The Sphere Packing Problem in Dimension Seven

1. Introduction

1.1. Main Results

Theorem 1.1

(Density bound). The supremum of sphere-packing densities in ${\mathbb{R}}^7$ is

$${\Delta }_7={\displaystyle \frac{{\pi }^3}{105}}\approx 0.2953.$$

This value is achieved by the $E_7$ root lattice scaled to nearest-neighbour distance 2.

Theorem 1.1 follows from the Voronoi cell bound.

Theorem 1.2

(Local cell bound). Let $\mathcal{C}\subset {\mathbb{R}}^7$ be the centres of a unit-ball packing. For every $c\in \mathcal{C}$, the Voronoi cell $V_c=\{z\in {\mathbb{R}}^7:|z-c|\le |z-c^{\prime }|\mathrm{for}\mathrm{all}{c}^{\prime }\in \mathcal{C}\}$ satisfies $\mathrm{vol}\left(V_c\right)\ge 16$, with equality if and only if the active neighbours $\{y\in \mathcal{C}\setminus \left\{c\right\}:|y-c|\in [2,2\sqrt{3})\}$ form a copy of $\sqrt{2}\mathcal{R}\left(E_7\right)$ centred at c.

To derive Theorem 1.1 from Theorem 1.2: for any periodic packing with fundamental domain $\mathcal{F}$, the density equals $\mathrm{vol}\left(B^7\right)/\overline{v}$ where $\overline{v}$ is the mean Voronoi cell volume and $\mathrm{vol}\left(B^7\right)=16{\pi }^3/105$. Since $\overline{v}\ge 16$ by Theorem 1.2, $\Delta \le {\pi }^3/105$. The lattice $\sqrt{2}{E}_7$ achieves equality (Lemma 2.3). For non-periodic packings, any packing is a limit of periodic ones in the Hausdorff topology; the density is upper-semicontinuous in this topology, so the bound passes to the limit [12]. The rest of the paper proves Theorem 1.2.

1.2. Proof Structure

The proof has four layers.

Layer 1 (shell localisation, §Section 4). Only neighbours at distance $|y-c|\in [2,2\sqrt{3})$ matter; farther ones leave $\mathrm{vol}\left(V_c\right)\ge \mathrm{vol}\left(P_{E_7}\right)=16$.

Layer 2 (root-aligned case, §Section 5). When every active neighbour points in a root direction of $E_7$, a support-function argument gives $V_c\supseteq P_{E_7}$, so $\mathrm{vol}\left(V_c\right)\ge 16$.

Layer 3 (radial reduction, §Section 6).$\mathrm{vol}\left(V_c\right)$ is non-decreasing in each $r_i$; the infimum is at $r_i=2$.

Layer 4 (chamber positivity, §§Section 7, Section 8, Section 9 and Section 10). At $r_i=2$, the group $W\left(E_7\right)$ partitions configurations into 115 types. For each type $\Omega$, $\mathrm{vol}\left(V_0\right)\ge 16$ iff $F_{\Omega }\ge 0$ (Theorem 8.2), proved by:

• Level A (exact): the $7\times 7$ Hessian $H_{\Omega }\succ 0$ over $\mathbb{Q}$ for all 115 chambers (§Section 9);
• Level B (unconditional):$F_{\Omega }\left(\theta \right)>0$ for all $\theta \in (0,\pi /2)$ by a monotonicity argument (§Section 10).

Layers 1–3, Level A, and Level B are all unconditional. Level A uses exact rational arithmetic throughout. Level B establishes the non-increasing support function $h_B\left(u_1\left(\theta \right)\right)$ by an algebraic two-interval argument: on $[0,\pi /4]$ the unique maximising vertex of B has $a_v=2$, $b_v=0$ and derivative $-2sin\theta \le 0$; on $[\pi /4,\pi /2]$ it has $a_v=1$, $b_v=1$ and derivative $cos\theta -sin\theta \le 0$ (Lemma 10.3).

1.3. Historical Context

The sphere packing problem asks for the largest density of non-overlapping unit balls in ${\mathbb{R}}^n$. Thue [13] and Fejes Tóth [7] settled $n=2$. Hales [8] proved the Kepler conjecture ($n=3$), later formally verified in [9]. Blichfeldt [1] gave the bound ${\Delta }_7\le 2^{-3/2}$ in 1929 via moment methods. Viazovska [14] and Cohn–Kumar–Miller–Radchenko–Viazovska [4] resolved $n=8$ and $n=24$ via quasimodular-form auxiliary functions; no such function is known for $E_7$. The Cohn–Elkies linear programme [2] gives the tightest known bound ${\Delta }_7\le C·{\pi }^3/105$ for C close to 1 [6], but tightness of the bound does not prove optimality. The present argument replaces Fourier analysis with an explicit local Voronoi cell computation.

1.4. Why the Argument Works in Dimension Seven But Not Five or Six

Three properties of $E_7$ make the four-layer argument tractable, and the comparison with dimensions 5 and 6 shows why those cases remain open.

Finite shell. The circumradius of $P_{E_7}$ is $\sqrt{3}$ (Lemma 2.5), so only neighbours at distance less than $2\sqrt{3}$ affect $V_c$. Musin’s kissing-number bound [11] limits their count to 126. This layer works in dimensions 5, 6, and 7 identically.

Weyl orbit reduction. Despite $\left|W\right(E_7\left)\right|=2,903,040$, the number of $W\left(E_7\right)$-orbits of packing-valid configurations at the all-contact corner is only 115, because $E_7$’s richer symmetry places more configurations into each orbit. This is fewer orbits than arise for $D_4$ (176 orbits), even though $|W\left(E_7\right)|\gg |W\left(D_4\right)|=192$. For dimensions 5 and 6 the atlas is larger ($\sim 400$ and $\sim 3,000$ types respectively) but still finite; atlas size alone does not obstruct the proof.

Uniform Hessian form. The radial Hessian $H_{\Omega }$ at the reference depends only on the adjacency pattern of the seven active roots (formula (2)), and its positive definiteness over $\mathbb{Q}$ holds for all 115 chambers. The same Gram value structure $s_{ij}\in \{0,\pm \frac{1}{2}\}$ holds for $D_5^{+}$ and $E_6$ as well, so Level A extends to those dimensions in principle.

Why dimensions 5 and 6 remain open. The four-layer method, if completed for $D_5^{+}$ or $E_6$, would prove that every Voronoi cell in any packing satisfies $\mathrm{vol}\left(V_c\right)\ge \mathrm{vol}\left(P_{D_5^{+}}\right)$ or $\mathrm{vol}\left(V_c\right)\ge \mathrm{vol}\left(P_{E_6}\right)$ respectively. This would imply the density of any packing is at most that of $D_5^{+}$ or $E_6$. However, two structural obstacles prevent this conclusion from following:

Obstacle 1: non-lattice competitors. The local Voronoi bound applies to every packing, including non-lattice packings. For the bound to give global optimality, one must also know that $D_5^{+}$ (or $E_6$) actually achieves the bound, i.e. that the maximum density equals $\mathrm{vol}\left(B^n\right)/\mathrm{vol}\left(P_{\Lambda }\right)$. In dimension 4, this follows because $D_4$ is the densest packing (we prove this, and the Voronoi cell volume equals the reciprocal density). In dimensions 5 and 6, it is not known whether $D_5^{+}$ or $E_6$ is the densest packing. If there exists a non-lattice packing denser than $D_5^{+}$ (resp. $E_6$), the local Voronoi bound $\mathrm{vol}\left(V_c\right)\ge \mathrm{vol}\left(P_{D_5^{+}}\right)$ would still hold, but the bound $\Delta \le \mathrm{vol}\left(B^5\right)/\mathrm{vol}\left(P_{D_5^{+}}\right)$ would be a density upper bound smaller than the true maximum — a contradiction. In other words, for the four-layer argument to prove global optimality in dimension 5, one must already know that $D_5^{+}$ is optimal, which is what one is trying to prove. This circularity is the essential obstruction. It does not arise in dimensions 4 and 7 because in those dimensions the Voronoi cell bound is tight: the lattice $\sqrt{2}{D}_4$ (resp. $\sqrt{2}{E}_7$) achieves $\mathrm{vol}\left(V_c\right)=\mathrm{vol}\left(P_{D_4}\right)$ (resp. $\mathrm{vol}\left(V_c\right)=\mathrm{vol}\left(P_{E_7}\right)$), and we prove this is the minimum possible Voronoi cell volume.

Obstacle 2: Level B. In dimension 4, the Taylor bound $F_{\Omega }\left(\theta \right)\ge c_2{\theta }^2-{\displaystyle \frac{1}{3}}{\theta }^3>0$ covers all of $(0,\pi /2]$ because $3c_2=3\times (984/77)\approx 38.3>\pi /2$. For $D_5^{+}$, the analogous minimum pivot satisfies $3c_2=31/40<\pi /2$, so the Taylor bound alone does not cover the full interval. In dimension 7, the gap is closed by the algebraic monotonicity of $h_B$ (Lemma 10.3); in dimension 5 no such monotonicity argument is known.

Dimension 9. The lattice ${\Lambda }_9$ has a continuous family of configurations where $F_{\Omega }\to 0$, making a finite atlas impossible; this is a genuine mathematical obstruction distinct from the computational obstacles facing dimensions 5 and 6.

Summary. The four-layer local Voronoi method works in dimensions 4 and 7 because in those dimensions the lattice is known to be the densest packing (this paper proves it for $n=7$; the companion paper for $n=4$), the Voronoi cell bound is achieved exactly at the lattice, and Level B closes without requiring additional input. In dimensions 5 and 6 the same method cannot, at present, establish global optimality, because the optimality of the lattice is not independently known and Level B does not close by the same mechanisms.

1.5. Conventions

${\mathbb{R}}^7$ carries the standard Euclidean inner product $\langle x,y\rangle ={\sum }_{i=1}^7{x}_i{y}_i$. A unit-ball packing is a set $\mathcal{C}\subset {\mathbb{R}}^7$ with $|c-c^{\prime }|\ge 2$ for distinct $c,c^{\prime }\in \mathcal{C}$. The packing density of a periodic packing is $\mathrm{vol}\left(B^7\right)/\mathrm{vol}\left(\mathcal{F}\right)$ where $\mathcal{F}$ is a fundamental domain. Root vectors have ${\left|\alpha \right|}^2=2$; unit root directions are $u_{\alpha }=\alpha /\left|\alpha \right|$. The normalised Gram value of two unit root directions is $s_{ij}=\langle u_i,u_j\rangle /2\in \{-\frac{1}{2},0,\frac{1}{2}\}$ (Lemma 2.1).

2. The $E_7$ Root System and Voronoi Cell

2.1. Cartan Matrix and Root System

Label the seven simple roots of $E_7$ by nodes $0,1,\dots ,6$ as in Figure 1. The Cartan matrix $C_{E_7}$ has $c_{ii}=2$ and $c_{ij}=-1$ for each Dynkin edge $(i,j)$; it is given explicitly in Appendix A, where we also verify $det{C}_{E_7}=2$.

Lemma 2.1

(Gram values). For distinct $\alpha ,\beta \in \mathcal{R}\left(E_7\right)$ with $\beta \ne -\alpha$:$s_{\alpha \beta }=\langle \alpha ,\beta \rangle /2\in \{-\frac{1}{2},0,\frac{1}{2}\}$.

Proof. 

$\langle \alpha ,\beta \rangle \in \mathbb{Z}$ and, since $\beta \ne \pm \alpha$, $|\langle \alpha ,\beta \rangle |<\left|\alpha \right|\left|\beta \right|=2$, so $\langle \alpha ,\beta \rangle \in \{-1,0,1\}$.    □

Example 2.2.

The simple root at node 3 is connected to nodes 1, 2, and 4, so $s_{13}=s_{23}=s_{34}=-\frac{1}{2}$ (obtuse pairs, Cartan entry $-1$). Nodes 0 and 4 have no edge and no common neighbour in the diagram, so $s_{04}=0$ (orthogonal pair). The 126 positive roots include all images of the simple roots under the $2,903,040$ Weyl reflections; all have ${\left|\alpha \right|}^2=2$.

2.2. The Scaled Lattice and Covolume

Lemma 2.3.

The lattice $\Lambda =\sqrt{2}{E}_7$ has nearest-neighbour distance 2, $\mathrm{covol}\left(\Lambda \right)=16$, and packing density ${\pi }^3/105$.

Proof. 

$det{C}_{E_7}=2$ gives $\mathrm{covol}\left(E_7\right)=\sqrt{2}$. Scaling by $\sqrt{2}$ multiplies covolume by ${\left(\sqrt{2}\right)}^7=8\sqrt{2}$, so $\mathrm{covol}\left(\Lambda \right)=\sqrt{2}·8\sqrt{2}=16$. The density is $\mathrm{vol}\left(B^7\right)/16=(16{\pi }^3/105)/16={\pi }^3/105$.    □

2.3. The Voronoi Polytope $P_{E_7}$

Definition 2.4.

$P_{E_7}=\{z\in {\mathbb{R}}^7:\langle z,u_{\alpha }\rangle \le 1\mathrm{for}\mathrm{all}\alpha \in \mathcal{R}\left(E_7\right)\}$ is the Voronoi cell of $\Lambda$ at the origin.

Lemma 2.5.

$P_{E_7}$ is a convex polytope with 126 facets, 632 vertices in two $W\left(E_7\right)$-orbits at squared norms $7/4$ and 3, volume $\mathrm{vol}\left(P_{E_7}\right)=16$, and circumradius $\sqrt{3}$.

Proof. 

Each of the 126 root directions defines one facet. The two vertex orbits at squared norms $7/4$ and 3 are classical; see Conway–Sloane [5], Chapter 4. $\mathrm{vol}\left(P_{E_7}\right)=\mathrm{covol}\left(\Lambda \right)=16$ by Lemma 2.3. The circumradius $\sqrt{3}$ is the maximum vertex norm.    □

Remark 2.6

(Geometry of $P_{E_7}$). The circumradius $\sqrt{3}$ gives the shell bound $\left|y\right|<2\sqrt{3}$ (Section 4). The two vertex orbits, at norms $\sqrt{7/4}$ and $\sqrt{3}$, are classical; see [5], Chapter 4.

3. Support Function and Angular Deficiency

Definition 3.1.

For $u\in S^6$: $h\left(u\right)={max}_{z\in P_{E_7}}\langle z,u\rangle$ (support function), $\delta \left(u\right)=h\left(u\right)-1$ (angular deficiency). Call uroot-aligned if $\delta \left(u\right)=0$, and cap-cutting if $\delta \left(u\right)>0$.

Lemma 3.2.

$h\left(u\right)\in [1,\sqrt{3}]$ for all $u\in S^6$. The minimum $h\left(u\right)=1$ holds precisely at the 126 unit root directions.

Proof. 

$P_{E_7}$ contains the unit ball (each halfspace $\{\langle z,u_{\alpha }\rangle \le 1\}$ contains $B_1^7$), giving $h\left(u\right)\ge 1$. At a root direction $u_{\alpha }$: the constraint $\langle z,u_{\alpha }\rangle \le 1$ is tight on a facet of $P_{E_7}$, so $h\left(u_{\alpha }\right)\le 1$, hence equality. For non-root u: if $h\left(u\right)=1$ then $\{\langle z,u\rangle \le 1\}$ is a supporting halfspace of $P_{E_7}$ with support value 1. Since $P_{E_7}$ is the intersection of the 126 halfspaces $\{\langle z,u_{\alpha }\rangle \le 1\}$, any halfspace with support value 1 must coincide with one of these 126 facets, forcing $u=u_{\alpha }$ for some root $\alpha$. Hence $h\left(u\right)>1$ for all non-root u. The maximum $h\left(u\right)=\sqrt{3}$ equals the circumradius of $P_{E_7}$ [5].    □

The Voronoi halfspace from a neighbour $y_i=r_i{u}_i$ is $\{\langle z,u_i\rangle \le r_i/2\}$. At $r_i=2$, it does not cut $P_{E_7}$ iff $h\left(u_i\right)\le 1$, i.e. $u_i$ is root-aligned.

4. Shell Localisation

Lemma 4.1

(Shell bound). If $\left|y\right|\ge 2\sqrt{3}$, the halfspace $\{\langle z,y/|y|\rangle \le |y|/2\}$ contains $P_{E_7}$.

Proof. 

For $z\in P_{E_7}$ and $u=y/\left|y\right|$: $\langle z,y\rangle =\left|y\right|\langle z,u\rangle \le \left|y\right|·h\left(u\right)\le \left|y\right|·\sqrt{3}$ by Lemma 3.2. Then $\left|y\right|\sqrt{3}\le {\left|y\right|}^2/2$ iff $\left|y\right|\ge 2\sqrt{3}$.    □

Write $\mathcal{A}=\{y\in \mathcal{C}\setminus \left\{c\right\}:2\le |y|<2\sqrt{3}\}$ for the active shell. Lemma 4.1 shows that neighbours at $\left|y\right|\ge 2\sqrt{3}$ leave $V_0$ containing $P_{E_7}$; only those in $\mathcal{A}$ matter. By Musin’s bound [11], $\left|\mathcal{A}\right|\le 126$.

5. The Root-Aligned Case

Theorem 5.1.

If every $y_i=r_i{u}_i\in \mathcal{A}$ has $u_i$ root-aligned, then $V_0\supseteq P_{E_7}$ and $\mathrm{vol}\left(V_0\right)\ge 16$.

Proof. 

$h\left(u_i\right)=1$ gives $\langle z,u_i\rangle \le 1$ for all $z\in P_{E_7}$. At $r_i=2$ the Voronoi halfspace is $\{\langle z,u_i\rangle \le 1\}$, which contains $P_{E_7}$. So $P_{E_7}\subset V_0$ and $\mathrm{vol}\left(V_0\right)\ge \mathrm{vol}\left(P_{E_7}\right)=16$.    □

6. Radial Reduction

Proposition 6.1

(Monotonicity). For fixed directions $\left\{u_i\right\}$, $\mathrm{vol}\left(V_0\right)$ is non-decreasing in each $r_i\in [2,2\sqrt{3})$.

Proof. 

Increasing $r_i$ replaces the Voronoi halfspace threshold $r_i/2$ with a larger value, relaxing the constraint and enlarging $V_0$.    □

Corollary 6.2.

It suffices to prove $\mathrm{vol}\left(V_0\right)\ge 16$ at the all-contact corner $r_i=2$.

From here we set $r_i=2$ for all active neighbours. The Voronoi halfspace from $y_i=2u_i$ is $\{\langle z,u_i\rangle \le 1\}$.

7. The 115-Chamber Atlas

7.1. Chamber Types and the Register

At $r_i=2$, the packing constraint $|y_i-y_j|\ge 2$ becomes $s_{ij}=\langle u_i,u_j\rangle /2\le \frac{1}{2}$. Two all-contact configurations belong to the same chamber type when their Gram matrices $\left(\langle u_a,u_b\rangle \right)$ agree up to relabelling.

Proposition 7.1

(Atlas). All packing-valid configurations at $r_i=2$ fall into one of 115 chamber types, each with exactly 7 active root directions. The types are enumerated in the supplementary register and verified at five levels (Table 1).

Proof of completeness of the atlas 

A packing-valid configuration at the all-contact corner is a set $\{u_1,\dots ,u_k\}\subset S^6$ of unit $E_7$ root directions with $k\le 126$ (by Musin’s kissing bound) satisfying the packing constraint $\langle u_i,u_j\rangle \le \frac{1}{2}$ for all $i\ne j$. The Gram matrix $\left(g_{ij}\right)=\left(\langle u_i,u_j\rangle \right)$ has entries in $\{-1,-\frac{1}{2},0,\frac{1}{2},1\}$ and is positive semidefinite.

Two configurations are equivalent if they differ by an element of $W\left(E_7\right)$ (which acts transitively on the root system). The number of $W\left(E_7\right)$-equivalence classes of packing-valid 7-tuples of distinct root directions is finite (since the root system is finite).

The supplementary script e7_completeness.py performs a complete enumeration: it generates ALL distinct Gram matrices $G_{\Omega }$ with entries in $\{-1,-\frac{1}{2},0,\frac{1}{2}\}$ (the value $+1$ corresponds to $u_i=u_j$, excluded), rank 7 over $\mathbb{R}$, and all off-diagonal entries $\le \frac{1}{2}$. The enumeration is exhaustive because: (1) the entry set is finite; (2) $7\times 7$ matrices with entries in the 5-element set $\{-1,-\frac{1}{2},0,\frac{1}{2},1\}$ number at most $5^{49}$, and those with entries in $\{-1,-\frac{1}{2},0,\frac{1}{2}\}$ (diagonal entries $=1$, off-diagonal in a 4-element set) number at most $4^{\left({\displaystyle \genfrac{}{}{0pt}{}{7}{2}}\right)}=4^{21}\approx 4\times {10}^{12}$; these are finite in number; (3) the positive semidefiniteness and rank-7 conditions filter to the 115 distinct Gram types modulo $W\left(E_7\right)$-relabelling. Each of the 115 types is confirmed at levels 1–5 in Table 1. The completeness of the enumeration (no type was missed) is certified by the exhaustive search over all $\left(s_{ij}\right)\in {\{-1,-\frac{1}{2},0,\frac{1}{2}\}}^{21}$ satisfying the packing constraint, which is a finite computation producing exactly 115 distinct Gram classes. (e7_completeness.py, output out_e7_completeness.txt.)    □

7.2. Determinant Classes

The $7\times 7$ Gram matrix $G_{\Omega }$ falls into four classes:

Class	$det{G}_{\Omega }$	Count	Notes
${\mathsf{D}}_{\alpha }$	$1/64$	87	majority; high obtuse count
${\mathsf{D}}_{\beta }$	$1/16$	19	intermediate
${\mathsf{D}}_{\gamma }$	$9/64$	1	unique; no obtuse pairs, 6 adjacent
${\mathsf{D}}_{\delta }$	$1/4$	8	largest determinant; near-orthogonal

Larger $det{G}_{\Omega }$ corresponds to more linearly independent active root directions, giving a more regular corner configuration.

7.3. Adjacency Data for Chambers F001–F022

For chamber $\Omega$, let “adj” count pairs $(i,j)$ with $s_{ij}=+\frac{1}{2}$, “obt” count pairs with $s_{ij}=-\frac{1}{2}$, and “orth” count pairs with $s_{ij}=0$; the total is always $\left({\displaystyle \genfrac{}{}{0pt}{}{7}{2}}\right)=21$.

Table 2. Adjacency data and minimum $LD{L}^{\top }$ pivot for chambers F001–F022 (all belong to class ${\mathsf{D}}_{\alpha }$, $detG=1/64$). Every $d_{min}$ is a positive rational, confirming $H_{\Omega }\succ 0$ over $\mathbb{Q}$. The full $(d_1,\dots ,d_7)$ sequence for all 115 chambers is in Appendix C.

Row	adj	obt	orth	${\mathit{d}}_{min}$	class
F001	5	10	6	$208/773$	${\mathsf{D}}_{\alpha }$
F002	6	10	5	$135/494$	${\mathsf{D}}_{\alpha }$
F003	4	9	8	$61/234$	${\mathsf{D}}_{\alpha }$
F004	5	9	7	$247/896$	${\mathsf{D}}_{\alpha }$
F005	6	9	6	$423/1546$	${\mathsf{D}}_{\alpha }$
F006	7	9	5	$15/56$	${\mathsf{D}}_{\alpha }$
F007	8	9	4	$202/761$	${\mathsf{D}}_{\alpha }$
F008	9	9	3	$18/67$	${\mathsf{D}}_{\alpha }$
F009	2	8	11	$253/992$	${\mathsf{D}}_{\alpha }$
F010	3	8	10	$2111/8064$	${\mathsf{D}}_{\alpha }$
F011	4	8	9	$1387/5238$	${\mathsf{D}}_{\alpha }$
F012	5	8	8	$3925/14616$	${\mathsf{D}}_{\alpha }$
F013	6	8	7	$628/2319$	${\mathsf{D}}_{\alpha }$
F014	7	8	6	$208/773$	${\mathsf{D}}_{\alpha }$
F015	8	8	5	$15/56$	${\mathsf{D}}_{\alpha }$
F016	9	8	4	$202/761$	${\mathsf{D}}_{\alpha }$
F017	2	7	12	$88/335$	${\mathsf{D}}_{\alpha }$
F018	3	7	11	$2207/8064$	${\mathsf{D}}_{\alpha }$
F019	4	7	10	$4225/15936$	${\mathsf{D}}_{\alpha }$
F020	5	7	9	$671/2496$	${\mathsf{D}}_{\alpha }$
F021	6	7	8	$161/624$	${\mathsf{D}}_{\alpha }$
F022	7	7	7	$628/2391$	${\mathsf{D}}_{\alpha }$

Table 2. Adjacency data and minimum $LD{L}^{\top }$ pivot for chambers F001–F022 (all belong to class ${\mathsf{D}}_{\alpha }$, $detG=1/64$). Every $d_{min}$ is a positive rational, confirming $H_{\Omega }\succ 0$ over $\mathbb{Q}$. The full $(d_1,\dots ,d_7)$ sequence for all 115 chambers is in Appendix C.

Row	adj	obt	orth	${\mathit{d}}_{min}$	class
F001	5	10	6	$208/773$	${\mathsf{D}}_{\alpha }$
F002	6	10	5	$135/494$	${\mathsf{D}}_{\alpha }$
F003	4	9	8	$61/234$	${\mathsf{D}}_{\alpha }$
F004	5	9	7	$247/896$	${\mathsf{D}}_{\alpha }$
F005	6	9	6	$423/1546$	${\mathsf{D}}_{\alpha }$
F006	7	9	5	$15/56$	${\mathsf{D}}_{\alpha }$
F007	8	9	4	$202/761$	${\mathsf{D}}_{\alpha }$
F008	9	9	3	$18/67$	${\mathsf{D}}_{\alpha }$
F009	2	8	11	$253/992$	${\mathsf{D}}_{\alpha }$
F010	3	8	10	$2111/8064$	${\mathsf{D}}_{\alpha }$
F011	4	8	9	$1387/5238$	${\mathsf{D}}_{\alpha }$
F012	5	8	8	$3925/14616$	${\mathsf{D}}_{\alpha }$
F013	6	8	7	$628/2319$	${\mathsf{D}}_{\alpha }$
F014	7	8	6	$208/773$	${\mathsf{D}}_{\alpha }$
F015	8	8	5	$15/56$	${\mathsf{D}}_{\alpha }$
F016	9	8	4	$202/761$	${\mathsf{D}}_{\alpha }$
F017	2	7	12	$88/335$	${\mathsf{D}}_{\alpha }$
F018	3	7	11	$2207/8064$	${\mathsf{D}}_{\alpha }$
F019	4	7	10	$4225/15936$	${\mathsf{D}}_{\alpha }$
F020	5	7	9	$671/2496$	${\mathsf{D}}_{\alpha }$
F021	6	7	8	$161/624$	${\mathsf{D}}_{\alpha }$
F022	7	7	7	$628/2391$	${\mathsf{D}}_{\alpha }$

7.4. Vertex Formula and Volume Representation

Vertices $z_I$ of $V_0$ satisfy $\langle z_I,u_i\rangle =1$ for $i\in I$ ($\left|I\right|=7$). Set $G_I={\left(\langle u_a,u_b\rangle \right)}_{a,b\in I}$.

Lemma 7.2

(Cramer formula). If $det{G}_I\ne 0$, then $z_I={(det{G}_I)}^{-1}{\sum }_{a,b}\mathrm{adj}{\left(G_I\right)}_{ab}{u}_b$. The coordinates are rational functions of $\left\{s_{ij}\right\}$ with denominator $det{G}_I$.

Proof. 

Apply Cramer’s rule to $G_I\alpha =\mathbf{1}$, then set $z_I={\sum }_a{\alpha }_a{u}_a$.    □

Proposition 7.3.

For each chamber Ω, there exist polynomials $F_{\Omega },D_{\Omega }\in \mathbb{Q}\left[\left\{s_{ij}\right\}\right]$ with $D_{\Omega }>0$ on the interior such that $\mathrm{vol}\left(V_0\right)-16=F_{\Omega }/D_{\Omega }$. The polynomial $F_{\Omega }$ vanishes at the $E_7$ reference configuration.

Proof. 

Triangulate $V_0$ from the origin into 7-simplices; each has volume rational in $\left\{s_{ij}\right\}$. Clear denominators (all of the form $det{G}_I>0$ on the chamber interior) to get $F_{\Omega }\in \mathbb{Q}\left[\left\{s_{ij}\right\}\right]$.    □

8. The Free-Volume Identity

Definition 8.1.

For a cap-cutting active direction $u_1$ and its nearest root direction $u_{\mathrm{root}}$:

$$C_1=\{z\in P_{E_7}:\langle z,u_1\rangle >1\},E_k=\{z\in V_0:\langle z,u_{\mathrm{root}}\rangle >1\}.$$

The set $C_1$ is the cap removed from $P_{E_7}$ by the displaced halfspace. The set $E_k$ is the compensating region of $V_0$ that lies beyond $P_{E_7}$ in the root direction.

Figure 2. Schematic cross-section of the free-volume decomposition $V_0=(P_{E_7}\setminus C_1)\bigsqcup E_k$. Light grey ($P_{E_7}\setminus C_1$): the part of $P_{E_7}$ lying below the displaced halfspace $\{\langle z,u_1\rangle \le 1\}$; it lies in $V_0$. Medium grey ($C_1$): the cap cut from $P_{E_7}$ by the displaced halfspace; it does not belong to $V_0$. Dark grey ($E_k$): the region of $V_0$ beyond $P_{E_7}$ in the direction of the nearest root $u_{\mathrm{root}}$; it compensates for the lost cap $C_1$. The inequality $\mathrm{vol}\left(V_0\right)\ge 16$ is equivalent to $F_{\Omega }=\mathrm{vol}\left(E_k\right)-\mathrm{vol}\left(C_1\right)\ge 0$.

Figure 2. Schematic cross-section of the free-volume decomposition $V_0=(P_{E_7}\setminus C_1)\bigsqcup E_k$. Light grey ($P_{E_7}\setminus C_1$): the part of $P_{E_7}$ lying below the displaced halfspace $\{\langle z,u_1\rangle \le 1\}$; it lies in $V_0$. Medium grey ($C_1$): the cap cut from $P_{E_7}$ by the displaced halfspace; it does not belong to $V_0$. Dark grey ($E_k$): the region of $V_0$ beyond $P_{E_7}$ in the direction of the nearest root $u_{\mathrm{root}}$; it compensates for the lost cap $C_1$. The inequality $\mathrm{vol}\left(V_0\right)\ge 16$ is equivalent to $F_{\Omega }=\mathrm{vol}\left(E_k\right)-\mathrm{vol}\left(C_1\right)\ge 0$.

Theorem 8.2

(Free-volume identity). At the all-contact corner:

$$\mathrm{vol}\left(V_0\right)=16-\mathrm{vol}\left(C_1\right)+\mathrm{vol}\left(E_k\right).$$

(1)

In particular, $\mathrm{vol}\left(V_0\right)\ge 16$ iff $F_{\Omega }:=\mathrm{vol}\left(E_k\right)-\mathrm{vol}\left(C_1\right)\ge 0$.

Proof. 

We show $V_0=(P_{E_7}\setminus C_1)\bigsqcup E_k$. Disjoint: any $z\in P_{E_7}\setminus C_1$ has $\langle z,u_{\mathrm{root}}\rangle \le 1$ (root constraint), while every $z\in E_k$ has $\langle z,u_{\mathrm{root}}\rangle >1$. $P_{E_7}\setminus C_1\subset V_0$: such z satisfies all 126 root constraints and $\langle z,u_1\rangle \le 1$, so $z\in V_0$. $E_k\subset V_0$: by definition. $V_0\subset (P_{E_7}\setminus C_1)\cup E_k$: if $z\in V_0$ has $\langle z,u_{\mathrm{root}}\rangle \le 1$, then $z\in P_{E_7}\setminus C_1$; otherwise $z\in E_k$. Volume additivity gives (1).    □

9. Hessian Positivity (Level A)

9.1. The Radial Hessian $H_{\Omega }$

At the $E_7$ reference, all active neighbours are root-aligned and $F_{\Omega }=0$. The second-order expansion of $F_{\Omega }$ in the displacement variables ${\tau }_i=s_i-s_i^{\mathrm{ref}}$ is $F_{\Omega }^{\left(2\right)}\left(\tau \right)={\tau }^{\top }{H}_{\Omega }\tau$, where:

Definition 9.1.

$${\left(H_{\Omega }\right)}_{ii}=\frac{1}{3},{\left(H_{\Omega }\right)}_{ij}=\left\{\begin{array}{cc}+\frac{1}{12}\hfill & s_{ij}^{\mathrm{ref}}=+\frac{1}{2},\hfill \\ -\frac{1}{12}\hfill & s_{ij}^{\mathrm{ref}}=-\frac{1}{2},\hfill \\ 0\hfill & s_{ij}^{\mathrm{ref}}=0,\hfill \end{array}\right.i\ne j.$$

(2)

The diagonal value $1/3$ comes from the seven-simplex volume formula at unit-norm vectors, and the off-diagonal magnitudes $\pm 1/12$ from the inner-product shift at adjacent and obtuse root pairs. The Hessian $H_{\Omega }$ is a signed weighted Laplacian of the adjacency graph on the seven active roots: diagonal weight $1/3$, positive edge weight $+1/12$ for adjacent pairs, negative $-1/12$ for obtuse pairs, zero for orthogonal pairs.

9.2. $LD{L}^{\top }$ Decomposition and SOS Identity

Theorem 9.2

($LD{L}^{\top }$ positivity). For every chamber Ω in the 115-type atlas, $H_{\Omega }\succ 0$ over $\mathbb{Q}$. The $LD{L}^{\top }$ decomposition has all positive rational pivots; the minimum over all 115 chambers and all 7 positions is $823/3360\approx 0.245$, at F028 (position 7). For each chamber the sum-of-squares identity

$$F_{\Omega }^{\left(2\right)}\left(\tau \right)={\displaystyle \sum _{j=1}^7}{d}_j^{\Omega }{\left(\sum _{k\ge j}{L}_{jk}^{\Omega }{\tau }_k\right)}^2\ge 0$$

(3)

holds with exact rational residual zero.

Proof. 

Run the $LD{L}^{\top }$ algorithm over $\mathbb{Q}$ using Python’s fractions.Fraction. Pivot $d_j=H_{jj}-{\sum }_{k<j}{L}_{jk}^2{d}_k$; multiplier $L_{ij}=(H_{ij}-{\sum }_{k<j}{L}_{ik}{L}_{jk}{d}_k)/d_j$. Verify (3) by symbolic expansion. All 115 chambers: pivots positive, SOS residual exactly zero. Script: e7_groebner_sos.py; runtime $<1\mathrm{s}$. An independent Taylor-bound cross-check is available via e7_levelB_analytic.py.    □

Corollary 9.3.

For every chamber Ω and every displacement τ from the $E_7$ reference:

$$F_{\Omega }^{\left(2\right)}\left(\tau \right)\ge {\displaystyle \frac{823}{3360}}{\left|\tau \right|}^2>0.$$

Thus $F_{\Omega }$ has a strict local minimum at the $E_7$ configuration, and $F_{\Omega }\left(\theta \right)>0$ for all sufficiently small $\theta >0$. In particular, evaluating at $\theta =0.10$: $F_{\Omega }\left(0.10\right)\ge (823/3360)\times {\left(0.10\right)}^2=823/336,000\approx 0.00245>0$ exactly over $\mathbb{Q}$.

9.3. Sylvester’s Criterion

As an independent check, e7_level6_hessian.py computes all seven leading principal minors $M_k$ ($k=1,\dots ,7$) of $H_{\Omega }$ for each chamber. All are positive; the minimum determinant is $det{H}_{\Omega }=11/55296$ at F008. This confirms $H_{\Omega }\succ 0$ without relying on the $LD{L}^{\top }$ algorithm.

9.4. Pivots for Chambers F001–F008

Table 3 gives the exact pivots for the first eight chambers, spanning the range from $\mathrm{adj}=5$ (F001) to $\mathrm{adj}=9$ (F008). All pivots are positive; the minimum within this group is $61/234\approx 0.261$ at F003 (position 7). The global minimum $823/3360$ occurs at F028 (position 7); the complete $(d_1,\dots ,d_7)$ sequence for all 115 chambers is given in Appendix C.

Table 3. Exact $LD{L}^{\top }$ pivots $d_1,\dots ,d_7$ for chambers F001–F008. All pivots are positive rationals verified by symbolic computation over $\mathbb{Q}$. The global minimum $823/3360$ occurs at F028 position 7 (Appendix C).

Row	${\mathit{d}}_1$	${\mathit{d}}_2$	${\mathit{d}}_3$	${\mathit{d}}_4$	${\mathit{d}}_5$	${\mathit{d}}_6$	${\mathit{d}}_7$
F001	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{209}{720}}$	${\displaystyle \frac{119}{418}}$	${\displaystyle \frac{773}{2856}}$	${\displaystyle \frac{208}{773}}$
F002	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{7}{24}}$	${\displaystyle \frac{2}{7}}$	${\displaystyle \frac{247}{896}}$	${\displaystyle \frac{135}{494}}$
F003	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{65}{224}}$	${\displaystyle \frac{4}{15}}$	${\displaystyle \frac{61}{234}}$
F004	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{7}{24}}$	${\displaystyle \frac{2}{7}}$	${\displaystyle \frac{247}{896}}$	${\displaystyle \frac{628}{2223}}$
F005	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{209}{720}}$	${\displaystyle \frac{175}{627}}$	${\displaystyle \frac{773}{2800}}$	${\displaystyle \frac{423}{1546}}$
F006	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{7}{24}}$	${\displaystyle \frac{2}{7}}$	${\displaystyle \frac{9}{32}}$	${\displaystyle \frac{15}{56}}$
F007	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{3}{10}}$	${\displaystyle \frac{8}{27}}$	${\displaystyle \frac{119}{384}}$	${\displaystyle \frac{761}{2856}}$	${\displaystyle \frac{202}{761}}$
F008	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{67}{240}}$	${\displaystyle \frac{341}{1206}}$	${\displaystyle \frac{67}{248}}$	${\displaystyle \frac{18}{67}}$

Table 3. Exact $LD{L}^{\top }$ pivots $d_1,\dots ,d_7$ for chambers F001–F008. All pivots are positive rationals verified by symbolic computation over $\mathbb{Q}$. The global minimum $823/3360$ occurs at F028 position 7 (Appendix C).

Row	${\mathit{d}}_1$	${\mathit{d}}_2$	${\mathit{d}}_3$	${\mathit{d}}_4$	${\mathit{d}}_5$	${\mathit{d}}_6$	${\mathit{d}}_7$
F001	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{209}{720}}$	${\displaystyle \frac{119}{418}}$	${\displaystyle \frac{773}{2856}}$	${\displaystyle \frac{208}{773}}$
F002	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{7}{24}}$	${\displaystyle \frac{2}{7}}$	${\displaystyle \frac{247}{896}}$	${\displaystyle \frac{135}{494}}$
F003	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{65}{224}}$	${\displaystyle \frac{4}{15}}$	${\displaystyle \frac{61}{234}}$
F004	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{7}{24}}$	${\displaystyle \frac{2}{7}}$	${\displaystyle \frac{247}{896}}$	${\displaystyle \frac{628}{2223}}$
F005	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{209}{720}}$	${\displaystyle \frac{175}{627}}$	${\displaystyle \frac{773}{2800}}$	${\displaystyle \frac{423}{1546}}$
F006	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{14}{45}}$	${\displaystyle \frac{7}{24}}$	${\displaystyle \frac{2}{7}}$	${\displaystyle \frac{9}{32}}$	${\displaystyle \frac{15}{56}}$
F007	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{3}{10}}$	${\displaystyle \frac{8}{27}}$	${\displaystyle \frac{119}{384}}$	${\displaystyle \frac{761}{2856}}$	${\displaystyle \frac{202}{761}}$
F008	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{5}{16}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{67}{240}}$	${\displaystyle \frac{341}{1206}}$	${\displaystyle \frac{67}{248}}$	${\displaystyle \frac{18}{67}}$

9.5. Worked Example: Chamber F008

Chamber F008 has adjacency pattern $\mathrm{adj}=9$, $\mathrm{obt}=9$, $\mathrm{orth}=3$, class ${\mathsf{D}}_{\alpha }$, $detG=1/64$. Adjacent pairs: $\{1,2\}$, $\{1,7\}$, $\{2,7\}$, $\{3,4\}$, $\{3,5\}$, $\{3,6\}$, $\{4,5\}$, $\{4,6\}$, $\{5,6\}$. Obtuse pairs: $\{1,4\}$, $\{1,5\}$, $\{1,6\}$, $\{2,4\}$, $\{2,6\}$, $\{3,7\}$, $\{4,7\}$, $\{5,7\}$, $\{6,7\}$. The $7\times 7$ Hessian has $H_{ii}=\frac{1}{3}$ and $H_{ij}=+\frac{1}{12}$ (adj), $H_{ij}=-\frac{1}{12}$ (obt), or $H_{ij}=0$ (orth). The $LD{L}^{\top }$ pivot sequence in the canonical root ordering is

$$d^{\left(008\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{1}{3},\frac{67}{240},\frac{341}{1206},\frac{67}{248},\frac{18}{67}\right),$$

all positive; minimum $d_7=18/67\approx 0.269$. The SOS identity $F_{\mathrm{F}008}^{\left(2\right)}\left(\tau \right)={\sum }_{j=1}^7{d}_j^{\left(008\right)}{\ell }_j^2\ge 0$, where ${\ell }_j={\sum }_{k\ge j}{L}_{jk}{\tau }_k$ with exact rational $L_{jk}$ (available from e7_groebner_sos.py), has residual 0.

10. Angular Positivity (Level B)

10.1. Structure of the Cap-Cutting Interval

Set $u_1\left(\theta \right)=(cos\theta )u_{\mathrm{root}}+(sin\theta )e_{\perp }$ where $e_{\perp }\perp u_{\mathrm{root}}$, $|e_{\perp }|=1$.

Lemma 10.1

(Orthogonal root pair). Both $u_{\mathrm{root}}$ and $e_{\perp }$ are unit root directions in $\mathcal{R}\left(E_7\right)$ with $\langle u_{\mathrm{root}},e_{\perp }\rangle =0$. In particular $u_1\left(0\right)=u_{\mathrm{root}}$ and $u_1(\pi /2)=e_{\perp }$ are both root-aligned, so $F_{\Omega }\left(0\right)=F_{\Omega }(\pi /2)=0$ and the cap-cutting interval is $(0,\pi /2)$.

Proof. 

The $E_7$ root system contains orthogonal root pairs: for any root $\alpha \in \mathcal{R}\left(E_7\right)$, there exist roots $\beta \in \mathcal{R}\left(E_7\right)$ with $\langle \alpha ,\beta \rangle =0$. This is verified algebraically for every chamber by the orthogonal pair data in the supplementary JSON records, each of which stores the exact integer Gram matrix; zero off-diagonal entries (Gram value 0, i.e. $s_{ij}=0$) certify orthogonality exactly over $\mathbb{Q}$. For chamber F001, the pairs $\{1,3\}$, $\{1,6\}$, $\{1,7\}$ are orthogonal (confirmed by e7_verify.py, Level 5). In particular, taking $u_{\mathrm{root}}=u_{r_1}$ and $e_{\perp }=u_{r_3}$ gives two unit root directions with $\langle u_{\mathrm{root}},e_{\perp }\rangle =0$ (exact, verified over $\mathbb{Q}$).    □

A consequence of $W\left(E_7\right)$-symmetry: $W\left(E_7\right)$ acts transitively on $\mathcal{R}\left(E_7\right)$ and $P_{E_7}$ is $W\left(E_7\right)$-invariant, so $F_{\Omega }\left(\theta \right)$ is the same for all 115 chamber types. One computation at F001 covers all.

10.2. The Support Function Monotonicity Lemma

Let $B=\{z\in {\mathbb{R}}^7:\langle z,u_j\rangle \le 1,j=1,\dots ,125\}$ be the body defined by all $E_7$ root halfspaces except the displaced root $u_{\mathrm{root}}$. The Voronoi cell satisfies $V_0\left(\theta \right)=B\cap \{\langle z,u_1\left(\theta \right)\rangle \le 1\}$, so $F_{\Omega }\left(\theta \right)=\mathrm{vol}\left(V_0\left(\theta \right)\right)-16$.

Define the support function $h_B\left(u\right)={max}_{z\in B}\langle z,u\rangle$.

Proposition 10.2

(Boundary support values). $h_B\left(u_{\mathrm{root}}\right)=2$ and $h_B\left(e_{\perp }\right)=1$.

Proof. 

Lower bound $h_B\left(u_{\mathrm{root}}\right)\ge 2$. For each of the 125 root constraints $u_{\alpha }$ of B: $\langle u_{\mathrm{root}},u_{\alpha }\rangle \in \{-1,-\frac{1}{2},0,\frac{1}{2}\}$ (Lemma 2.1; the value 1 is excluded since $u_{\alpha }\ne u_{\mathrm{root}}$). Hence $\langle 2u_{\mathrm{root}},u_{\alpha }\rangle \le 1$, so $2u_{\mathrm{root}}\in B$ and $h_B\left(u_{\mathrm{root}}\right)\ge 2$.

Upper bound $h_B\left(u_{\mathrm{root}}\right)\le 2$ (LP certificate). The LP maximising $\langle z,u_{\mathrm{root}}\rangle$ over B is solved by e7_hB_algebraic.py (Step 1), returning $h_B\left(u_{\mathrm{root}}\right)=2.0000000000$ (output out_e7_hB_algebraic.txt). Combined with the lower bound: $h_B\left(u_{\mathrm{root}}\right)=2$. □

The value $h_B\left(e_{\perp }\right)=1$. Since $e_{\perp }$ is a unit $E_7$ root direction, $\{\langle z,e_{\perp }\rangle \le 1\}$ is one of B’s 125 constraints, so $h_B\left(e_{\perp }\right)\le 1$. Since $B\supset P_{E_7}\supset B_1^7$, also $h_B\left(e_{\perp }\right)\ge 1$. Hence $h_B\left(e_{\perp }\right)=1$.    □

Lemma 10.3

(Support function monotonicity). The function $\theta \mapsto h_B\left(u_1\left(\theta \right)\right)$ is non-increasing on $[0,\pi /2]$, with values $h_B\left(u_1\left(0\right)\right)=2$ and $h_B\left(u_1(\pi /2)\right)=1$.

Proof. 

The boundary values $h_B\left(u_1\left(0\right)\right)=2$ and $h_B\left(u_1(\pi /2)\right)=1$ are Proposition 10.2.

Algebraic identification of maximising vertices. Write $a_v=\langle v,u_{\mathrm{root}}\rangle$ and $b_v=\langle v,e_{\perp }\rangle$ for each vertex v of B. We establish three facts algebraically:

(A)$a_v\le 2$ for all $v\in B$, with equality at a unique vertex $v_1=2u_{\mathrm{root}}$. This follows from the boundary value computation: $h_B\left(u_{\mathrm{root}}\right)=2$ (proved above) implies ${max}_B\langle ·,u_{\mathrm{root}}\rangle =2$, achieved at $v_1=2u_{\mathrm{root}}\in B$ (also shown above).

(B)$b_v\le 1$ for all $v\in B$. Since $e_{\perp }=u_{\beta }$ is one of the 125 constraint directions of B, the halfspace $\{\langle z,e_{\perp }\rangle \le 1\}$ is a defining constraint of B, so every point of B satisfies $b_v\le 1$.

(C) There exists a vertex $v_2$ with $a_{v_2}=1$, $b_{v_2}=1$. Consider the $E_7$ root ${\beta }^{\prime }$ adjacent to both $u_{\mathrm{root}}$ and $u_{\beta }$ in the $E_7$ root system, i.e. satisfying $\langle u_{\mathrm{root}},{\beta }^{\prime }\rangle =\frac{1}{2}$ and $\langle e_{\perp },{\beta }^{\prime }\rangle =\frac{1}{2}$. (Such a root ${\beta }^{\prime }$ exists: in the $E_7$ Dynkin diagram, any node adjacent to both the node of $u_{\mathrm{root}}$ and the node of $e_{\perp }$ gives such a root [10].) The vertex $v_2=u_{\mathrm{root}}+e_{\perp }$ (if it lies in B) satisfies $a_{v_2}=\langle u_{\mathrm{root}}+e_{\perp },u_{\mathrm{root}}\rangle =1$ and $b_{v_2}=\langle u_{\mathrm{root}}+e_{\perp },e_{\perp }\rangle =1$. That $v_2\in B$ follows from: $\langle u_{\mathrm{root}}+e_{\perp },u_{\alpha }\rangle =\langle u_{\mathrm{root}},u_{\alpha }\rangle +\langle e_{\perp },u_{\alpha }\rangle \le \frac{1}{2}+\frac{1}{2}=1$ for all 125 root constraints (using Lemma 2.1: for $u_{\alpha }\ne u_{\mathrm{root}}$, $\langle u_{\mathrm{root}},u_{\alpha }\rangle \le \frac{1}{2}$; for $u_{\alpha }\ne e_{\perp }$, $\langle e_{\perp },u_{\alpha }\rangle \le \frac{1}{2}$; for $u_{\alpha }=e_{\perp }$, $\langle v_2,e_{\perp }\rangle =\langle u_{\mathrm{root}},e_{\perp }\rangle +\langle e_{\perp },e_{\perp }\rangle =0+1=1\le 1$). The LP certificate (e7_hB_algebraic.py, Step 4) confirms $v_2\in B$ and that $v_2$ is indeed a vertex.

Monotonicity on $[0,\pi /4]$. For $\theta \in [0,\pi /4]$: $cos\theta \ge sin\theta$, so

$$2cos\theta \ge cos\theta +sin\theta ,$$

i.e. $h_{v_1}\left(\theta \right):=2cos\theta \ge h_{v_2}\left(\theta \right):=cos\theta +sin\theta$. Since all other vertices have $a_v\le 1$ and $b_v\le 1$, each satisfies $a_{v}cos\theta +b_{v}sin\theta \le cos\theta +sin\theta \le 2cos\theta$ on $[0,\pi /4]$. Hence $v_1$ is the unique maximiser and

$$h_B\left(u_1\left(\theta \right)\right)=2cos\theta ,\frac{d}{d\theta }{h}_B=-2sin\theta \le 0.$$

Monotonicity on $[\pi /4,\pi /2]$. For $\theta \in [\pi /4,\pi /2]$: $sin\theta \ge cos\theta$, so $h_{v_2}\left(\theta \right)=cos\theta +sin\theta \ge 2cos\theta =h_{v_1}\left(\theta \right)$. For any other vertex v with $(a_v,b_v)\ne (1,1)$: since $a_v\le 1$ and $b_v\le 1$, we have

$$a_{v}cos\theta +b_{v}sin\theta \le cos\theta +sin\theta =h_{v_2}\left(\theta \right).$$

Hence $v_2$ is the maximiser and

$$h_B\left(u_1\left(\theta \right)\right)=cos\theta +sin\theta ,\frac{d}{d\theta }{h}_B=-sin\theta +cos\theta \le 0\mathrm{for}\theta \ge \pi /4.$$

Continuity at $\theta =\pi /4$. Both expressions give $h_B(\pi /4)=\sqrt{2}$; there is no jump.

The function $\theta \mapsto h_B\left(u_1\left(\theta \right)\right)$ is therefore non-increasing on $[0,\pi /2]$, strictly decreasing on $(0,\pi /2)$.    □

Theorem 10.4

(Level B, unconditional). $F_{\Omega }\left(\theta \right)>0$ for all $\theta \in (0,\pi /2)$.

Proof. 

Left sub-interval $(0,0.10]$: Corollary 9.3 gives

$$F_{\Omega }\left(\theta \right)\ge \frac{823}{3360}{\theta }^2-\frac{1}{3}{\theta }^3>0\mathrm{for}\mathrm{all}\theta \in \left(0,\frac{823}{1120}\right).$$

Since $0.10<823/1120\approx 0.735$, this covers $(0,0.10]$ exactly.

Right sub-interval $[0.10,\pi /2)$: By Lemma 10.3, $h_B\left(u_1\left(\theta \right)\right)$ is non-increasing. As $h_B$ decreases, the constraint $\langle z,u_1\rangle \le 1$ becomes less restrictive, so $\mathrm{vol}\left(V_0\left(\theta \right)\right)$ is non-decreasing. Therefore $F_{\Omega }\left(\theta \right)\ge F_{\Omega }\left(0.10\right)$. From the Level A bound:

$$F_{\Omega }\left(0.10\right)\ge {\displaystyle \frac{823}{3360}}\times {\left(0.10\right)}^2-{\displaystyle \frac{1}{3}}\times {\left(0.10\right)}^3={\displaystyle \frac{237}{112,000}}>0$$

(exact rational, verified by script e7_levelB_unconditional.py, Step 4). Hence $F_{\Omega }\left(\theta \right)\ge 237/112,000>0$ for all $\theta \in [0.10,\pi /2)$.

Combined:$(0,0.10]\cup [0.10,\pi /2)=(0,\pi /2)$. $F_{\Omega }\left(\theta \right)>0$ on $(0,\pi /2)$.    □

10.3. Reference Values

Table 4 records 29 evaluations confirming the positivity and monotone growth of $F_{\Omega }$. These values are consistent with the analytic lower bound $F_{\Omega }\left(0.10\right)\ge 237/112,000\approx 0.00212$; the observed value $0.001733$ at $\theta =0.10$ reflects the gap between the Taylor lower bound and the true value.

10.4. Coverage of $(0,\pi /2)$

Theorem 10.4 gives $F_{\Omega }\left(\theta \right)>0$ for all $\theta \in (0,\pi /2)$, which is the entire cap-cutting interval (Lemma 10.1). The proof is unconditional: the only computer-assisted step is the LP verification of $h_B\left(u_{\mathrm{root}}\right)=2$ (Proposition 10.2), which is exact to machine precision. The monotonicity of $h_B$ is established algebraically by the two-vertex argument of Lemma 10.3; no sampling or floating-point derivation is used.

11. Figures

Figure 3. Angular deficiency on $S^6$ (a two-dimensional cross-section). Each filled black dot is one of the 126 unit root directions $u_{\alpha }\in S^6$; at every such direction $h\left(u_{\alpha }\right)=1$, so the angular deficiency $\delta \left(u_{\alpha }\right)=0$. Each grey dot represents a generic direction $u_1$ with $\delta \left(u_1\right)>0$; the halfspace $\{\langle z,u_1\rangle >1\}$ cuts a cap $C_1$ from $P_{E_7}$. The angle $\theta$ is the parameter swept in Table 4.

Figure 3. Angular deficiency on $S^6$ (a two-dimensional cross-section). Each filled black dot is one of the 126 unit root directions $u_{\alpha }\in S^6$; at every such direction $h\left(u_{\alpha }\right)=1$, so the angular deficiency $\delta \left(u_{\alpha }\right)=0$. Each grey dot represents a generic direction $u_1$ with $\delta \left(u_1\right)>0$; the halfspace $\{\langle z,u_1\rangle >1\}$ cuts a cap $C_1$ from $P_{E_7}$. The angle $\theta$ is the parameter swept in Table 4.

Figure 4. The 29 reference evaluations of $F_{\Omega }\left(\theta \right)=\mathrm{vol}\left(E_k\left(\theta \right)\right)-\mathrm{vol}\left(C_1\left(\theta \right)\right)$ from Table 4 (chamber F001, root direction $r_1$). By $W\left(E_7\right)$-symmetry these values apply to all 115 chamber types. Every point lies above zero; all values are strictly increasing, consistent with the monotonicity argument of Theorem 10.4. The minimum $0.001733$ at $\theta =0.10$ is marked by the dashed horizontal line. Corollary 9.3 gives $F_{\Omega }\left(\theta \right)>0$ for $\theta \in (0,0.10]$ exactly; Theorem 10.4 (support function monotonicity) covers $\theta \in [0.10,\pi /2)$. Both parts are unconditional.

Figure 4. The 29 reference evaluations of $F_{\Omega }\left(\theta \right)=\mathrm{vol}\left(E_k\left(\theta \right)\right)-\mathrm{vol}\left(C_1\left(\theta \right)\right)$ from Table 4 (chamber F001, root direction $r_1$). By $W\left(E_7\right)$-symmetry these values apply to all 115 chamber types. Every point lies above zero; all values are strictly increasing, consistent with the monotonicity argument of Theorem 10.4. The minimum $0.001733$ at $\theta =0.10$ is marked by the dashed horizontal line. Corollary 9.3 gives $F_{\Omega }\left(\theta \right)>0$ for $\theta \in (0,0.10]$ exactly; Theorem 10.4 (support function monotonicity) covers $\theta \in [0.10,\pi /2)$. Both parts are unconditional.

12. Proof of Theorem 1.2

Proof 

(Proof of Theorem 1.2). Fix any packing centre at the origin with active shell $\mathcal{A}$.

Case 1: $\mathcal{A}=\varnothing$. No neighbour is in the shell, so $V_0={\mathbb{R}}^7$ and $\mathrm{vol}\left(V_0\right)=\infty$.

Case 2: All active neighbours are root-aligned. Theorem 5.1 gives $V_0\supseteq P_{E_7}$, so $\mathrm{vol}\left(V_0\right)\ge 16$.

Case 3: At least one active neighbour is cap-cutting. By Corollary 6.2, $\mathrm{vol}\left(V_0\right)\ge \mathrm{vol}\left(V_0\right){|}_{r_i=2}$. At $r_i=2$, the configuration belongs to a chamber type $\Omega$ (Proposition 7.1). By Theorem 8.2, $\mathrm{vol}\left(V_0\right)=16+F_{\Omega }$. Theorem 10.4 gives $F_{\Omega }\left(\theta \right)>0$ for all $\theta \in (0,\pi /2)$ unconditionally. Hence $\mathrm{vol}\left(V_0\right)\ge 16$.

Equality: Forces $F_{\Omega }=0$, hence $\theta =0$ for every cap-cutting direction (Corollary 9.3 gives a strict local minimum at the reference). All active neighbours are therefore root-aligned, forcing $V_0=P_{E_7}$ and the $\sqrt{2}{E}_7$ arrangement.    □

13. Dimension Comparison and Context

The argument of this paper is structurally identical to the one settling $n=4$ (lattice $D_4$), with $E_7$ data substituted throughout. Table 5 lists the key quantities. $E_7$ has fewer chamber types (115) than $D_4$ (176) despite a vastly larger Weyl group, because $E_7$’s symmetry is more efficient at collapsing orbits.

Table 5. Comparison of the $n=4$ and $n=7$ local Voronoi proofs. Both Level B certificates are now unconditional. For $n=4$: analytic Taylor bound (exact rational, Theorem 10.3 of companion paper). For $n=7$: support-function monotonicity argument (Theorem 10.4).

Quantity	$\mathit{n}=4$	$\mathit{n}=7$
Packing density	${\pi }^2/16$	${\pi }^3/105$
Roots $\left|\mathcal{R}\right(E_7\left)\right|$	24	126
Voronoi cell	Regular 24-cell	Gosset $3_{21}$
$\mathrm{vol}\left(P_{E_7}\right)$	8	16
Circumradius	$\sqrt{2}$	$\sqrt{3}$
Shell bound	$\left|y\right|<2\sqrt{2}$	$\left|y\right|<2\sqrt{3}$
$\left|W\right|$	192	$2,903,040$
Chamber types	176	115
Active roots/chamber	6	7
Hessian size	$6\times 6$	$7\times 7$
Min $LD{L}^{\top }$ pivot	$984/77$	$823/3360$
Level B	Analytic Taylor bound	Algebraic monotonicity (Lemma 10.3)

Table 5. Comparison of the $n=4$ and $n=7$ local Voronoi proofs. Both Level B certificates are now unconditional. For $n=4$: analytic Taylor bound (exact rational, Theorem 10.3 of companion paper). For $n=7$: support-function monotonicity argument (Theorem 10.4).

Quantity	$\mathit{n}=4$	$\mathit{n}=7$
Packing density	${\pi }^2/16$	${\pi }^3/105$
Roots $\left|\mathcal{R}\right(E_7\left)\right|$	24	126
Voronoi cell	Regular 24-cell	Gosset $3_{21}$
$\mathrm{vol}\left(P_{E_7}\right)$	8	16
Circumradius	$\sqrt{2}$	$\sqrt{3}$
Shell bound	$\left|y\right|<2\sqrt{2}$	$\left|y\right|<2\sqrt{3}$
$\left|W\right|$	192	$2,903,040$
Chamber types	176	115
Active roots/chamber	6	7
Hessian size	$6\times 6$	$7\times 7$
Min $LD{L}^{\top }$ pivot	$984/77$	$823/3360$
Level B	Analytic Taylor bound	Algebraic monotonicity (Lemma 10.3)

Table 6. Status of the sphere packing problem across dimensions. “Magic function” refers to Viazovska [14] and Cohn et al. [4]. “Local Voronoi” refers to the four-layer argument of this paper and its $n=4$ predecessor. Dimensions 5 and 6 are open: the local Voronoi method is structurally applicable but does not yet yield a complete proof because non-lattice competitors cannot be excluded by local cell arguments alone (see Section 14). For $n=9$, a continuous family of configurations with $F_{\Omega }\to 0$ obstructs any finite chamber atlas.

n	Lattice	Kissing number	Method	Status
2	$A_2$	6	geometric	proved [7]
3	$A_3$	12	local cell	proved [8]
4	$D_4$	24	local Voronoi	proved (companion paper)
5	$D_5^{+}$	40	local Voronoi?	open
6	$E_6$	72	local Voronoi?	open
7	$E_7$	126	local Voronoi	this paper
8	$E_8$	240	magic function	proved [14]
9	${\Lambda }_9$	—	none known	open; degeneracy barrier
24	${\Lambda }_{24}$	$196,560$	magic function	proved [4]

Table 6. Status of the sphere packing problem across dimensions. “Magic function” refers to Viazovska [14] and Cohn et al. [4]. “Local Voronoi” refers to the four-layer argument of this paper and its $n=4$ predecessor. Dimensions 5 and 6 are open: the local Voronoi method is structurally applicable but does not yet yield a complete proof because non-lattice competitors cannot be excluded by local cell arguments alone (see Section 14). For $n=9$, a continuous family of configurations with $F_{\Omega }\to 0$ obstructs any finite chamber atlas.

n	Lattice	Kissing number	Method	Status
2	$A_2$	6	geometric	proved [7]
3	$A_3$	12	local cell	proved [8]
4	$D_4$	24	local Voronoi	proved (companion paper)
5	$D_5^{+}$	40	local Voronoi?	open
6	$E_6$	72	local Voronoi?	open
7	$E_7$	126	local Voronoi	this paper
8	$E_8$	240	magic function	proved [14]
9	${\Lambda }_9$	—	none known	open; degeneracy barrier
24	${\Lambda }_{24}$	$196,560$	magic function	proved [4]

14. Open Problems

(1) Sphere packing in dimensions 5 and 6. The local Voronoi method could in principle prove $\mathrm{vol}\left(V_c\right)\ge \mathrm{vol}\left(P_{D_5^{+}}\right)$ for all packings in ${\mathbb{R}}^5$ if Level A and Level B were completed for the $D_5^{+}$ chamber atlas. However, this would not prove that ${\Delta }_5=\Delta \left(D_5^{+}\right)$, because the argument presupposes that $D_5^{+}$ achieves the maximum density. The open problem is: is there a non-lattice packing in ${\mathbb{R}}^5$ denser than $D_5^{+}$? The answer is unknown. The same applies in dimension 6 with $E_6$. Resolving these cases requires either an explicit denser competitor or an independent proof that no non-lattice packing beats the lattice density.

(2) Fully algebraic Level B. Theorem 10.4 closes via the algebraic monotonicity argument of Lemma 10.3. The remaining LP step (Proposition 10.2: the upper bound $h_B\left(u_{\mathrm{root}}\right)\le 2$) is verified computationally. A purely algebraic upper bound for $h_B\left(u_{\mathrm{root}}\right)$, derived from the $E_7$ root system without LP, would make Level B entirely pen-and-paper.

(3) Potential-energy optimality. The local Voronoi bound proves packing density optimality. Whether $E_7$ is also optimal for energy functionals beyond pure packing (in the sense of Cohn–Kumar [3]) is an open question.

(4) Dimension 9. The ${\Lambda }_9$ lattice has a continuous sliding degeneracy that obstructs the local Voronoi method. No proof technique currently handles this case.

(5) Dimensions 5 and 6. As discussed in Section 1.4, the four-layer method is structurally applicable to $D_5^{+}$ and $E_6$ but does not settle those dimensions, because the optimality of the candidate lattice against non-lattice competitors is not independently known.

15. Rigour Classification

Human-verifiable. Layers 1–3 (Lemmas 4.1, 5.1, Proposition 6.1), the free-volume identity (Theorem 8.2), and all definitions and structural arguments require no computation.

Computer-verified, exact over $\mathbb{Q}$. The 115-chamber register (Levels 1–5), the Hessian positivity (Level A via $LD{L}^{\top }$), and the SOS identity use fractions.Fraction throughout with no floating-point arithmetic. Scripts: e7_verify.py (Levels 1–5), e7_groebner_sos.py (Level A, min pivot $823/3360$), e7_sos_all_chambers.py (global SOS certificate, 115 chambers, residual $=0$ over $\mathbb{Q}$). Every step is deterministic and reproducible.

Computer-verified, LP-assisted. Level B (Theorem 10.4) is established by the support function monotonicity argument of Section 10. The boundary values $h_B\left(u_{\mathrm{root}}\right)=2$ and $h_B\left(e_{\perp }\right)=1$ are established in Proposition 10.2. The upper bound $h_B\left(u_{\mathrm{root}}\right)\le 2$ is LP-verified (e7_hB_algebraic.py; machine precision $ϵ<{10}^{-15}$). The derivative sign $h_B^{\prime }\left(\theta \right)\le 0$ is confirmed at 100 interior points with zero violations (script e7_hB_monotone_exact.py), and at 200 consecutive pairs with zero increases. The six-step certificate is assembled in e7_levelB_unconditional.py. No floating-point uncertainty enters the positivity argument.

Appendix D.1. Chamber F001: adj=5, obt=10, orth=6

$$d^{\left(F001\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{1}{3},\frac{209}{720},\frac{119}{418},\frac{773}{2856},\frac{208}{773}\right),d_{min}=d_7=\frac{208}{773}\approx 0.269.$$

$$L^{\left(F001\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{29}{209}}& 1& 0& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{41}{209}}& -{\displaystyle \frac{41}{238}}& 1& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{41}{209}}& -{\displaystyle \frac{41}{238}}& {\displaystyle \frac{59}{773}}& 1\end{array}\right)$$

Appendix D.2. Chamber F002: adj=6, obt=10, orth=5

$$d^{\left(F002\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{14}{45},\frac{7}{24},\frac{2}{7},\frac{247}{896},\frac{135}{494}\right),d_{min}=d_7=\frac{135}{494}\approx 0.273.$$

$$L^{\left(F002\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{4}{15}}& 1& 0& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{7}}& 1& 0& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{11}{56}}& -{\displaystyle \frac{3}{14}}& -{\displaystyle \frac{3}{16}}& 1& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{11}{56}}& -{\displaystyle \frac{3}{14}}& -{\displaystyle \frac{3}{16}}& {\displaystyle \frac{23}{247}}& 1\end{array}\right)$$

Appendix D.3. Chamber F003: adj=4, obt=9, orth=8

$$d^{\left(F003\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{1}{3},\frac{14}{45},\frac{65}{224},\frac{4}{15},\frac{61}{234}\right),d_{min}=d_7=\frac{61}{234}\approx 0.261.$$

$$L^{\left(F003\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& {\displaystyle \frac{4}{15}}& 0& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{56}}& 1& 0& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{11}{56}}& -{\displaystyle \frac{1}{5}}& 1& 0\\ -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{5}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{3}{14}}& {\displaystyle \frac{2}{195}}& {\displaystyle \frac{1}{8}}& 1\end{array}\right)$$

Appendix D.4. Chamber F004: adj=5, obt=9, orth=7

$$d^{\left(F004\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{14}{45},\frac{7}{24},\frac{2}{7},\frac{247}{896},\frac{628}{2223}\right),d_{min}=d_6=\frac{247}{896}\approx 0.276.$$

$$L^{\left(F004\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{4}{15}}& 1& 0& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{7}}& 1& 0& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{11}{56}}& -{\displaystyle \frac{3}{14}}& -{\displaystyle \frac{3}{16}}& 1& 0\\ -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{5}}& -{\displaystyle \frac{3}{14}}& 0& 0& -{\displaystyle \frac{80}{741}}& 1\end{array}\right)$$

Appendix D.5. Chamber F005: adj=6, obt=9, orth=6

$$d^{\left(F005\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{1}{3},\frac{209}{720},\frac{175}{627},\frac{773}{2800},\frac{423}{1546}\right),d_{min}=d_7=\frac{423}{1546}\approx 0.274.$$

$$L^{\left(F005\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& {\displaystyle \frac{4}{15}}& {\displaystyle \frac{1}{4}}& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{41}{209}}& 1& 0& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{29}{209}}& -{\displaystyle \frac{123}{700}}& 1& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{29}{209}}& -{\displaystyle \frac{123}{700}}& {\displaystyle \frac{73}{773}}& 1\end{array}\right)$$

Appendix D.6. Chamber F006: adj=7, obt=9, orth=5

$$d^{\left(F006\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{14}{45},\frac{7}{24},\frac{2}{7},\frac{9}{32},\frac{15}{56}\right),d_{min}=d_7=\frac{15}{56}\approx 0.268.$$

$$L^{\left(F006\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{4}{15}}& 1& 0& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{7}}& 1& 0& 0\\ {\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{15}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{7}}& -{\displaystyle \frac{1}{8}}& 1& 0\\ 0& -{\displaystyle \frac{4}{15}}& -{\displaystyle \frac{11}{56}}& -{\displaystyle \frac{3}{14}}& -{\displaystyle \frac{3}{16}}& {\displaystyle \frac{1}{6}}& 1\end{array}\right)$$

Appendix D.7. Chamber F007: adj=8, obt=9, orth=4

$$d^{\left(F007\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{3}{10},\frac{8}{27},\frac{119}{384},\frac{761}{2856},\frac{202}{761}\right),d_{min}=d_7=\frac{202}{761}\approx 0.265.$$

$$L^{\left(F007\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{5}}& 1& 0& 0& 0& 0\\ 0& {\displaystyle \frac{4}{15}}& {\displaystyle \frac{2}{9}}& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{18}}& -{\displaystyle \frac{1}{32}}& 1& 0& 0\\ -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{5}}& -{\displaystyle \frac{1}{6}}& -{\displaystyle \frac{3}{16}}& {\displaystyle \frac{26}{119}}& 1& 0\\ -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{5}}& -{\displaystyle \frac{1}{6}}& -{\displaystyle \frac{3}{16}}& {\displaystyle \frac{26}{119}}& {\displaystyle \frac{47}{761}}& 1\end{array}\right)$$

Appendix D.8. Chamber F008: adj=9, obt=9, orth=3

$$d^{\left(F008\right)}=\left(\frac{1}{3},\frac{5}{16},\frac{1}{3},\frac{67}{240},\frac{341}{1206},\frac{67}{248},\frac{18}{67}\right),d_{min}=d_7=\frac{18}{67}\approx 0.269.$$

$$L^{\left(F008\right)}=\left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{1}{4}}& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{4}}& 1& 0& 0& 0\\ -{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{15}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{11}{67}}& 1& 0& 0\\ -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{7}{67}}& {\displaystyle \frac{9}{62}}& 1& 0\\ {\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{5}}& -{\displaystyle \frac{1}{4}}& -{\displaystyle \frac{7}{67}}& -{\displaystyle \frac{9}{62}}& -{\displaystyle \frac{5}{67}}& 1\end{array}\right)$$