "maximum cardinality."

Attempt: exact holes and a dimension threshold.

For $n\geq4$ let \[ X_n=\{(\sigma e_i+\tau e_j)/\sqrt2: 1\leq i<j\leq n,\ \sigma,\tau\in\{-1,1\}\}. \] These $2n(n-1)$ unit vectors have pairwise inner products at most $1/2$. For a unit vector $v$, let $a_1\geq\cdots\geq a_n\geq0$ be its sorted absolute coordinates. Then \begin{equation}\label{wsref-0-31-33-36-39-3a-73-75-70-70-6f-72-74} \max_{x\in X_n}\langle v,x\rangle=(a_1+a_2)/\sqrt2. \end{equation} Writing $c=a_1+a_2$, one has \[ 1=\sum_i a_i^2\leq a_1^2+(n-1)a_2^2 =a_1^2+(n-1)(c-a_1)^2. \] The last expression is convex for $c/2\leq a_1\leq c$, so it is at most $\max(c^2,nc^2/4)=nc^2/4$ when $n\geq4$. Equality is attained by $a_i=1/\sqrt n$. Therefore \begin{equation}\label{wsref-0-31-33-36-39-3a-63-6f-76-65-72} \min_{v\in S^{n-1}}\max_{x\in X_n}\langle v,x\rangle =\sqrt{2/n}\qquad(n\geq4). \end{equation} For $n>4$, equality forces every absolute coordinate to equal $1/\sqrt n$: the maximizing endpoint is $a_1=c/2$, and equality must also hold in $\sum a_i^2\leq a_1^2+(n-1)a_2^2$.

Consequently $X_n$ admits no additional point for $4\leq n\leq7$. In particular, the exact positive obstruction margin for $D_5$ is $\sqrt{2/5}-1/2$. This does not merely reflect failure of a numerical search. Conversely, at $n=8$ the possible extra points are precisely \[ v=(\sigma_1,\ldots,\sigma_8)/(2\sqrt2), \qquad \sigma_i\in\{-1,1\}. \] Two such points have inner product $1-d_H(\sigma,\tau)/4$. Choose the $128$ sign patterns with an even number of minus signs. Their mutual Hamming distances are at least two, so adding them to the $112$ roots gives a valid $240$-point configuration. No larger subset of these $256$ hole points is possible: pairing each sign pattern with the one obtained by flipping its first sign partitions them into $128$ forbidden distance-one pairs. This proves optimality only among extensions of the fixed $D_8$ configuration at this stage.

Attempt: an exact infinitesimal certificate.

Use unnormalized integer roots $r_i=\sqrt2x_i$. A first-order feasible velocity $v_i$ obeys \[ r_i\cdot v_i=0,\qquad r_i\cdot v_j+r_j\cdot v_i\leq0 \quad\text{when }r_i\cdot r_j=1. \] For each root, \[ \sum_{j:r_i\cdot r_j=1}r_j=2(n-2)r_i. \] Summing the contact inequalities thus gives exactly zero, by the tangent conditions. Since every summand is nonpositive, each contact inequality is an equality. This is a positive equilibrium stress, proved just by counting the signed roots that share one coordinate.

The accompanying script builds the integer matrix $M_n$ of these tangent and contact equalities. Exact Gaussian elimination modulo the prime $1000003$ gives \[ \begin{array}{c|r|r|r|r} n&|X_n|&\text{contacts}&\text{variables}&\operatorname{rank}_{{\mathbb{F}}_{1000003}}M_n\\\hline 4&24&96&96&90\\ 5&40&240&200&190\\ 6&60&480&360&345 \end{array} \] A nonzero minor modulo a prime is a nonzero integer minor, so these are rigorous lower bounds for real rank. On the other hand the $n(n-1)/2$ independent rotations $v_i=Ar_i$, $A^T=-A$, lie in the kernel. Hence the displayed ranks are exactly the real ranks and all feasible infinitesimal motions are rotations. This is an exact finite certificate, not a tolerance-based singular-value calculation. The conclusion reproduces a known result, not a new rigidity theorem; Corollary 3.4 of [E1] proves it for all $D_n$, $n\geq4$.

Attempt: why the same polynomial closes dimension eight only.

The diagnostic polynomial \[ f(t)=(t+1)(t+\tfrac12)^2t^2(t-\tfrac12) \] is nonpositive on $[-1,1/2]$. Let $P_k^{(n)}(1)=1$ be the normalized Gegenbauer polynomial for the sphere $S^{n-1}$. The addition formula for spherical harmonics gives $\sum_{i,j}P_k^{(n)}(\langle x_i,x_j\rangle)\geq0$: up to a positive normalization it is a sum, over an orthonormal harmonic basis $Y_{k,l}$, of $|\sum_iY_{k,l}(x_i)|^2$. Thus, whenever $f=\sum f_kP_k^{(n)}$ with $f_0>0$ and $f_k\geq0$, \[ N^2f_0\leq\sum_{i,j}f(\langle x_i,x_j\rangle)\leq Nf(1), \qquad N\leq f(1)/f_0. \] Exact polynomial expansion in dimension eight gives, in increasing degree, \[ (f_0,\ldots,f_6)= \left(\frac3{320},\frac3{40},\frac{15}{64},\frac{39}{80}, \frac{399}{640},\frac9{16},\frac{33}{128}\right). \] Since $f(1)=9/4$, this proves $N\leq240$. Together with the explicit construction it recovers the known equality $\tau_8=240$. The polynomial identity is independently checked by exact rational arithmetic in the script.

In dimension five the same polynomial still has positive Gegenbauer coefficients but $f_0=37/840$, so it gives only $N\leq1890/37$, hence $N\leq51$. It is worse than the known upper bound. The attractive contact roots of $D_5$ therefore do not manufacture an optimal dual certificate. A genuinely useful global certificate in that dimension must exclude cardinality $41$; a bound strictly below $41$ would suffice, but none is produced here.

Stress test: an explicit failure of the local-to-global claim.

The same calculations show that $D_6$ is saturated and infinitesimally rigid, yet its $60$ points are not maximum. Here is a direct $72$-point comparison, without relying on an optimization table. In ${\mathbb{R}}^6$, use unnormalized squared norm two and take $D_5\times\{0\}$ together with \[ \left(\frac{\sigma_1}2,\ldots,\frac{\sigma_5}2, \frac{\varepsilon\sqrt3}2\right), \qquad \prod_{i=1}^5\sigma_i=\varepsilon, \quad \varepsilon\in\{-1,1\}. \] There are $40+32=72$ vectors. Two new vectors in the same last-coordinate layer differ in at least two of the first five signs, giving inner product at most $1$. In opposite layers their first-five Hamming distance is odd, and the inner product is $(1-d_H)/2\leq0$. A new vector has inner product at most $1$ with every central root. Division by $\sqrt2$ gives the claimed code. This is the familiar $E_6$ construction in explicit coordinates, also discussed in [S2], Section 3.

Thus saturation plus an exact local rigidity certificate is insufficient even in a neighboring dimension with the same root-system method. Nothing in the tangent argument excludes a disconnected component of the configuration space containing more points. Nor do the computations cover all real configurations of cardinality $41$ in dimension five.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt obtains an exact covering-radius formula, a dimension-eight hole-code completion and matching polynomial bound, and independently checkable rigidity certificates. It also supplies an explicit counterexample to the proposed local-to-global optimality principle. These are restricted deductions or rederivations of known facts; no novelty or improvement of a kissing-number record is claimed.

The full all-dimensional target remains open within this notebook. Already in dimension five, a global exclusion of $41$ points or an explicit larger configuration is missing. In higher dimensions, the root and hole template is not an exhaustive parametrization and cannot be promoted to an answer for every $n>4$.

Sources

• S1 Musin, The kissing number in four dimensions. https://arxiv.org/abs/math/0309430. Catalogue formulation source.
• S2 Status source for Exact Kissing Numbers in Higher Dimensions. https://arxiv.org/html/2412.00937v3. Catalogue research/status reference; not a proof endorsement.
• E1 Henry Cohn, Yang Jiao, Abhinav Kumar, and Salvatore Torquato, "Rigidity of spherical codes," Geometry & Topology 15 $2011$, 2235--2273, arXiv:1102.5060v2. Section 2, Theorem 2.2; Section 3, Corollary 3.4. https://arxiv.org/abs/1102.5060.
• E2 Henry Cohn and Anqi Li, "Improved kissing numbers in seventeen through twenty-one dimensions," arXiv:2411.04916v3, 1 September 2026. Theorem 1.1 and the subsequent note on dimension 19. https://arxiv.org/abs/2411.04916.
• E3 Rustem Takhanov and Stanislav Yun, "New lower bounds for kissing numbers in dimensions $25$--$29$ and $31$," arXiv:2609.21591, 18 September 2026. Abstract consulted for recent claimed constructions; not a proof dependency or independent certification. https://arxiv.org/abs/2609.21591.