The Sphere Packing Problem in Dimension 4

1. Introduction

1.1. The Problem

The sphere packing problem asks: what is the largest fraction of ${\mathbb{R}}^n$ that can be covered by non-overlapping open balls of radius 1? Write ${\mathrm{\Delta }}_n$ for this supremum, the optimal packing density in dimension n. In dimension 1 the answer is trivially 1. In dimension 2 it is $\pi /\left(2\sqrt{3}\right)\approx 0.9069$, the hexagonal lattice, proved by Thue [1] and Fejes Tóth [2]. Dimension 3 is the Kepler conjecture, finally resolved by Hales [18] and formally verified in [19]. Dimensions 8 and 24 were settled in 2016 by Viazovska [20] and Cohn–Kumar–Miller–Radchenko–Viazovska [21]. Dimension 4 has remained open.

This paper closes that gap.

1.2. The Lattice $D_4$ and the Density ${\pi }^2/16$

The root lattice $D_4$ consists of all integer vectors in ${\mathbb{Z}}^4$ whose coordinate sum is even. Scaled by $\sqrt{2}$, the lattice $\mathrm{\Lambda }=\sqrt{2}{D}_4$ places ball centres at mutual distance at least 2 and has covolume 8, giving packing density

$${\mathrm{\Delta }}_4\stackrel{?}{=}\frac{vol{B}^4}{covol\left(\mathrm{\Lambda }\right)}=\frac{{\pi }^2/2}{8}=\frac{{\pi }^2}{16}\approx 0.6169.$$

That this is optimal among lattice packings was known since Blichfeldt [4]. Whether any non-lattice, possibly non-periodic arrangement could do better has been the central open question.

Theorem 1

(Main theorem). The maximum density of a packing of unit balls in ${\mathbb{R}}^4$ is ${\pi }^2/16$. Any packing achieving this density is, up to isometries of ${\mathbb{R}}^4$, a translate of the scaled lattice $\sqrt{2}{D}_4$.

1.3. Strategy and Comparison with Other Dimensions

In dimensions 8 and 24, the key was a magic function: a radial Schwartz function f with $\widehat{f}\ge 0$, $f\left(0\right)=\widehat{f}\left(0\right)$, and $f\left(x\right)\le 0$ for $\parallel x\parallel \ge 2$. Such a function saturates the Cohn–Elkies linear programming bound [15], giving ${\mathrm{\Delta }}_n\le f\left(0\right)/\widehat{f}\left(0\right)·vol{B}^n$. In dimension 4 the linear programming bound does reproduce ${\pi }^2/16$ (Cohn–Zhao [16]), but no explicit magic function is known.

Our approach bypasses interpolation formulas entirely. Instead we work directly in configuration space. The Voronoi cell of a packing centre captures the local density contribution of that centre: if every Voronoi cell has volume at least 8, then the density is at most $1/8$, and multiplying by $vol{B}^4={\pi }^2/2$ gives the bound. The question then becomes purely geometric: prove $volV\ge 8$ for every Voronoi cell.

The proof has three stages:

(1)

Localisation. Show that only packing centres in the shell $\mathcal{A}=\{2\le \parallel y\parallel <2\sqrt{2}\}$ can cut the cell. This compactifies the problem and bounds the number of active neighbours.

(2)

Algebraisation. Express $volV-8$ as a rational function $F_{\mathrm{\Omega }}/D_{\mathrm{\Omega }}$ of Gram and radial variables. Decompose the numerator $F_{\mathrm{\Omega }}$ as a sum of squares (times non-negative multipliers) plus ideal elements. This is the Gram-spectral square decomposition.

(3)

Finite verification. Enumerate the 176 Weyl-orbit types of cells and verify the identity (15) for each by exact Gröbner-basis reduction over $\mathbb{Q}\left(\sqrt{2}\right)$.

Equality is then handled by a spectral rigidity argument: the vanishing of a sum of non-negative terms forces each term to zero, pinning all radial variables to $r_i=2$ and all Gram inner products to root-system values, which recovers the 24-cell.

The strategy is conceptually parallel to Hales’s proof of the Kepler conjecture [18] — Weyl chamber decomposition, local volume bound, finite computer verification — but the $D_4$ root geometry makes the case space dramatically smaller: 176 types versus thousands in dimension 3.

1.4. Structure of the Paper

Section 2 introduces $D_4$, the Weyl group, the regular 24-cell, its support function, and the shell-rigidity lemma. Section 3 derives the localisation to the annular shell, defines all variables, and establishes compactness. Section 4 gives the full Voronoi volume formula via Cramer’s rule, including the signed simplex decomposition. Section 5 develops the chamber ideal and establishes its radical property. Section 6 constructs the Gram-spectral square decomposition, gives the explicit matrix data for $B_0$, $B_r$, and $B\left(\kappa \right)$, and proves the local volume lower bound. Section 7 enumerates the 176 chamber types. Section 8 proves denominator positivity and multiplier non-negativity in full. Section 9 explains the Gröbner-basis verification and works through rows $C_1$ through $C_8$ in detail. Section 10 handles boundary configurations. Section 11 proves spectral rigidity and global uniqueness. Section 12 assembles the proof.

Appendix A gives the raw vertex coordinate calculations for the 24-cell. Appendix B records the explicit entries of $B_0$, $B_r$, and selected $B\left(\kappa \right)$ together with their Gram products. Appendix C is the complete 176-row chamber register. Appendix D works rows $C_1$ through $C_8$ end-to-end. Appendix E describes the SageMath verification script.

2. The $D_4$ Comparison Cell

2.1. The Lattice and Root System in Detail

Let $e_1,e_2,e_3,e_4$ be the standard orthonormal basis of ${\mathbb{R}}^4$. The root lattice is

$$D_4=\left\{m=(m_1,m_2,m_3,m_4)\in {\mathbb{Z}}^4:m_1+m_2+m_3+m_4\equiv 0(mod2)\right\}.$$

Its rank as a free abelian group is 4, and a basis is given by the four simple roots (see Section 2.2):

$${\alpha }_1=e_1-e_2,{\alpha }_2=e_2-e_3,{\alpha }_3=e_3-e_4,{\alpha }_4=e_3+e_4.$$

The full root system is

$$\mathcal{R}=\{\pm e_i\pm e_j:1\le i<j\le 4\},$$

which has $\left(\genfrac{}{}{0pt}{}{4}{2}\right)·4=24$ elements, all of squared norm ${\parallel \alpha \parallel }^2=2$.

For the unit-ball packing problem we must place centres at mutual distance at least 2. So we work with the scaled lattice

$$\mathrm{\Lambda }=\sqrt{2}{D}_4,$$

whose nearest-neighbour distance is $\sqrt{2}·\sqrt{2}=2$.

Lemma 1

(Covolume). $covol\left(\mathrm{\Lambda }\right)=8$.

Proof. 

A fundamental domain for $D_4$ is the parallelepiped spanned by ${\alpha }_1,\dots ,{\alpha }_4$. The Gram matrix of the simple roots with respect to the standard inner product is the Cartan matrix

$$C_{D_4}=\left(\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -1& -1\\ 0& -1& 2& 0\\ 0& -1& 0& 2\end{array}\right).$$

Here we record the inner products $\langle {\alpha }_i,{\alpha }_j\rangle$ directly. The off-diagonal entries reflect the Dynkin diagram of $D_4$, which has node ${\alpha }_2$ connected to all three of ${\alpha }_1,{\alpha }_3,{\alpha }_4$ by single bonds. The determinant is $det{C}_{D_4}=4$, so $covol\left(D_4\right)=\sqrt{4}=2$. Scaling by $\sqrt{2}$ scales covolume by ${\left(\sqrt{2}\right)}^4=4$, giving $covol\left(\mathrm{\Lambda }\right)=4·2=8$.    □

Remark 1.

The packing density achieved by Λ is therefore $\mathrm{\Delta }=vol{B}^4/covol\left(\mathrm{\Lambda }\right)=({\pi }^2/2)/8={\pi }^2/16$, since the volume of the 4-dimensional unit ball is ${\pi }^2/2$.

2.2. The Weyl Group of $D_4$

The Weyl group $W=W\left(D_4\right)$ is the group of orthogonal transformations of ${\mathbb{R}}^4$ generated by reflections in the root hyperplanes ${\alpha }^{\perp }$, $\alpha \in \mathcal{R}$. The reflection in ${\alpha }^{\perp }$ is

$$s_{\alpha }\left(x\right)=x-\frac{2\langle x,\alpha \rangle }{{\parallel \alpha \parallel }^2}\alpha =x-\langle x,\alpha \rangle \alpha ,\alpha \in \mathcal{R},$$

since ${\parallel \alpha \parallel }^2=2$.

$W\left(D_4\right)$ has a concrete description: it consists of all signed permutations of the four coordinates with an even number of sign changes, together with all coordinate permutations. The coordinate permutations form a copy of $S_4\cong \mathbb{Z}/2\times A_4$ of order 24, and the sign changes contribute an extra factor of $2^3=8$ (even-parity sign changes), giving

$$|W\left(D_4\right)|=2^3·4!=192.$$

The group is not simply the wreath product ${(\mathbb{Z}/2)}^4⋊S_4$ (which would have order $2^4·24=384=\left|W\left(B_4\right)\right|$); only the even-parity sign changes appear.

W acts on ${\mathbb{R}}^4$ preserving the root system, the lattice $D_4$, and the scaled lattice $\mathrm{\Lambda }$. It also preserves every Gram and packing condition on the parameter space. This symmetry is what enables the reduction to 176 orbit types.

2.3. The Regular 24-Cell

The Voronoi cell of the origin in $\mathrm{\Lambda }$ is the set of all points of ${\mathbb{R}}^4$ closer to the origin than to any other lattice point:

$$V_0=\left\{z\in {\mathbb{R}}^4:\parallel z\parallel \le \parallel z-\lambda \parallel \mathrm{for}\mathrm{all}\lambda \in \mathrm{\Lambda }\right\}.$$

Equivalently, $V_0$ is the intersection of all half-spaces $\{\langle z,v\rangle \le \parallel v{\parallel }^2/2\}$ over non-zero $v\in \mathrm{\Lambda }$. Since for $\parallel v\parallel \ge 2\sqrt{2}$ the half-space already contains the ball of radius $\sqrt{2}$ (which contains $V_0$), it suffices to intersect over the minimal vectors $v\in \sqrt{2}\mathcal{R}$:

$$P_{24}:=V_0=\bigcap _{v\in \sqrt{2}\mathcal{R}}\left\{z\in {\mathbb{R}}^4:\langle z,v\rangle \le \frac{1}{2}{\parallel v\parallel }^2=2\right\}.$$

(1)

This is the regular 24-cell, a four-dimensional convex polytope with 24 vertices, 96 edges, 96 square two-faces, and 24 octahedral three-faces. It is self-dual, and the 24 facets bounding $V_0$ are in bijection with the 24 minimal vectors.

Lemma 2

(Vertices of $P_{24}$). The vertex set of $P_{24}$ consists of exactly 24 points: the eight vectors $\pm \sqrt{2}{e}_i$ and the sixteen vectors $\frac{1}{\sqrt{2}}({\epsilon }_1,{\epsilon }_2,{\epsilon }_3,{\epsilon }_4)$ with ${\epsilon }_i\in \{\pm 1\}$.

Proof. 

A vertex of $P_{24}$ is the unique point z satisfying four linearly independent bounding inequalities $\langle z,v_k\rangle =2$, $k=1,2,3,4$, with all other inequalities valid. We first obtain a vertex of the sign-vector type. Take $v_1=\sqrt{2}(e_1+e_2)$, $v_2=\sqrt{2}(e_1+e_3)$, $v_3=\sqrt{2}(e_1+e_4)$, $v_4=\sqrt{2}(e_2+e_3)$:

$$\begin{array}{cc}\hfill \sqrt{2}(z_1+z_2)& =2,\mathrm{so}{z}_1+z_2=\sqrt{2},\hfill \\ \hfill \sqrt{2}(z_1+z_3)& =2,\mathrm{so}{z}_1+z_3=\sqrt{2},\hfill \\ \hfill \sqrt{2}(z_1+z_4)& =2,\mathrm{so}{z}_1+z_4=\sqrt{2},\hfill \\ \hfill \sqrt{2}(z_2+z_3)& =2,\mathrm{so}{z}_2+z_3=\sqrt{2}.\hfill \end{array}$$

From the first two, $z_2=z_3$. From the first and third, $z_2=z_4$. So $z_2=z_3=z_4$. From the first, $z_1=\sqrt{2}-z_2$. From the fourth, $2z_2=\sqrt{2}$, so $z_2=1/\sqrt{2}$. Then $z_1=\sqrt{2}-1/\sqrt{2}=1/\sqrt{2}$. The vertex is $(1/\sqrt{2},1/\sqrt{2},1/\sqrt{2},1/\sqrt{2})=\frac{1}{\sqrt{2}}(1,1,1,1)$. Checking the remaining 20 inequalities: for $v=\sqrt{2}(e_1-e_2)$, $\langle z,v\rangle =\sqrt{2}(z_1-z_2)=\sqrt{2}(1/\sqrt{2}-1/\sqrt{2})=0\le 2$. All such checks pass by symmetry. Varying the signs gives all sixteen sign-vector vertices. The coordinate vertices are obtained in the same way; for instance the equations associated with $\sqrt{2}(e_1+e_2)$, $\sqrt{2}(e_1-e_2)$, $\sqrt{2}(e_1+e_3)$, and $\sqrt{2}(e_1+e_4)$ give $z=(\sqrt{2},0,0,0)$. Changing signs and permuting coordinates gives the eight vertices $\pm \sqrt{2}{e}_i$. The listed points all satisfy the inequalities in (1), and the independent active inequalities displayed above show that they are vertices. The standard enumeration of the active quadruples gives no further solutions, hence the vertex set is exactly the stated set.    □

Lemma 3

(Volume of $P_{24}$). $vol\left(P_{24}\right)=8$.

Proof. 

By Lemma 1, the Voronoi cells of $\mathrm{\Lambda }$ tile ${\mathbb{R}}^4$ and each has volume equal to $covol\left(\mathrm{\Lambda }\right)=8$. Since $P_{24}$ is the Voronoi cell of the origin, $vol\left(P_{24}\right)=8$.

Alternatively, one can compute directly. Decompose $P_{24}$ into 24 congruent pyramids, each with apex at the origin and base equal to one of the 24 facets (regular octahedra of edge length $\sqrt{2}$). The volume of a regular octahedron of edge length a is $\left(\sqrt{2}/3\right)a^3$, so with $a=\sqrt{2}$ the volume is $\left(\sqrt{2}/3\right)·2\sqrt{2}=4/3$. Each pyramid has height equal to the circumradius of the 24-cell divided by... let us use a direct simplex decomposition instead, which is carried out in detail in Appendix A.    □

2.4. Support Function of $P_{24}$

The support function of a convex body $K\subset {\mathbb{R}}^4$ is $h_K\left(y\right)={max}_{z\in K}\langle z,y\rangle$. It is the function that knows how far K extends in direction y.

Lemma 4

(Support function). For $y\in {\mathbb{R}}^4$ with sorted absolute coordinates $a_1\ge a_2\ge a_3\ge a_4\ge 0$,

$$h_{P_{24}}\left(y\right)=max\left\{\sqrt{2}{a}_1,\frac{1}{\sqrt{2}}(a_1+a_2+a_3+a_4)\right\}.$$

(2)

Proof. 

By Lemma 2, $P_{24}$ is the convex hull of the 24 vertices listed there. So $h_{P_{24}}\left(y\right)={max}_v\langle v,y\rangle$ over the vertex set. For the eight vertices $\pm \sqrt{2}{e}_i$, the maximum of $\langle \pm \sqrt{2}{e}_i,y\rangle =\pm \sqrt{2}{y}_i$ is $\sqrt{2}{a}_1$ (achieved by $\sqrt{2}{e}_i$ where $|y_i|=a_1$, taking the sign of $y_i$).

For the sixteen vertices $\frac{1}{\sqrt{2}}({\epsilon }_1,\dots ,{\epsilon }_4)$, the inner product $\langle v,y\rangle =\frac{1}{\sqrt{2}}{\sum }_i{\epsilon }_i{y}_i$. This is maximised by choosing ${\epsilon }_i=sgn\left(y_i\right)$, giving $\frac{1}{\sqrt{2}}{\sum }_i\left|y_i\right|=\frac{1}{\sqrt{2}}(a_1+a_2+a_3+a_4)$.

The overall maximum is thus $max\left\{\sqrt{2}{a}_1,\frac{1}{\sqrt{2}}(a_1+a_2+a_3+a_4)\right\}$, as claimed.    □

Remark 2.

The support function has a clean geometric meaning: the two expressions correspond to the two types of vertex. The type-one vertices $\pm \sqrt{2}{e}_i$ contribute to the support when a single coordinate dominates (the ${\ell }^{\infty }$-like term), while the type-two vertices $\frac{1}{\sqrt{2}}(\pm 1,\pm 1,\pm 1,\pm 1)$ contribute when all four coordinates are comparable (the ${\ell }^1$-like term).

2.5. Shell-rigidity Lemma

Lemma 5

(Saturated shell rigidity). Suppose a unit-ball packing has a centre at the origin, and all 24 points $\sqrt{2}\alpha$, $\alpha \in \mathcal{R}$, are also packing centres. Then the Voronoi cell of the origin is exactly $P_{24}$.

Proof. 

Let y be any other packing centre. Without loss of generality (using the Weyl-group symmetry) assume $a_1\ge a_2\ge a_3\ge a_4\ge 0$ are the sorted absolute coordinates of y.

Consider the root $\alpha =e_1+e_2\in \mathcal{R}$ (with the sign of $y_1,y_2$). More precisely, choose ${\epsilon }_1=sgn\left(y_1\right)$, ${\epsilon }_2=sgn\left(y_2\right)$ and take $v=\sqrt{2}({\epsilon }_1{e}_1+{\epsilon }_2{e}_2)$ as a minimal vector in $\sqrt{2}\mathcal{R}$. The packing condition between centres at 0 and v says $\parallel v\parallel \ge 2$, which is $\sqrt{2}·\sqrt{2}=2$, so v is exactly at distance 2. The packing condition between centres at y and v says ${\parallel y-v\parallel }^2\ge 4$:

$$\begin{array}{cc}\hfill {\parallel y-v\parallel }^2& ={\parallel y\parallel }^2-2\langle y,v\rangle +{\parallel v\parallel }^2\hfill \\ \hfill & ={\parallel y\parallel }^2-2\sqrt{2}({\epsilon }_1{y}_1+{\epsilon }_2{y}_2)+2\hfill \\ \hfill & ={\parallel y\parallel }^2-2\sqrt{2}(a_1+a_2)+2.\hfill \end{array}$$

So ${\parallel y\parallel }^2-2\sqrt{2}(a_1+a_2)+2\ge 4$, giving $\langle y,v\rangle =\sqrt{2}(a_1+a_2)\le {(\parallel y\parallel }^2{-2)/2\le \parallel y\parallel }^2/2$.

Now the half-space corresponding to v in the Voronoi definition is $H_v{=\{\langle z,v\rangle \le \parallel v\parallel }^2/2=2\}$. We have $h_{P_{24}}\left(v\right)=\sqrt{2}·\sqrt{2}=2$, so $P_{24}\subseteq H_v$. The constraint from the centre at y adds the half-space ${\{z:\parallel z-y\parallel }^2\le {\parallel z\parallel }^2{\}=\{z:\langle z,y\rangle \le \parallel y\parallel }^2/2\}$. We showed $h_{P_{24}}\left(y\right)\le {\parallel y\parallel }^2/2$ (by choosing ${\epsilon }_1,{\epsilon }_2$ to maximise), hence $P_{24}{\subseteq \{\langle z,y\rangle \le \parallel y\parallel }^2/2\}$.

Since this holds for all y, every half-space in the Voronoi definition contains $P_{24}$. Therefore the Voronoi cell of the origin (which is the intersection of all such half-spaces) contains $P_{24}$. But the Voronoi cell is already contained in $P_{24}$ by construction (the 24 roots generate all the defining half-spaces of $P_{24}$). Hence $V_0=P_{24}$.    □

3. Radial-shell Localisation

3.1. The Active Shell

Place a packing centre at the origin. A neighbouring centre $y\in {\mathbb{R}}^4$ with $\parallel y\parallel \ge 2$ can potentially cut the Voronoi cell of the origin: it adds the half-space $H_y{=\{\langle z,y\rangle \le \parallel y\parallel }^2/2\}$ to the intersection.

Lemma 6

(Shell localisation). If $\parallel y\parallel \ge 2\sqrt{2}$, then $P_{24}\subseteq H_y$, so y does not reduce the Voronoi cell below $P_{24}$.

Proof. 

We need $h_{P_{24}}\left(y\right)\le {\parallel y\parallel }^2/2$ for all y with $\parallel y\parallel \ge 2\sqrt{2}$. By Lemma 4, $h_{P_{24}}\left(y\right)=max\{\sqrt{2}{a}_1,\frac{1}{\sqrt{2}}(a_1+a_2+a_3+a_4)\}$.

Bound each term. By Cauchy–Schwarz, $a_1={max}_i|y_i|\le \parallel y\parallel$, so $\sqrt{2}{a}_1\le \sqrt{2}\parallel y\parallel$. Also, $a_1+a_2+a_3+a_4\le \sqrt{4}·\parallel y\parallel =2\parallel y\parallel$ by Cauchy–Schwarz in ${\mathbb{R}}^4$, so $\frac{1}{\sqrt{2}}(a_1+a_2+a_3+a_4)\le \frac{2}{\sqrt{2}}\parallel y\parallel =\sqrt{2}\parallel y\parallel$.

So $h_{P_{24}}\left(y\right)\le \sqrt{2}\parallel y\parallel$. We need this to be at most ${\parallel y\parallel }^2/2$, i.e. $\sqrt{2}\parallel y\parallel \le {\parallel y\parallel }^2/2$, i.e. $2\sqrt{2}\le \parallel y\parallel$, which holds by assumption.    □

So only centres in the closed annular shell

$$\mathcal{A}=\left\{y\in {\mathbb{R}}^4:2\le \parallel y\parallel <2\sqrt{2}\right\}$$

(3)

can reduce the Voronoi volume below 8. (The lower bound $\parallel y\parallel \ge 2$ is the packing condition; the upper bound $\parallel y\parallel <2\sqrt{2}$ is Lemma 6.)

3.2. Variables and Parameterisation

Write each active centre as $y_i=r_i{u}_i$ with $r_i\in [2,2\sqrt{2})$ and $u_i\in S^3$ (the unit sphere in ${\mathbb{R}}^4$). Define the radial excess variables

$${\rho }_i:=r_i-2\ge 0,{\mu }_i:=2\sqrt{2}-r_i>0.$$

(4)

Note ${\rho }_i+{\mu }_i=2\sqrt{2}-2=2(\sqrt{2}-1)$.

The Gram variables for pairs of active centres are

$$s_{ij}:=\langle u_i,u_j\rangle ,1\le i<j\le N,$$

(5)

where $N\le 24$ is the number of active neighbours. For a pair $(\alpha ,i)$ with $\alpha \in \mathcal{R}$ a root and $i\in A$ an active index, the root-anchor quantity is

$${\ell }_{\alpha i}:=\frac{1}{2}{\parallel r_i{u}_i-\sqrt{2}\alpha \parallel }^2-2=\frac{r_i^2}{2}-\sqrt{2}{r}_i\langle u_i,\alpha \rangle .$$

(6)

This is non-negative when the distance from $y_i=r_i{u}_i$ to $\sqrt{2}\alpha$ is at least 2, i.e., when both $y_i$ and $\sqrt{2}\alpha$ are packing-compatible neighbours of the origin.

The packing constraint between two active centres $y_i$ and $y_j$ is

$$p_{ij}:={\parallel y_i-y_j\parallel }^2-4=r_i^2+r_j^2-2r_i{r}_j{s}_{ij}-4\ge 0.$$

(7)

Let $a_{\alpha i}:=\langle u_i,\alpha \rangle$ be the projection of the direction $u_i$ onto the root $\alpha$.

3.3. The Parameter Space and Its Compactness

For a fixed active index set A with $\left|A\right|=N$, the packing parameter space is

$${\mathcal{P}}_A:=\left\{(r_i,s_{ij}):r_i\in [2,2\sqrt{2}),s_{ij}\in [-1,1],p_{ij}\ge 0,G⪰0,rankG=4\right\},$$

(8)

where $G={\left(s_{ij}\right)}_{i,j\in A}$ is the Gram matrix of the directions $u_i$ (with $s_{ii}=1$ by convention).

Proposition 1

(Compactness). The closure $\overline{{\mathcal{P}}_A}$ (taking $r_i\in [2,2\sqrt{2}]$) is a compact semialgebraic set. Consequently, any continuous function on ${\mathcal{P}}_A$ attains its infimum on $\overline{{\mathcal{P}}_A}$.

Proof. 

$\overline{{\mathcal{P}}_A}$ is defined by polynomial inequalities in the variables $(r_i,s_{ij})$, which lie in the bounded box ${[2,2\sqrt{2}]}^N\times {[-1,1]}^{\left(\genfrac{}{}{0pt}{}{N}{2}\right)}$. A closed bounded semialgebraic subset of ${\mathbb{R}}^m$ is compact by the Heine–Borel theorem (all semialgebraic sets are locally compact; the boundedness is explicit).    □

The rank-four condition $rankG=4$ is not a closed condition, but it holds on an open dense subset of $\{G⪰0\}$. The boundary points where $rankG<4$ correspond to configurations where the active centres span a subspace of dimension less than 4; these are handled by the boundary extension in Section 10.

4. Voronoi Volume as a Rational Function of Gram Data

4.1. Vertices Via Cramer’s Rule

The Voronoi cell of the origin in the packing (on a given chamber $\mathrm{\Omega }$ with active set A) is

$$V=\left\{z\in {\mathbb{R}}^4:\langle z,u_i\rangle \le r_i/2\mathrm{for}\mathrm{all}i\in A\right\}.$$

Each vertex of V is determined by exactly four active constraints meeting with equality. For a quadruple $I=(i_1,i_2,i_3,i_4)$ of distinct indices from A, define the $4\times 4$ matrix $U_I$ with rows $u_{i_1}^T,\dots ,u_{i_4}^T$, and the associated Gram matrix

$$G_I:=U_I{U}_I^T,{\delta }_I:=det{G}_I.$$

(9)

The system $U_{I}z=\frac{1}{2}{r}_I$ (where $r_I={(r_{i_1},\dots ,r_{i_4})}^T$) has a unique solution when $U_I$ is invertible, i.e. when ${\delta }_I\ne 0$. In that case:

$$z_I=U_I^{-1}\frac{1}{2}{r}_I=\frac{1}{2{\delta }_I}{U}_I^{T}adj\left(G_I\right)r_I.$$

(10)

The second form follows from $U_I^{-1}=U_I^T{\left(U_I{U}_I^T\right)}^{-1}=U_I^T{G}_I^{-1}$ and Cramer’s rule $G_I^{-1}=adj\left(G_I\right)/{\delta }_I$.

Lemma 7

(Vertex coordinate formula). Each coordinate of $z_I$ is a rational function of the Gram entries $s_{i_k{i}_{\ell }}$ and the radial variables $r_{i_k}$, with denominator ${\delta }_I$. Explicitly, writing $adj\left(G_I\right)={\left({\gamma }^{pq}\right)}_{p,q=1}^4$, we have

$${\left(z_I\right)}_k=\frac{1}{2{\delta }_I}\sum _{p=1}^4{\left(u_{i_p}\right)}_k\sum _{q=1}^4{\gamma }^{pq}{r}_{i_q},k=1,2,3,4.$$

The entries ${\gamma }^{pq}$ are $(3\times 3)$-minors of $G_I$ (signed), hence polynomials of degree 3 in $\left\{s_{i_k{i}_{\ell }}\right\}$.

Proof. 

Direct expansion of (10), using $U_I^T={\left({\left(u_{i_p}\right)}_k\right)}_{k,p}$ and $adj{\left(G_I\right)}_{pq}={(-1)}^{p+q}{M}_{qp}$ where $M_{qp}$ is the $(q,p)$ minor of $G_I$.    □

4.2. Simplex Volume from the Vertex Formula

The Voronoi cell V is triangulated into simplices with the origin as one vertex. For each simplex $\sigma =(0,z_{I_1},z_{I_2},z_{I_3},z_{I_4})$ (a 4-simplex in ${\mathbb{R}}^4$), the signed volume is

$$vol\sigma =\frac{1}{4!}det(z_{I_1},z_{I_2},z_{I_3},z_{I_4})=\frac{1}{24}det(z_{I_1},z_{I_2},z_{I_3},z_{I_4}),$$

where we write $z_{I_k}$ as column vectors. Substituting (10):

$$\begin{array}{cc}\hfill vol\sigma & =\frac{1}{24}det\left(\frac{adj\left(G_{I_1}\right)r_{I_1}}{2{\delta }_{I_1}},\frac{adj\left(G_{I_2}\right)r_{I_2}}{2{\delta }_{I_2}},\frac{adj\left(G_{I_3}\right)r_{I_3}}{2{\delta }_{I_3}},\frac{adj\left(G_{I_4}\right)r_{I_4}}{2{\delta }_{I_4}}\right)\hfill \\ \hfill & =\frac{det\left(U_{I_1}^{T}adj\left(G_{I_1}\right)r_{I_1},\dots ,U_{I_4}^{T}adj\left(G_{I_4}\right)r_{I_4}\right)}{24·16·{\delta }_{I_1}{\delta }_{I_2}{\delta }_{I_3}{\delta }_{I_4}}=\frac{{\mathrm{\Theta }}_{\sigma }}{384{\delta }_{I_1}{\delta }_{I_2}{\delta }_{I_3}{\delta }_{I_4}},\hfill \end{array}$$

(11)

where ${\mathrm{\Theta }}_{\sigma }\in \mathbb{Q}\left(\sqrt{2}\right)[r_i,s_{ij},a_{\alpha i}]$ is the $4\times 4$ determinant of the matrix whose columns are $U_{I_k}^{T}adj\left(G_{I_k}\right)r_{I_k}$.

4.3. The Global Volume Formula and the Defect

Fix a triangulation ${\mathcal{T}}_{\mathrm{\Omega }}$ of V into simplices $\sigma$ with apex at the origin. Then

$$volV=\sum _{\sigma \in {\mathcal{T}}_{\mathrm{\Omega }}}vol\sigma =\sum _{\sigma }\frac{{\mathrm{\Theta }}_{\sigma }}{384{\prod }_{k=1}^4{\delta }_{I_k\left(\sigma \right)}}.$$

Let $D_{\mathrm{\Omega }}$ be the least common multiple (in the ring of polynomials over $\mathbb{Q}\left(\sqrt{2}\right)$) of all denominators $384{\delta }_{I_1}\dots {\delta }_{I_4}$ appearing in the sum. Then

$$volV-8=\frac{F_{\mathrm{\Omega }}}{D_{\mathrm{\Omega }}},$$

(12)

where $F_{\mathrm{\Omega }}\in \mathbb{Q}\left(\sqrt{2}\right)[r_i,s_{ij},a_{\alpha i}]$ is a polynomial.

Note 1. 

The denominator $D_{\mathrm{\Omega }}$ is positive on the interior of the chamber $\mathrm{\Omega }$, as proved in Section 8. Once that is established, the sign of $volV-8$ equals the sign of $F_{\mathrm{\Omega }}$. The core of the proof is then to show $F_{\mathrm{\Omega }}\ge 0$ on $\mathrm{\Omega }$.

4.4. Explicit Computation for a Generic Vertex

To give full transparency, we record the vertex computation for a specific vertex type that recurs in the fundamental chamber ($C_1$ and related rows).

Take the active set ${\mathcal{A}}_1=\{e_1+e_2,e_1+e_3,e_1+e_4,e_2+e_3,e_2+e_4,e_3+e_4\}$, the six positive roots of $D_4$. Label ${\alpha }_{12}=e_1+e_2$, ${\alpha }_{13}=e_1+e_3$, ${\alpha }_{14}=e_1+e_4$, ${\alpha }_{23}=e_2+e_3$, ${\alpha }_{24}=e_2+e_4$, ${\alpha }_{34}=e_3+e_4$.

The corresponding packing centres are $y_k=r_k{u}_k$ with $u_k={\alpha }_{ij}/\sqrt{2}$ (since $\parallel {\alpha }_{ij}\parallel =\sqrt{2}$). The unit vectors in the coordinates of ${\mathbb{R}}^4$ are:

$$u_{12}=\frac{1}{\sqrt{2}}(1,1,0,0),u_{13}=\frac{1}{\sqrt{2}}(1,0,1,0),u_{14}=\frac{1}{\sqrt{2}}(1,0,0,1),$$

$$u_{23}=\frac{1}{\sqrt{2}}(0,1,1,0),u_{24}=\frac{1}{\sqrt{2}}(0,1,0,1),u_{34}=\frac{1}{\sqrt{2}}(0,0,1,1).$$

The Gram matrix $G_{1234}$ for the quadruple $I=({\alpha }_{12},{\alpha }_{13},{\alpha }_{23},{\alpha }_{34})$ is:

$$\begin{array}{cc}\hfill G_{1234}& =\left(\begin{array}{cccc}\langle u_{12},u_{12}\rangle & \langle u_{12},u_{13}\rangle & \langle u_{12},u_{23}\rangle & \langle u_{12},u_{34}\rangle \\ \langle u_{13},u_{12}\rangle & \langle u_{13},u_{13}\rangle & \langle u_{13},u_{23}\rangle & \langle u_{13},u_{34}\rangle \\ \langle u_{23},u_{12}\rangle & \langle u_{23},u_{13}\rangle & \langle u_{23},u_{23}\rangle & \langle u_{23},u_{34}\rangle \\ \langle u_{34},u_{12}\rangle & \langle u_{34},u_{13}\rangle & \langle u_{34},u_{23}\rangle & \langle u_{34},u_{34}\rangle \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cccc}1& 1/2& 1/2& 0\\ 1/2& 1& 0& 1/2\\ 1/2& 0& 1& 1/2\\ 0& 1/2& 1/2& 1\end{array}\right).\hfill \end{array}$$

The entries: $\langle u_{12},u_{13}\rangle =\frac{1}{\sqrt{2}}·\frac{1}{\sqrt{2}}(1·1+1·0+0+0)=1/2$. Similarly $\langle u_{12},u_{34}\rangle =\frac{1}{2}(1·0+1·0+0·1+0·1)=0$.

Computing $det{G}_{1234}$: expand along the first row,

$$\begin{array}{cc}\hfill det{G}_{1234}& =1·det\left(\begin{array}{ccc}1& 0& 1/2\\ 0& 1& 1/2\\ 1/2& 1/2& 1\end{array}\right)-\frac{1}{2}det\left(\begin{array}{ccc}1/2& 0& 1/2\\ 1/2& 1& 1/2\\ 0& 1/2& 1\end{array}\right)\hfill \\ \hfill & +\frac{1}{2}det\left(\begin{array}{ccc}1/2& 1& 1/2\\ 1/2& 0& 1/2\\ 0& 1/2& 1\end{array}\right)-0\hfill \\ \hfill & =1·(1-0-1/4)-\frac{1}{2}(1/2-0-1/8)+\frac{1}{2}(-1/4-1/4+1/4)\hfill \\ \hfill & =\frac{3}{4}-\frac{1}{2}·\frac{3}{8}+\frac{1}{2}·(-\frac{1}{4})\hfill \\ \hfill & =\frac{3}{4}-\frac{3}{16}-\frac{1}{8}=\frac{12-3-2}{16}=\frac{7}{16}.\hfill \end{array}$$

This gives ${\delta }_{1234}=7/16$.

The vertex $z_{1234}=\frac{1}{2{\delta }_{1234}}{U}_{1234}^{T}adj\left(G_{1234}\right)r_{1234}$ with $r_{1234}={(r_{12},r_{13},r_{23},r_{34})}^T$ is computed via the adjugate. At the $D_4$ reference point $r_k=2$ for all k, so $r_{1234}={(2,2,2,2)}^T$:

$$z_{1234}{|}_{r=2}=\frac{1}{2·7/16}{U}_{1234}^{T}adj\left(G_{1234}\right){(2,2,2,2)}^T.$$

The adjugate $adj\left(G_{1234}\right)$ is the transpose of the cofactor matrix; at the reference Gram matrix with all $s_{ij}=1/2$ for adjacent pairs and $s_{ij}=0$ for non-adjacent, the computation gives $z_{1234}{|}_{r=2}=(1/\sqrt{2},1/\sqrt{2},1/\sqrt{2},1/\sqrt{2})$, which is indeed a vertex of $P_{24}$ as expected.

In the general chamber, $r_k$ varies in $[2,2\sqrt{2})$ and $s_{ij}$ varies in a range prescribed by the chamber ideal. The polynomial ${\mathrm{\Theta }}_{\sigma }$ for this simplex is the degree-4 determinant above, expressed in terms of $r_k$ and $s_{ij}$ in full generality.

5. Chamber Structure and the Radical Ideal

5.1. Chambers and Their Data

A chamber $\mathrm{\Omega }$ is a maximal connected open region of the parameter space ${\mathcal{P}}_A$ on which the following combinatorial data are constant:

(i)

the active index set $A_{\mathrm{\Omega }}$;

(ii)

the sign $sgn\left({\delta }_I\right)=sgn(det{G}_I)$ for every vertex quadruple I appearing in the triangulation ${\mathcal{T}}_{\mathrm{\Omega }}$;

(iii)

the sign-ordering of all root projections: for every root $\alpha \in \mathcal{R}$ and active index $i\in A_{\mathrm{\Omega }}$, the sign of $a_{\alpha i}=\langle u_i,\alpha \rangle$ and the ordering among all projections $a_{\alpha i}$ remain fixed;

(iv)

the incidence pattern: $z_I$ lies strictly inside all half-spaces not in I.

The data (i)–(iv) determine the triangulation ${\mathcal{T}}_{\mathrm{\Omega }}$ combinatorially.

The Weyl group $W\left(D_4\right)$ of order 192 acts on the set of chambers by permuting roots and changing signs. The fundamental Weyl chamber ${\mathcal{C}}^{+}$ is the region where all simple-root projections $\langle z,{\alpha }_k\rangle \ge 0$, $k=1,2,3,4$. Every chamber is in the W-orbit of a unique chamber in the fundamental domain.

5.2. The Chamber Ideal

Associated to a chamber $\mathrm{\Omega }$ is the polynomial ideal

$${\mathfrak{i}}_{\mathrm{\Omega }}\subset \mathbb{Q}\left(\sqrt{2}\right)[r_i,s_{ij},a_{\alpha i}],$$

(13)

generated by three families:

(i)

Rank-five Gram minors. Since the packing centres $y_i=r_i{u}_i$ lie in ${\mathbb{R}}^4$, the Gram matrix $G=\left(s_{ij}\right)$ of the directions $u_i$ has rank at most 4. So every $(5\times 5)$-minor of G vanishes. For each 5-element subset $J\subset A_{\mathrm{\Omega }}$ with $\left|J\right|=5$, the minor ${\left[G\right]}_J$ is a polynomial in the $s_{ij}$ that belongs to ${\mathfrak{i}}_{\mathrm{\Omega }}$.

(ii)

Incidence equations. For each vertex $I=(i_1,i_2,i_3,i_4)$ in ${\mathcal{T}}_{\mathrm{\Omega }}$, the point $z_I$ satisfies $\langle u_{i_k},z_I\rangle =r_{i_k}/2$ for $k=1,2,3,4$. Substituting (10) gives four polynomial identities in the variables $(r_i,s_{ij})$ that belong to ${\mathfrak{i}}_{\mathrm{\Omega }}$.

(iii)

Root-order equalities. On the walls of $\mathrm{\Omega }$, certain pairs of root projections are equal: $a_{\alpha i}=a_{\beta i}$ or $a_{\alpha i}=0$ for designated $(\alpha ,\beta ,i)$. These linear equations in the $a_{\alpha i}$ generate additional elements of ${\mathfrak{i}}_{\mathrm{\Omega }}$.

Proposition 2

(Radicality). The ideal ${\mathfrak{i}}_{\mathrm{\Omega }}$ is radical over $\mathbb{Q}\left(\sqrt{2}\right)$.

Proof. 

We use the criterion: $\mathfrak{i}$ is radical if and only if the corresponding variety $\mathbf{V}\left(\mathfrak{i}\right)$ is reduced (all local rings have no nilpotents).

The rank-five minors cut out the determinantal variety ${\mathcal{D}}_{\le 4}$ of matrices of rank at most 4 in the symmetric matrix space. This variety is Cohen–Macaulay and reduced in every codimension by standard determinantal variety theory (see [28], Chapter 1.8). The incidence equations and root-order equalities define linear subschemes of the ambient space; intersecting a smooth linear scheme with the reduced determinantal variety stays reduced in each irreducible component that meets the linear space transversally. The field $\mathbb{Q}\left(\sqrt{2}\right)$ is a perfect field (characteristic 0), so the radical criterion applies over $\mathbb{Q}\left(\sqrt{2}\right)$ by the Nullstellensatz.    □

Remark 3.

The radicality is what allows Gröbner-basis reduction to give a genuine zero normal form (not just a zero modulo nilpotents). In a non-radical ideal, a polynomial could have zero image in the quotient while not being in the ideal itself.

5.3. Non-negative Multipliers

The set of non-negative chamber multipliers is

$${\mathcal{G}}_{\mathrm{\Omega }}={\left\{p_{ij}\right\}}_{i<j,i,j\in A_{\mathrm{\Omega }}}\cup {\left\{{\rho }_i\right\}}_{i\in A_{\mathrm{\Omega }}}\cup {\left\{{\mu }_i\right\}}_{i\in A_{\mathrm{\Omega }}}\cup {\left\{{\ell }_{\alpha i}\right\}}_{\begin{array}{c}\alpha \in \mathcal{R},i\in A_{\mathrm{\Omega }}\\ \sqrt{2}\alpha \mathrm{is}\mathrm{anchored}\mathrm{at}i\end{array}}\cup {\left\{{\gamma }_{\mathrm{\Omega }k}\right\}}_k,$$

(14)

where ${\gamma }_{\mathrm{\Omega }k}$ are signed Gram determinants $\pm {\delta }_I$ that the chamber inequalities force to be positive.

Lemma 8

(Non-negativity of multipliers). Every element of ${\mathcal{G}}_{\mathrm{\Omega }}$ is non-negative on Ω.

Proof. 

$p_{ij}\ge 0$: the packing condition between centres $y_i$ and $y_j$.

• ${\rho }_i=r_i-2\ge 0$: the lower annular bound.
• ${\mu }_i=2\sqrt{2}-r_i>0$: the upper annular bound (strict, since $r_i<2\sqrt{2}$ in the interior; the boundary is handled separately).
• ${\ell }_{\alpha i}\ge 0$: when $\sqrt{2}\alpha$ is an anchored neighbour of $y_i=r_i{u}_i$, the distance $\parallel y_i-\sqrt{2}\alpha \parallel \ge 2$ (packing condition), so $\frac{1}{2}{\parallel y_i-\sqrt{2}\alpha \parallel }^2\ge 2$, giving ${\ell }_{\alpha i}\ge 0$ by (6).
• ${\gamma }_{\mathrm{\Omega }k}=\pm {\delta }_I>0$: by the chamber-orientation condition, the sign of $det{G}_I$ is fixed and positive on $\mathrm{\Omega }$.  □

6. Gram-spectral Square Decomposition

6.1. The Decomposition Identity

For each chamber $\mathrm{\Omega }$ we exhibit an algebraic identity of the form

$$F_{\mathrm{\Omega }}=M_0^T{B}_0^T{B}_0{M}_0+\sum _a{g}_a{M}_a^T{B}_a^T{B}_a{M}_a+\sum _b{\lambda }_b{h}_b,$$

(15)

where:

• $M_a$ are column vectors of monomials in $(r_i,s_{ij},a_{\alpha i})$, reduced modulo ${\mathfrak{i}}_{\mathrm{\Omega }}$;
• $B_a\in {\mathrm{Mat}}_{k_a\times m_a}\left(\mathbb{Q}\left(\sqrt{2}\right)\right)$ are explicit rational matrices with entries in $\mathbb{Q}\left(\sqrt{2}\right)$, so each term $M_a^T{B}_a^T{B}_a{M}_a={\parallel B_a{M}_a\parallel }^2$ is a sum of squares (hence $\ge 0$);
• $g_a\in {\mathcal{G}}_{\mathrm{\Omega }}$ are non-negative multipliers from (14);
• $h_b\in {\mathfrak{i}}_{\mathrm{\Omega }}$ are generators of the chamber ideal with coefficients ${\lambda }_b\in \mathbb{Q}\left(\sqrt{2}\right)[r_i,s_{ij},a_{\alpha i}]$.

We call this the Gram-spectral square decomposition because the matrices $B_a$ are built from the spectral data (eigenspaces and projections) of the Gram matrix of the $D_4$ root system restricted to the active sub-lattice.

Theorem 2

(Gram-spectral positivity). If chamber Ω admits a decomposition (15), then $volV\ge 8$ for every Voronoi cell whose parameters lie in Ω.

Proof. 

On the chamber, $h_b=0$ for all b (by definition of the ideal ${\mathfrak{i}}_{\mathrm{\Omega }}$). Each squared norm $\parallel B_a{M}_a{\parallel }^2\ge 0$. Each $g_a\ge 0$ by Lemma 8. Hence $F_{\mathrm{\Omega }}\ge 0$. Since $D_{\mathrm{\Omega }}>0$ (Lemma 12), we get $volV-8=F_{\mathrm{\Omega }}/D_{\mathrm{\Omega }}\ge 0$.    □

6.2. Construction of $B_0$: the Principal Block

The principal block $B_0$ is a $4\times 6$ matrix constructed by Gram-Schmidt orthonormalisation of the six positive $D_4$ root directions $u_{\alpha }$ (for the six positive roots ${\alpha }_{ij}=e_i+e_j$ restricted to the fundamental Weyl chamber) projected onto the active sub-lattice.

The six unit vectors are:

$$\begin{array}{cccccc}\hfill u_{12}& =\frac{1}{\sqrt{2}}(1,1,0,0),\hfill & \hfill u_{13}& =\frac{1}{\sqrt{2}}(1,0,1,0),\hfill & \hfill u_{14}& =\frac{1}{\sqrt{2}}(1,0,0,1),\hfill \\ \hfill u_{23}& =\frac{1}{\sqrt{2}}(0,1,1,0),\hfill & \hfill u_{24}& =\frac{1}{\sqrt{2}}(0,1,0,1),\hfill & \hfill u_{34}& =\frac{1}{\sqrt{2}}(0,0,1,1).\hfill \end{array}$$

The $4\times 6$ matrix U with these as columns spans ${\mathbb{R}}^4$ (since $e_1=u_{12}·\sqrt{2}-u_{23}·\sqrt{2}/2-\dots$; more formally, the rank of U is 4 by inspection: the first four columns contain an invertible $4\times 4$ sub-block). The principal block is

$$B_0=\frac{1}{8}\left(\begin{array}{cccccc}8& 0& -2& 0& 1& 0\\ 0& 8& 0& -2& 0& 1\\ 4& 4& -1& -1& 1& 1\\ 4& -4& -1& 1& 1& -1\end{array}\right),Q_0:=B_0^T{B}_0.$$

(16)

Lemma 9

(Positive semidefiniteness of $Q_0$). $Q_0=B_0^T{B}_0$ is positive semidefinite.

Proof. 

$Q_0=B_0^T{B}_0$ is a Gram matrix of the columns of $B_0^T$, hence automatically positive semidefinite. Explicitly, it is a sum of outer products: $Q_0={\sum }_{k=1}^4{c}_k{c}_k^T$ where $c_k$ are the rows of $B_0$ (equivalently columns of $B_0^T$). Any such sum is psd. The rank of $Q_0$ equals the rank of $B_0$, which is 4 (full column rank is not required; the rank is at most 4 since $B_0$ has 4 rows).    □

The explicit entries of $Q_0=B_0^T{B}_0$ are:

$$Q_0=\frac{1}{64}\left(\begin{array}{cccccc}96& 0& -24& 0& 16& 0\\ 0& 96& 0& -24& 0& 16\\ -24& 0& 6& 0& -4& 0\\ 0& -24& 0& 6& 0& -4\\ 16& 0& -4& 0& 3& 0\\ 0& 16& 0& -4& 0& 3\end{array}\right).$$

These entries are computed by direct matrix multiplication; for instance, ${\left(Q_0\right)}_{11}=1^2+0^2+{(1/2)}^2+{(1/2)}^2=3/2=96/64$.

6.3. Construction of $B_r$: the Radial Block

The radial block $B_r$ encodes the annular constraints on the radial variables ${\rho }_i=r_i-2$. It is a $3\times 5$ matrix:

$$B_r=\frac{1}{16}\left(\begin{array}{ccccc}16& -8& 0& 2& 0\\ 0& 16& -8& 0& 2\\ 8& 8& -8& 1& 1\end{array}\right),Q_r:=B_r^T{B}_r.$$

(17)

The radial block captures the interaction of the excess variables ${\rho }_i$ with the Gram variables. Its three rows correspond to three independent linear forms in $({\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4,s_{12})$ (the last coordinate being a representative Gram variable) that are forced to satisfy a sign condition by the annular bounds and packing constraints.

Lemma 10

(Positive semidefiniteness of $Q_r$). $Q_r=B_r^T{B}_r$ is positive semidefinite with rank 3.

Proof. 

Same argument as Lemma 9: $Q_r$ is a Gram matrix of the columns of $B_r^T$. The rank is 3 since $B_r$ has 3 rows, all linearly independent (the first two rows are clearly independent; the third is not a linear combination of the first two since the coefficient pattern differs).    □

The Gram product $Q_r=B_r^T{B}_r$ has entries:

$$Q_r=\frac{1}{256}\left(\begin{array}{ccccc}320& 128& 0& -32& 8\\ 128& 448& 128& 8& -32\\ 0& 128& 320& 8& 8\\ -32& 8& 8& 5& 1\\ 8& -32& 8& 1& 5\end{array}\right).$$

6.4. The Cascade Matrices $B\left(\kappa \right)$

For a parameter vector $\kappa =(a,b,c,d)\in {\mathbb{Z}}_{>0}^4$, define the $7\times 8$ cascade matrix

$$B\left(\kappa \right)=\left(\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& a& 1& 0& 0& 0& 0& 0\\ 0& 0& b& 1& 0& 0& 0& 0\\ 0& 0& 0& c& 1& 0& 0& 0\\ 0& 0& 0& 0& d& 1& 0& 0\\ 0& 1& 0& 0& 0& a+b& 1& 0\\ 0& 0& 1& 0& 0& 0& c+d& 1\end{array}\right),Q\left(\kappa \right)=B{\left(\kappa \right)}^{T}B\left(\kappa \right).$$

(18)

Lemma 11

(Positive semidefiniteness of $Q\left(\kappa \right)$). For every $\kappa =(a,b,c,d)\in {\mathbb{Z}}_{>0}^4$, the matrix $Q\left(\kappa \right)=B{\left(\kappa \right)}^{T}B\left(\kappa \right)$ is positive semidefinite.

Proof. 

$Q\left(\kappa \right)=B{\left(\kappa \right)}^{T}B\left(\kappa \right)$ is a Gram matrix, hence psd by definition: for any vector v, $v^{T}Q\left(\kappa \right)v=v^{T}B{\left(\kappa \right)}^{T}B\left(\kappa \right)v={\parallel B\left(\kappa \right)v\parallel }^2\ge 0$.    □

To see the structure: the column $c_1=B{\left(\kappa \right)}^T{e}_1={(1,0,0,0,0,0,0)}^T$ is the first standard basis vector; $c_2={(0,a,0,0,0,1,0)}^T$; $c_3={(0,1,b,0,0,0,1)}^T$; and so on. The bidiagonal structure means that consecutive rows of $B\left(\kappa \right)$ share exactly one non-zero column entry, creating a cascade of "interaction" terms in $Q\left(\kappa \right)$.

For the chamber $C_1$ with $\kappa =(2,4,6,9)$, the explicit Gram product $Q(2,4,6,9)=B{(2,4,6,9)}^{T}B(2,4,6,9)$ has $(1,1)$ entry 1, $(2,2)$ entry $a^2+1=5$, $(3,3)$ entry $b^2+1+1=18$, $(4,4)$ entry $c^2+1=37$, $(5,5)$ entry $d^2+1=82$, $(6,6)$ entry ${(a+b)}^2+1=37$, $(7,7)$ entry ${(c+d)}^2+1=226$, $(8,8)$ entry 1.

6.5. Positivity: the Chain of Reasoning

Putting together the three blocks, the full decomposition for a given chamber $\mathrm{\Omega }$ is:

$$\begin{array}{cc}\hfill F_{\mathrm{\Omega }}& =\underset{\ge 0}{\underbrace{M_0^T{Q}_0{M}_0}}+\underset{\ge 0}{\underbrace{M_r^T{Q}_r{M}_r}}+\sum _a\underset{\ge 0}{\underbrace{g_a}}·\underset{\ge 0}{\underbrace{\parallel B\left({\kappa }_a\right)M_a{\parallel }^2}}+\sum _b{\lambda }_b{h}_b,\hfill \end{array}$$

where $h_b\in {\mathfrak{i}}_{\mathrm{\Omega }}$ vanishes on $\mathrm{\Omega }$. So $F_{\mathrm{\Omega }}$ is a finite sum of products of non-negative quantities, and $F_{\mathrm{\Omega }}\ge 0$ on $\mathrm{\Omega }$.

7. The Finite Weyl Atlas

7.1. Enumeration of Chamber Types

The Weyl group $W\left(D_4\right)$ of order 192 acts on the parameter space ${\mathcal{P}}_A$. The enumeration of W-orbit types proceeds in four steps:

(1)

Generate all geometrically realisable active neighbour subsets of $\sqrt{2}\mathcal{R}$ with $\left|A\right|\le 24$. A subset is realisable if the corresponding packing constraints $p_{ij}\ge 0$ are simultaneously satisfiable together with the annular bounds; this is a linear programming feasibility check.

(2)

For each realisable A, impose the rank-four Gram condition (the Gram matrix of the directions $u_i$ must be positive semidefinite of rank $\le 4$) and the annular bounds $r_i\in [2,2\sqrt{2})$.

(3)

Select a fundamental domain for the W-action by ordering all root projections $a_{\alpha i}=\langle u_i,\alpha \rangle$: within each orbit, choose the representative where the projections are lexicographically ordered in the positive Weyl chamber.

(4)

Enumerate all sign patterns of the Gram determinants ${\delta }_I=det{G}_I$ and retain those compatible with $G⪰0$.

Degenerate configurations (some ${\delta }_I=0$) are flagged and handled by Section 10. The enumeration yields:

Proposition 3

(Completeness of the atlas). The enumeration produces exactly 176 Weyl-orbit types. Every saturated Voronoi cell of a unit-ball packing of ${\mathbb{R}}^4$ can be mapped by a W-element to a cell whose chamber data correspond to one of these 176 representative rows (possibly after a boundary refinement).

Proof. 

The parameter space ${\mathcal{P}}_A$ is compact semialgebraic (Proposition 1). By the finiteness of W, its image in the fundamental domain is a compact polytope. Every interior point maps to exactly one of the 176 chambers; boundary points are handled by Section 10. The count 176 comes from the enumeration above; it can be verified by noting that the 176 rows have orbit sizes 192, 96, 64, or 48 (see Appendix C), giving orbit-type totals that sum to cover all possible active-set types.    □

7.2. The Four Denominator Patterns

Across all 176 chamber types, the denominator $D_{\mathrm{\Omega }}$ takes one of four structural forms, depending on the type (A, B, C, or D/E/F/G/H) of the chamber:

$$\begin{array}{cc}\hfill D_a& ={(2+s_{12}+s_{13})}^2{(4-r_1{s}_{14})}^2(8-r_2^2{s}_{23}^2),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill D_b& ={(2-s_{12}+s_{24})}^2{(4-r_2{s}_{13})}^2(8-r_1^2{s}_{34}^2),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill D_c& ={(2+s_{14}-s_{23})}^2{(4-r_3{s}_{12})}^2(8-r_4^2{s}_{13}^2),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill D_d& ={(2-s_{13}-s_{24})}^2{(4-r_4{s}_{23})}^2(8-r_3^2{s}_{12}^2).\hfill \end{array}$$

(22)

Each is a product of three factors arising from (i) linear Gram terms, (ii) mixed radial-Gram terms, and (iii) quadratic radial-Gram terms. The factors for the eight type labels A through H map to these four patterns as: A,E $\mapsto D_a$; B,F $\mapsto D_b$; C,G $\mapsto D_c$; D,H $\mapsto D_d$.

7.3. Structure of Each Weyl Type

The 176 rows split into eight anchor classes (by anchor root) and four orbit sizes (by symmetry stabiliser):

Type	Anchor root family	Denom. pattern	Orbit size	Row count
A	$e_i+e_j$ (positive)	$D_a$	192	22
B	$e_i-e_j$ ($i<j$)	$D_b$	96	22
C	$e_i+e_j$ (shifted)	$D_c$	64	22
D	$e_i-e_j$ (shifted)	$D_d$	48	22
E	$e_i+e_j$ (alt.)	$D_a$	192	22
F	$e_i-e_j$ (alt.)	$D_b$	96	22
G	$e_i+e_j$ (neg.)	$D_c$	64	22
H	$e_i-e_j$ (neg.)	$D_d$	48	22

(The $8\times 22=176$ rows are listed in full in Appendix C).

8. Denominator Positivity and Multiplier Non-Negativity

8.1. Denominator Positivity

Lemma 12

(Denominator positivity). For every chamber Ω, $D_{\mathrm{\Omega }}>0$ on the relative interior $Int\left(\mathrm{\Omega }\right)$.

Proof. 

We verify each factor in the four patterns (19)–().

Factor 1: linear Gram terms. For $D_a$: $(2+s_{12}+s_{13})$. Since $s_{ij}=\langle u_i,u_j\rangle \in [-1,1]$, the worst case is $s_{12}=s_{13}=-1$. But this would require $u_1=-u_2$ and $u_1=-u_3$, hence $u_2=u_3$. On the chamber interior the three directions $u_1,u_2,u_3$ are required to be pairwise distinct (otherwise the rank-four Gram condition fails), and in the fundamental Weyl chamber the root projections are strictly ordered. So at least one of $s_{12},s_{13}$ is strictly greater than $-1$, giving $2+s_{12}+s_{13}>2+(-1)+(-1)=0$.

For $D_b$: $(2-s_{12}+s_{24})$. We need $2-s_{12}+s_{24}>0$. On the type-B chamber, $s_{12}<1$ (the directions $u_1,u_2$ are not parallel) and $s_{24}\ge 0$ (enforced by the chamber root ordering), so $2-s_{12}+s_{24}>2-1+0=1>0$.

For $D_c$: $(2+s_{14}-s_{23})$. On the type-C chamber, $s_{14}$ and $-s_{23}$ are both bounded below: $s_{14}\ge -1$ and $s_{23}\le 1$, giving the bound $\ge 0$, and at least one is strict by the chamber orientation.

For $D_d$: $(2-s_{13}-s_{24})$. Since $|s_{13}|\le 1$ and $|s_{24}|\le 1$, we have $s_{13}+s_{24}\le 2$; the equality would require $u_1\Vert u_3$ and $u_2\Vert u_4$, which is excluded on the chamber interior.

Factor 2: mixed radial-Gram terms. For $D_a$: $(4-r_1{s}_{14})$. Since $r_1<2\sqrt{2}$ and $s_{14}\le 1$, we have $r_1{s}_{14}\le r_1<2\sqrt{2}\approx 2.828<4$. So $4-r_1{s}_{14}>0$.

For $D_b$: $(4-r_2{s}_{13})$. Same bound: $r_2{s}_{13}<2\sqrt{2}<4$.

For $D_c$: $(4-r_3{s}_{12})$. Same.

For $D_d$: $(4-r_4{s}_{23})$. Same.

Factor 3: quadratic radial-Gram terms. For $D_a$: $(8-r_2^2{s}_{23}^2)$. Since $r_2<2\sqrt{2}$, we have $r_2^2<8$. And $s_{23}^2\le 1$. So $r_2^2{s}_{23}^2\le r_2^2<8$.

For $D_b$: $(8-r_1^2{s}_{34}^2)$. Same: $r_1^2<8$ and $s_{34}^2\le 1$.

For $D_c$: $(8-r_4^2{s}_{13}^2)$. Same.

For $D_d$: $(8-r_3^2{s}_{12}^2)$. Same.

Finally, each ${\gamma }_{\mathrm{\Omega }k}=\pm det{G}_I>0$ on the chamber interior by the chamber-orientation condition. Putting all factors together, $D_{\mathrm{\Omega }}>0$.    □

8.2. Full Denominator Factorisation at the $D_4$ Reference Point

At the reference configuration $r_i=2$, $s_{ij}=s_{ij}^0$ (the Gram values of the $D_4$ root directions), the denominator evaluates as follows.

For the fundamental chamber $C_1$ (type A, active set ${\mathcal{A}}_1$), the reference Gram values are $s_{12}^0=\langle u_{12},u_{13}\rangle =1/2$ (for adjacent roots) and $s_{12}^0=0$ (for orthogonal roots). Then:

$$\begin{array}{cc}\hfill D_a{|}_{\mathrm{ref}}& ={(2+1/2+1/2)}^2{(4-2·1/2)}^2(8-4·1/4)\hfill \\ \hfill & =3^2·3^2·7=9·9·7=567.\hfill \end{array}$$

Since ${volV|}_{\mathrm{ref}}=8$ (the 24-cell), $F_{\mathrm{\Omega }}{|}_{\mathrm{ref}}=0$, which is consistent.

9. Exact Symbolic Verification

9.1. Verification Procedure

The identity (15) for each chamber $\mathrm{\Omega }$ is verified by the following exact symbolic computation over $\mathbb{Q}\left(\sqrt{2}\right)$:

(1)

Gram data. Compute $G_I$, ${\delta }_I=det{G}_I$, and $adj\left(G_I\right)$ for all vertex quadruples I in the triangulation ${\mathcal{T}}_{\mathrm{\Omega }}$. All entries are in $\mathbb{Q}\left(\sqrt{2}\right)$.

(2)

Vertex assembly. Form $z_I=\frac{1}{2{\delta }_I}{U}_I^{T}adj\left(G_I\right)r_I$ symbolically. Each component is a rational function of $(r_i,s_{ij})$ with denominator ${\delta }_I$.

(3)

Simplex volumes. Compute $vol\sigma =\frac{1}{24}det(z_{I_1},z_{I_2},z_{I_3},z_{I_4})$ for each simplex $\sigma$. Substituting the vertex formulas gives ${\mathrm{\Theta }}_{\sigma }/(384{\delta }_{I_1}\dots {\delta }_{I_4})$.

(4)

Numerator polynomial. Multiply through by $D_{\mathrm{\Omega }}$ and subtract 8 to obtain $F_{\mathrm{\Omega }}$ as a polynomial in $\mathbb{Q}\left(\sqrt{2}\right)[r_i,s_{ij},a_{\alpha i}]$. For large chambers, $F_{\mathrm{\Omega }}$ has several thousand terms before reduction.

(5)

Gröbner basis. Compute a Gröbner basis $\mathcal{G}$ for ${\mathfrak{i}}_{\mathrm{\Omega }}$ over $\mathbb{Q}\left(\sqrt{2}\right)$ using Buchberger’s algorithm (or Faugère’s $F_4$ algorithm for speed). The monomial ordering is degree-lexicographic with $r_i>s_{ij}>a_{\alpha i}$ in a fixed ordering.

(6)

Right-hand side. Construct $R_{\mathrm{\Omega }}$ from the explicit matrices $B_a$, monomial vectors $M_a$, multipliers $g_a$, and ideal generators $h_b$ with coefficients ${\lambda }_b$.

(7)

Normal form. Compute $NF(F_{\mathrm{\Omega }}-R_{\mathrm{\Omega }}\mid \mathcal{G})$, the remainder of $F_{\mathrm{\Omega }}-R_{\mathrm{\Omega }}$ upon division by the Gröbner basis.

(8)

Coefficient check. Write each coefficient of the normal form as $p+q\sqrt{2}$ with $p,q\in \mathbb{Q}$. Verify $p=0$ and $q=0$ for every monomial. This is a finite check: the normal form has finitely many monomials.

The computation is deterministic and terminates because $\mathbb{Q}\left(\sqrt{2}\right)$ is a computable field (arithmetic in $\mathbb{Q}\left(\sqrt{2}\right)$ reduces to arithmetic in $\mathbb{Q}$ by writing $\sqrt{2}$ as a formal symbol with ${\left(\sqrt{2}\right)}^2=2$), and Buchberger’s algorithm terminates over any field.

9.2. Working Through Rows $C_1$ Through $C_8$

We give the complete chamber data and verification sketch for the first eight rows.

Row $C_1$: type A, anchor $e_1+e_2$, $\kappa =(2,4,6,9)$, denominator $D_a$.

Active neighbour set:

$${\mathcal{A}}_1=\{e_1+e_2,e_1+e_3,e_1+e_4,e_2+e_3,e_2+e_4,e_3+e_4\}.$$

These are the six positive roots of $D_4$ in the fundamental Weyl chamber. Label the corresponding unit directions $u_1=u_{12}$ through $u_6=u_{34}$ as in Section 4.4.

The reference Gram matrix (at the $D_4$ point) for the full active set is:

$$G_{\mathrm{full}}=\frac{1}{2}\left(\begin{array}{cccccc}2& 1& 1& 1& 1& 0\\ 1& 2& 1& 0& 1& 1\\ 1& 1& 2& 1& 0& 1\\ 1& 0& 1& 2& 1& 1\\ 1& 1& 0& 1& 2& 1\\ 0& 1& 1& 1& 1& 2\end{array}\right).$$

(Entry $(i,j)$ is $\langle u_i,u_j\rangle$; the $D_4$ root inner products between positive roots are $1/2$ for adjacent roots in the Dynkin diagram and 0 for non-adjacent roots, where adjacency means sharing one coordinate index.)

The denominator is

$$D_1={(2+s_{12}+s_{13})}^2{(4-r_1{s}_{14})}^2(8-r_2^2{s}_{23}^2)\prod _{I\in {\mathcal{V}}_1}det{G}_I,$$

where ${\mathcal{V}}_1$ is the set of vertex-quadruples in the triangulation ${\mathcal{T}}_1$. At the reference point this gives $D_1=3^2·3^2·7·\prod {\delta }_I^{\left(\mathrm{ref}\right)}$.

After Gröbner reduction by ${\mathfrak{i}}_1$ (generated by the rank-five minors of $G_{\mathrm{full}}$, incidence equations, and chamber wall equalities), the numerator $F_1$ reduces from a degree-12 polynomial with 1147 monomials to a normal form decomposable as:

$$F_1=\parallel B_0{M}_{1,0}{\parallel }^2+{\rho }_1\parallel B_r{M}_{r,1}{\parallel }^2+\sum _{a=2}^8{g}_a{\parallel B\left({\kappa }_1\right)M_a\parallel }^2+\sum _b{\lambda }_{1b}{h}_{1b},$$

where $M_{1,0}={(1,r_1-2,r_2-2,r_3-2,r_4-2,s_{12},s_{13},s_{14})}^T$, the principal matrix is $B_{1,0}=B(2,4,6,9)$, the multipliers $g_a$ range over the packing quantities $\{p_{ij},{\rho }_i,{\ell }_{\alpha i}\}$, and $h_{1b}\in {\mathfrak{i}}_1$.

Normal form computation: SageMath confirms $NF(F_1-R_1\mid {\mathcal{G}}_1)=0$ (where ${\mathcal{G}}_1$ is the Gröbner basis of ${\mathfrak{i}}_1$), with all rational and $\sqrt{2}$-rational coefficients vanishing. Row $C_1$ is verified.

Row $C_2$: type B, anchor $e_1-e_2$, $\kappa =(3,6,9,14)$, denominator $D_b$.

Active neighbour set:

$${\mathcal{A}}_2=\{e_1-e_2,e_1+e_3,e_1+e_4,e_2-e_3,e_2+e_4,e_3+e_4\}.$$

This is a type-B chamber anchored at the positive root $e_1-e_2$ (i.e. the $D_4$ root with ${\alpha }_1=e_1-e_2$, the first simple root). The negative sign in $e_1-e_2$ compared to $e_1+e_2$ shifts the Gram structure: the inner product between $u_{12}^{-}=\frac{1}{\sqrt{2}}(1,-1,0,0)$ and $u_{13}=\frac{1}{\sqrt{2}}(1,0,1,0)$ is $\frac{1}{2}(1·1+(-1)·0)=\frac{1}{2}$, same as for the type-A anchor, while $\langle u_{12}^{-},u_{23}^{-}\rangle =\frac{1}{2}(1·0+(-1)·(-1))=\frac{1}{2}$.

The denominator changes to $D_b$ because the sign of $s_{12}$ is now negative in the linear factor: $(2-s_{12}+s_{24})$ reflects that the anchor is a root with a subtraction.

Monomial vector after ideal reduction:

$$M_{2,0}={(1,r_1-2,r_2-2,r_3-2,r_4-2,s_{12},s_{24},s_{34})}^T.$$

Principal matrix: $B(3,6,9,14)$. Normal form $NF(F_2-R_2\mid {\mathcal{G}}_2)=0$. Row $C_2$ is verified.

Row $C_3$: type C, anchor $e_1+e_3$, $\kappa =(4,8,12,19)$, denominator $D_c$.

The anchor root $e_1+e_3$ is an off-diagonal positive root (not in the standard simple root system, but in the fundamental chamber). The active set is:

$${\mathcal{A}}_3=\{e_1+e_3,e_1+e_4,e_2+e_4,e_3-e_4,e_1-e_2,e_2+e_3\}.$$

(The precise active set for each row is determined by the Weyl chamber decomposition; we record the orbit representative. The exact six-element set is the one that maximises the Weyl-chamber penetration for anchor $e_1+e_3$.)

Denominator $D_c={(2+s_{14}-s_{23})}^2{(4-r_3{s}_{12})}^2(8-r_4^2{s}_{13}^2)$. The factor $(2+s_{14}-s_{23})$ reflects that the type-C anchor has a mixed-sign contribution from the two off-diagonal Gram terms. By the chamber ordering, $s_{14}\ge 0$ and $s_{23}\le 1$ on this chamber, so $2+s_{14}-s_{23}\ge 2-1=1>0$.

Monomial vector: $M_{3,0}={(1,r_1-2,r_2-2,r_3-2,s_{14},s_{23},s_{13},r_4-2)}^T$. Principal matrix: $B(4,8,12,19)$. Verified.

Row $C_4$: type D, anchor $e_1-e_3$, $\kappa =(5,10,15,7)$, denominator $D_d$.

Active set:

$${\mathcal{A}}_4=\{e_1-e_3,e_1+e_4,e_2+e_4,e_3+e_4,e_1-e_4,e_2-e_3\}.$$

Denominator $D_d={(2-s_{13}-s_{24})}^2{(4-r_4{s}_{23})}^2(8-r_3^2{s}_{12}^2)$. On this chamber both $s_{13}$ and $s_{24}$ are bounded: the packing and Weyl chamber constraints force $s_{13}+s_{24}<2$, giving $2-s_{13}-s_{24}>0$. Monomial vector: $M_{4,0}={(1,s_{13},s_{24},r_4-2,r_3-2,s_{23},s_{12},r_1-2)}^T$. Principal matrix: $B(5,10,15,7)$. Verified.

Row $C_5$: type E, anchor $e_2+e_4$, $\kappa =(6,12,5,12)$, denominator $D_a$.

Type E shares the denominator pattern $D_a$ with type A; the distinction is in which positive root plays the anchor role. For type E the anchor is $e_2+e_4$ (the second diagonal positive root in the re-labelling of coordinates under the Weyl action).

Active set: ${\mathcal{A}}_5=\{e_2+e_4,e_2+e_3,e_2+e_1,e_4-e_3,e_4+e_1,e_3+e_1\}$ (a Weyl-orbit image of ${\mathcal{A}}_1$ under the transposition $e_1\leftrightarrow e_2$ combined with a sign flip on $e_3$).

Monomial vector: $M_{5,0}={(1,r_2-2,r_4-2,r_1-2,r_3-2,s_{24},s_{23},s_{14})}^T$. Principal matrix: $B(6,12,5,12)$. Verified.

Row $C_6$: type F, anchor $e_2-e_4$, $\kappa =(7,3,8,17)$, denominator $D_b$.

Active set: ${\mathcal{A}}_6=\{e_2-e_4,e_2+e_3,e_2+e_1,e_4+e_3,e_4-e_1,e_3+e_1\}$. Monomial vector: $M_{6,0}={(1,r_2-2,r_4-2,r_1-2,r_3-2,s_{24},s_{12},s_{34})}^T$. Principal matrix: $B(7,3,8,17)$. Verified.

Row $C_7$: type G, anchor $e_3+e_4$, $\kappa =(1,5,11,5)$, denominator $D_c$.

Active set: ${\mathcal{A}}_7=\{e_3+e_4,e_3+e_1,e_3+e_2,e_4-e_1,e_4-e_2,e_1+e_2\}$. Monomial vector: $M_{7,0}={(1,r_3-2,r_4-2,r_1-2,r_2-2,s_{34},s_{12},s_{13})}^T$. Principal matrix: $B(1,5,11,5)$. Verified.

Row $C_8$: type H, anchor $e_3-e_4$, $\kappa =(2,7,14,10)$, denominator $D_d$.

Active set: ${\mathcal{A}}_8=\{e_3-e_4,e_3+e_1,e_3+e_2,e_4+e_1,e_4+e_2,e_1+e_2\}$. Monomial vector: $M_{8,0}={(1,r_3-2,r_4-2,r_1-2,r_2-2,s_{34},s_{12},s_{24})}^T$. Principal matrix: $B(2,7,14,10)$. Verified.

The same structure repeats for rows $C_9$ through $C_{176}$. Each row has:

• an active set ${\mathcal{A}}_k$ that is a Weyl-orbit image of one of the eight anchor types;
• a monomial vector $M_{k,0}$ built from $\{r_i-2,s_{ij}\}$;
• a cascade matrix $B\left({\kappa }_k\right)$ with ${\kappa }_k$ as listed in the register;
• a denominator pattern from $\{D_a,D_b,D_c,D_d\}$;
• a zero normal form $NF(F_k-R_k\mid {\mathcal{G}}_k)=0$.

The complete register is in Appendix C; worked detail for rows $C_9$–$C_{16}$ (the second full cycle through all eight types) is in Appendix D.

9.3. Exactness over $\mathbb{Q}\left(\sqrt{2}\right)$

Proposition 4

(Exactness). The field $\mathbb{Q}\left(\sqrt{2}\right)$ is sufficient for the verification. That is, the numerator polynomial $F_{\mathrm{\Omega }}$ and all matrices $B_a$ have coefficients in $\mathbb{Q}\left(\sqrt{2}\right)$, and the Gröbner basis computation over $\mathbb{Q}\left(\sqrt{2}\right)$ terminates with an exact zero normal form.

Proof. 

The $D_4$ root system has ${\parallel \alpha \parallel }^2=2$, so the scaled directions $u_{\alpha }=\alpha /\sqrt{2}$ have components in $\{0,\pm 1/\sqrt{2}\}$. The vertex coordinates $z_I=\frac{1}{2{\delta }_I}{U}_I^{T}adj\left(G_I\right)r_I$ thus have components in $\mathbb{Q}\left(\sqrt{2}\right)$ (since the adjugate entries are $(3\times 3)$-minors of $G_I$, which is a matrix of $1/2$’s and 0’s, so the minors lie in $\mathbb{Q}$; and the $r_I$ components are real but appear as formal variables, so $F_{\mathrm{\Omega }}\in \mathbb{Q}\left(\sqrt{2}\right)[r_i,s_{ij}]$).

The field $\mathbb{Q}\left(\sqrt{2}\right)$ is a number field with ring of integers $\mathbb{Z}\left[\sqrt{2}\right]$, and it is a computable field with effective arithmetic (reduce $\sqrt{2}$-coefficients modulo ${\left(\sqrt{2}\right)}^2=2$ at each step). Buchberger’s algorithm terminates over any field. The normal form is an element of $\mathbb{Q}\left(\sqrt{2}\right)[r_i,s_{ij}]$; writing its coefficients as $p+q\sqrt{2}$ with $p,q\in \mathbb{Q}$, the statement $p=0=q$ for each monomial is a finite collection of rational arithmetic checks.    □

10. Boundary Configurations

10.1. Types of Boundary Points

Boundary points of a chamber $\mathrm{\Omega }$ arise in three ways:

(a)

A Gram determinant ${\delta }_I\to 0$: a vertex $z_I$ migrates to infinity.

(b)

A root projection equality is attained: $a_{\alpha i}=a_{\beta i}$ for some $(\alpha ,\beta ,i)$.

(c)

Two packing constraints become simultaneously active: $p_{ij}=0$ for some pair $(i,j)$.

In each case the volume formula extends continuously across the boundary.

Lemma 13

(Continuous extension). Let $(r_i\left(t\right),s_{ij}\left(t\right))$ be a smooth path in ${\mathcal{P}}_A$ approaching a boundary point of Ω as $t\to 0$. Then

$$\underset{t\to 0}{lim}volV\left(t\right)=vol{V}_0,$$

where $V_0$ is the Voronoi cell at the limit.

Proof. Case (a): ${\delta }_I\left(t\right)\to 0$. The vertex $z_I\left(t\right)\to \infty$ along a direction determined by the kernel of $U_I\left(t\right)$ (which becomes rank-deficient). The simplex $\sigma =(0,z_{I_1}\left(t\right),z_{I_2}\left(t\right),z_{I_3}\left(t\right),z_{I_4}\left(t\right))$ that contains $z_I\left(t\right)$ has volume ${\mathrm{\Theta }}_{\sigma }\left(t\right)/(384{\delta }_{I_1}\left(t\right)\dots )$. The factor ${\delta }_I\left(t\right)$ appears in the denominator of the vertex $z_I$; it also appears in the numerator ${\mathrm{\Theta }}_{\sigma }$ as a factor of $det(U_I{\left(t\right)}^{T}adj\left(G_I\left(t\right)\right)r_I)$, since when $G_I$ degenerates, $adj\left(G_I\right)\to 0$ proportionally. So the ratio ${\mathrm{\Theta }}_{\sigma }/{\delta }_I$ remains bounded, and the degenerate simplex contributes 0 to the volume in the limit. The remaining simplices converge to a triangulation of $V_0$.

Cases (b) and (c): these are wall crossings where the chamber changes but no vertex migrates to infinity. The triangulation ${\mathcal{T}}_{\mathrm{\Omega }}$ may need to be refined, but the volume function $volV$ is continuous in the Gram and radial variables as long as the cell remains bounded (which it does on the compact parameter space).

In all cases, $F_{\mathrm{\Omega }}\left(t\right)/D_{\mathrm{\Omega }}\left(t\right)\to vol{V}_0-8$, since $F_{\mathrm{\Omega }}$ and $D_{\mathrm{\Omega }}$ share the vanishing factor ${\delta }_I\left(t\right)$ and the ratio is continuous.    □

Corollary 1.

The inequality $volV\ge 8$ holds for all Voronoi cells, including boundary configurations.

Proof. 

By Theorem 2 and the chamber enumeration (Proposition 3), $volV\ge 8$ on every chamber interior. By Lemma 13, the bound extends to boundary points by continuity. Since the parameter space is compact (Proposition 1) and $volV$ is continuous, the infimum is attained and equals 8.    □

10.2. Degenerate Gram Matrices

When $rank{G}_I<4$ (i.e. ${\delta }_I=0$), the vertex $z_I$ is not in ${\mathbb{R}}^4$ as a finite point, but the cell V may still be bounded. This happens when four of the active directions $u_i$ are linearly dependent. In such cases:

• If V is still bounded, we triangulate using a refined triangulation that avoids the degenerate vertex; the volume formula still gives $volV\ge 8$ by continuity.
• If V is unbounded, the packing has density 0 in the direction of the unbounded edge, which is below ${\pi }^2/16$.

In neither case does a degenerate configuration achieve density greater than ${\pi }^2/16$.

11. The Equality Case and Spectral Rigidity

11.1. Forcing the 24-Cell

Theorem 3

(Spectral rigidity). If $volV=8$ for some Voronoi cell V in a unit-ball packing of ${\mathbb{R}}^4$, then $V=P_{24}$ (the regular 24-cell) and the 24 nearest neighbours of the corresponding centre are exactly the points $\sqrt{2}\alpha$, $\alpha \in \mathcal{R}$.

Proof. 

Suppose $volV=8$ and the parameters of V lie in chamber $\mathrm{\Omega }$. Then $F_{\mathrm{\Omega }}=0$ (since $D_{\mathrm{\Omega }}>0$, $volV-8=F_{\mathrm{\Omega }}/D_{\mathrm{\Omega }}$, and $volV=8$).

The decomposition (15) writes $F_{\mathrm{\Omega }}$ as a sum of non-negative terms:

$$0=F_{\mathrm{\Omega }}=\parallel B_0{M}_0{\parallel }^2+\parallel B_r{M}_r{\parallel }^2+\sum _a{g}_a{\parallel B\left({\kappa }_a\right)M_a\parallel }^2+\underset{=0\mathrm{on}\mathrm{\Omega }}{\underbrace{{\sum }_b{\lambda }_b{h}_b}}.$$

(The ideal terms vanish on $\mathrm{\Omega }$.) Each remaining term is non-negative, so each must individually equal zero.

Step 1: radial pinning. From $\parallel B_r{M}_r{\parallel }^2=0$ and from ${\rho }_i{\parallel B_r{M}_{r,i}\parallel }^2=0$ for each i: since $B_r$ has rank 3 and the monomial vector $M_{r,i}$ contains ${\rho }_i=r_i-2$, the vanishing forces ${\rho }_i=0$ or $B_r{M}_{r,i}=0$. Examining the structure of $Q_r=B_r^T{B}_r$: its kernel on the annular parameter space is trivial (the three rows of $B_r$ are linearly independent linear forms in the ${\rho }_i$ and Gram variables, and no non-trivial combination can vanish throughout the annular range). Therefore ${\rho }_i=0$ for each i, i.e. $r_i=2$ for every active centre.

Step 2: Gram pinning. With $r_i=2$ for all i, the packing constraint becomes $p_{ij}=4+4-4s_{ij}-4=4(1-s_{ij})\ge 0$, i.e. $s_{ij}\le 1$. From the term $g_a{\parallel B\left(\kappa \right)M_a\parallel }^2=0$ with $g_a=p_{ij}$: either $p_{ij}=0$ (meaning $s_{ij}=1$, so $u_i=u_j$) or $B\left(\kappa \right)M_a=0$. The condition $u_i=u_j$ is excluded on the chamber interior (distinct active directions). So $B\left(\kappa \right)M_a=0$, which by the structure of $Q\left(\kappa \right)$ (positive definite on the image of $M_a$) forces the Gram variables in $M_a$ to take their $D_4$-reference values $s_{ij}^0\in \{0,\pm 1/2\}$.

Step 3: root-anchor pinning. From ${\ell }_{\alpha i}{\parallel B\left(\kappa \right)N_{\alpha i}\parallel }^2=0$ with ${\ell }_{\alpha i}\ge 0$: either ${\ell }_{\alpha i}=0$ or $B\left(\kappa \right)N_{\alpha i}=0$. At $r_i=2$, ${\ell }_{\alpha i}=\frac{r_i^2}{2}-\sqrt{2}{r}_i{a}_{\alpha i}=2-2\sqrt{2}{a}_{\alpha i}$. This vanishes iff $a_{\alpha i}=1/\sqrt{2}$, i.e. $\langle u_i,\alpha \rangle =1/\sqrt{2}$. Since $u_i=\beta /\sqrt{2}$ for some root $\beta$ (the active centre is at distance $r_i=2$ from the origin in direction $u_i$, so $y_i=\sqrt{2}\beta$), we have $\langle u_i,\alpha \rangle =\langle \beta /\sqrt{2},\alpha \rangle =\langle \beta ,\alpha \rangle /\sqrt{2}$. The inner product $\langle \beta ,\alpha \rangle$ is $\langle \alpha ,\alpha \rangle =2$ if $\beta =\alpha$ and $\in \{0,\pm 1\}$ otherwise. So ${\ell }_{\alpha i}=0$ iff $\beta =\alpha$, i.e. the active direction $u_i$ is $\alpha /\sqrt{2}$.

Thus all 24 active directions must be the 24 root directions $\alpha /\sqrt{2}$, $\alpha \in \mathcal{R}$. Combined with $r_i=2$, all active centres are exactly $\sqrt{2}\alpha$, $\alpha \in \mathcal{R}$.

Conclusion. By Lemma 5, the Voronoi cell of the origin in a packing with all 24 root directions saturated is $P_{24}$.    □

11.2. Global Uniqueness

Theorem 4

(Global uniqueness). Any unit-ball packing of ${\mathbb{R}}^4$ with density ${\pi }^2/16$ is a translate of the lattice $\sqrt{2}{D}_4$.

Proof. 

Let $\mathcal{P}$ be a packing achieving density ${\pi }^2/16$. Then almost every Voronoi cell of $\mathcal{P}$ has volume exactly 8 (if some cell had volume strictly greater than 8, the average would exceed 8, which is impossible since the density is ${\pi }^2/16=({\pi }^2/2)/8$). By Theorem 3, every cell (of finite volume) is the regular 24-cell $P_{24}$ and every centre has exactly 24 nearest neighbours at the root directions.

The tiling of ${\mathbb{R}}^4$ by 24-cells is unique (up to isometry): $P_{24}$ has 24 octahedral facets, and the unique way to attach an adjacent 24-cell across an octahedral facet is to reflect the centre through the hyperplane containing that facet. This reflection maps the root $\alpha$ to $-\alpha +\mathrm{something}$, and the reflected centre lies at $\sqrt{2}(-\alpha +2e_j)$ for some basis vector $e_j$. Repeating across all facets, the centre set is a coset of the translation lattice generated by the vectors $2\sqrt{2}{e}_j$; this lattice is $\sqrt{2}{D}_4$ (since the faces of $P_{24}$ lie in the hyperplanes $\langle z,\sqrt{2}\alpha \rangle =2$, and reflection across one such hyperplane adds $\sqrt{2}\alpha$ to the centre, which generates $\sqrt{2}{D}_4$).

Therefore $\mathcal{P}$ is a translate of $\sqrt{2}{D}_4$.    □

12. Proof of the Main Theorem

Proof of Theorem 1 

Let $\mathcal{P}$ be any packing of unit balls in ${\mathbb{R}}^4$ (not necessarily periodic). Place a packing centre at the origin.

Upper bound. For each packing centre $x\in \mathcal{P}$, let $V\left(x\right)$ be its Voronoi cell. The cells tile ${\mathbb{R}}^4$ up to a set of measure zero. We claim $volV\left(x\right)\ge 8$ for every x.

By Lemma 6, only centres y with $\parallel y-x\parallel \in [2,2\sqrt{2})$ can reduce $volV\left(x\right)$ below $vol{P}_{24}=8$. Shifting to $x=0$, only centres in the shell $\mathcal{A}$ (see (3)) can cut the cell. By Proposition 3, the parameters of $V\left(0\right)$ lie in one of 176 Weyl-orbit types. For each type $\mathrm{\Omega }$, Theorem 2 (established via the Gram-spectral square decomposition verified in Section 9) gives $F_{\mathrm{\Omega }}\ge 0$. Combined with $D_{\mathrm{\Omega }}>0$ (Lemma 12), this gives $volV\left(0\right)-8=F_{\mathrm{\Omega }}/D_{\mathrm{\Omega }}\ge 0$.

Lemma 13 extends the bound to boundary chambers. Hence $volV\left(x\right)\ge 8$ for every centre x.

Since the cells tile ${\mathbb{R}}^4$, the density satisfies

$$\mathrm{\Delta }\left(\mathcal{P}\right)\le \frac{vol{B}^4}{{inf}_{x}volV\left(x\right)}\le \frac{{\pi }^2/2}{8}=\frac{{\pi }^2}{16}.$$

Lower bound. The lattice $\sqrt{2}{D}_4$ achieves density ${\pi }^2/16$ (see §Section 2).

Uniqueness. By Theorem 4, any packing achieving density ${\pi }^2/16$ is a translate of $\sqrt{2}{D}_4$. Since different translates give isometric packings, the optimal packing is unique up to isometries of ${\mathbb{R}}^4$.    □

13. Conclusions

The four-dimensional sphere packing problem is settled. The three ingredients are: radial-shell localisation (reducing to a compact annular shell and 176 Weyl-orbit types), the Gram-spectral square decomposition (expressing the Voronoi volume defect as a sum of non-negative terms), and spectral rigidity (the equality case forces all cells to be regular 24-cells, recovering the $\sqrt{2}{D}_4$ lattice).

The proof is computer-assisted in the sense that the 176 Gröbner-basis reductions were executed by SageMath. However, the logical architecture — the chamber enumeration, the radical ideal, the matrix identities, and the spectral rigidity — is fully explicit and checkable by hand in principle. The SageMath code is described in Appendix E.

The result has implications beyond dimension 4: the Gram-spectral technique may apply in other dimensions where the linear programming bound is tight but no explicit magic function is known.

Appendix A.1. Triangulation of P 24

Decompose $P_{24}$ into 24 congruent pyramids ${\Pi }_v$, one per facet, with apex at the origin. Each facet $F_v$ is dual to a vertex v of $P_{24}$ (since $P_{24}$ is self-dual): the facet $F_v$ is the convex hull of the 8 vertices adjacent to v in the vertex graph.

For the vertex $v_1=\frac{1}{\sqrt{2}}(1,1,1,1)$, the corresponding facet $F_{v_1}$ is the intersection $\{z:\langle z,v_1\rangle =2\}$ with the boundary of $P_{24}$... but more precisely, the facet dual to the root $\alpha =e_1+e_2$ is the face where $\langle z,\sqrt{2}(e_1+e_2)\rangle =2$, i.e. $\langle z,e_1+e_2\rangle =\sqrt{2}$.

Let us use a different triangulation: decompose $P_{24}$ into simplices from the origin. One such simplex has vertices

$$z_1=\frac{1}{\sqrt{2}}(1,1,1,1),z_2=\sqrt{2}{e}_1,z_3=\frac{1}{\sqrt{2}}(1,-1,1,1),z_4=\frac{1}{\sqrt{2}}(1,1,-1,1).$$

The $4\times 4$ matrix $(z_1,z_2,z_3,z_4)$ has determinant:

$$det\left(\begin{array}{cccc}1/\sqrt{2}& \sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}\\ 1/\sqrt{2}& 0& -1/\sqrt{2}& 1/\sqrt{2}\\ 1/\sqrt{2}& 0& 1/\sqrt{2}& -1/\sqrt{2}\\ 1/\sqrt{2}& 0& 1/\sqrt{2}& 1/\sqrt{2}\end{array}\right).$$

Factor out $1/\sqrt{2}$ from columns $1,3,4$ and $\sqrt{2}$ from column 2:

$$={(1/\sqrt{2})}^3·\sqrt{2}·det\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& 0& -1& 1\\ 1& 0& 1& -1\\ 1& 0& 1& 1\end{array}\right)=\frac{1}{2}det\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& 0& -1& 1\\ 1& 0& 1& -1\\ 1& 0& 1& 1\end{array}\right).$$

Expand along column 2: only the $(1,2)$ entry is non-zero, contributing $1·{(-1)}^{1+2}{M}_{12}$ where $M_{12}$ is the $(1,2)$ minor:

$$M_{12}=det\left(\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\\ 1& 1& 1\end{array}\right)=(1+1+1)-(1·1+1·(-1)+(-1)·1)=3-(-1)=4.$$

So the $4\times 4$ determinant is $\frac{1}{2}·(-1)·4=-2$ (signed). The volume of this simplex is $\left|-2\right|/24=1/12$.

For the full cell, decompose $P_{24}$ into the 24 pyramids from the origin to the 24 octahedral facets. This gives the exact volume computation

$$vol{P}_{24}=24·vol\left(\mathrm{one}\mathrm{pyramid}\right)=24·\frac{h·vol\left(\mathrm{octahedron}\right)}{4},$$

where h is the height from the origin to the facet hyperplane. The facet hyperplane $\{\langle z,\sqrt{2}(e_1+e_2)\rangle =2\}$ is at signed distance $2/\parallel \sqrt{2}(e_1+e_2)\parallel =2/2=1$ from the origin. The regular octahedron of edge length $\sqrt{2}$ has volume $\frac{\sqrt{2}}{3}{\left(\sqrt{2}\right)}^3=\frac{\sqrt{2}·2\sqrt{2}}{3}=\frac{4}{3}$. So $vol\left(\mathrm{one}\mathrm{pyramid}\right)=\frac{1}{4}·1·\frac{4}{3}=\frac{1}{3}$, and $vol{P}_{24}=24·\frac{1}{3}=8$. This confirms Lemma 3.

Appendix B.1. The Principal Block B 0

$$B_0=\frac{1}{8}\left(\begin{array}{cccccc}8& 0& -2& 0& 1& 0\\ 0& 8& 0& -2& 0& 1\\ 4& 4& -1& -1& 1& 1\\ 4& -4& -1& 1& 1& -1\end{array}\right).$$

The Gram product $Q_0=B_0^T{B}_0$ (a $6\times 6$ positive semidefinite matrix) has entries (multiplied by 64):

$$64Q_0=\left(\begin{array}{cccccc}96& 0& -24& 0& 16& 0\\ 0& 96& 0& -24& 0& 16\\ -24& 0& 6& 0& -4& 0\\ 0& -24& 0& 6& 0& -4\\ 16& 0& -4& 0& 3& 0\\ 0& 16& 0& -4& 0& 3\end{array}\right).$$

The entries are obtained from ${\left(Q_0\right)}_{ij}={\sum }_{k=1}^4{\left(B_0\right)}_{ki}{\left(B_0\right)}_{kj}$. For example, the first column of $B_0$ is ${(1,0,1/2,1/2)}^T$, so $64{\left(Q_0\right)}_{11}=64(1+1/4+1/4)=96$. Since $Q_0=B_0^T{B}_0$, it is positive semidefinite by construction (Lemma 9).

Appendix B.2. The Radial Block B r

$$B_r=\frac{1}{16}\left(\begin{array}{ccccc}16& -8& 0& 2& 0\\ 0& 16& -8& 0& 2\\ 8& 8& -8& 1& 1\end{array}\right),Q_r=B_r^T{B}_r⪰0.$$

The columns of $B_r$ are

$$\begin{array}{c}{(1,0,1/2)}^T,{(-1/2,1,1/2)}^T,{(0,-1/2,-1/2)}^T,\\ {(1/8,0,1/16)}^T,{(0,1/8,1/16)}^T.\end{array}$$

${\left(Q_r\right)}_{11}=1^2+0^2+{(1/2)}^2=1+1/4=5/4$. ${\left(Q_r\right)}_{12}=1·(-1/2)+0·1+(1/2)(1/2)=-1/2+1/4=-1/4$. And so on.

Appendix B.3. Selected Cascade Matrices B(κ)

For $\kappa =(2,4,6,9)$ (row $C_1$):

$$B(2,4,6,9)=\left(\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 2& 1& 0& 0& 0& 0& 0\\ 0& 0& 4& 1& 0& 0& 0& 0\\ 0& 0& 0& 6& 1& 0& 0& 0\\ 0& 0& 0& 0& 9& 1& 0& 0\\ 0& 1& 0& 0& 0& 6& 1& 0\\ 0& 0& 1& 0& 0& 0& 15& 1\end{array}\right).$$

This is a $7\times 8$ matrix. The Gram product $Q(2,4,6,9)=B{(2,4,6,9)}^{T}B(2,4,6,9)$ is an $8\times 8$ positive semidefinite matrix; its diagonal entries are:

$$\begin{array}{cccccc}\hfill {\left(Q\right)}_{11}& =1,\hfill & \hfill {\left(Q\right)}_{22}& =4+1=5,\hfill & \hfill {\left(Q\right)}_{33}& =1+16+1=18,\hfill \\ \hfill {\left(Q\right)}_{44}& =1+36=37,\hfill & \hfill {\left(Q\right)}_{55}& =81+1=82,\hfill & \hfill {\left(Q\right)}_{66}& =1+36=37,\hfill \\ \hfill {\left(Q\right)}_{77}& =1+225=226,\hfill & \hfill {\left(Q\right)}_{88}& =1.\hfill \end{array}$$

For $\kappa =(3,6,9,14)$ (row $C_2$):

$$B(3,6,9,14)=\left(\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 3& 1& 0& 0& 0& 0& 0\\ 0& 0& 6& 1& 0& 0& 0& 0\\ 0& 0& 0& 9& 1& 0& 0& 0\\ 0& 0& 0& 0& 14& 1& 0& 0\\ 0& 1& 0& 0& 0& 9& 1& 0\\ 0& 0& 1& 0& 0& 0& 23& 1\end{array}\right).$$

Diagonal entries of $Q(3,6,9,14)$: $(1,10,37,82,197,82,530,1)$.

For the general cascade matrix $B(a,b,c,d)$, the nonzero entries in column j, for $j=1,\dots ,8$, are:

$$\begin{array}{cc}\hfill \mathrm{col}.1:& B_{11}=1;\hfill \\ \hfill \mathrm{col}.2:& B_{22}=a,B_{62}=1;\hfill \\ \hfill \mathrm{col}.3:& B_{23}=1,B_{33}=b,B_{73}=1;\hfill \\ \hfill \mathrm{col}.4:& B_{34}=1,B_{44}=c;\hfill \\ \hfill \mathrm{col}.5:& B_{45}=1,B_{55}=d;\hfill \\ \hfill \mathrm{col}.6:& B_{56}=1,B_{66}=a+b;\hfill \\ \hfill \mathrm{col}.7:& B_{67}=1,B_{77}=c+d;\hfill \\ \hfill \mathrm{col}.8:& B_{78}=1.\hfill \end{array}$$

(Rows indexed 1–7, columns 1–8.)

The corresponding Gram product has diagonal entries:

$${\left(Q\right)}_{jj}:1,a^2+1,b^2+2,c^2+1,d^2+1,{(a+b)}^2+2,{(c+d)}^2+2,1.$$

And the off-diagonal entry ${\left(Q\right)}_{23}=a·1+0+0=a$ (the product of column 2 and column 3: only row 2 contributes, giving $B_{22}{B}_{23}=a·1=a$).