We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (i, 0, \dots, 0), x_3 = (2i, 0, \dots, 0), x_4 = (1.5i, 1, 0, \dots, 0), \quad (200)$$

and get $z = \frac{1}{6}$, $\bar{z} = \frac{3}{2}$, which is in the range corresponding to class T. So we conclude that (1234) is in class T. The results of OPE convergence properties in the three channels are shown in table 7.

<table><caption>Table 7: OPE convergence properties of type 3 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td>(1234), (4321)</td><td>T</td><td>X</td><td>✓</td><td>X</td></tr><tr><td>(1243), (4312)</td><td>U</td><td>X</td><td>X</td><td>✓</td></tr><tr><td>(1324), (4231)</td><td>T</td><td>X</td><td>✓</td><td>X</td></tr><tr><td>(1342), (4213)</td><td>S</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>(1423), (4132)</td><td>U</td><td>X</td><td>X</td><td>✓</td></tr><tr><td>(1432), (4123)</td><td>S</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>(2134), (3421)</td><td>U</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(2143), (3412)</td><td>T</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(2314), (3241)</td><td>U</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(2341), (3214)</td><td>S</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>(2413), (3142)</td><td>T</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(2431), (3124)</td><td>S</td><td>✓</td><td>X</td><td>X</td></tr></tbody></table>

### C.4 Type 4

The type 4 causal ordering is given by

$$a \underset{c}{\overset{b}{\rightleftharpoons}} d. \qquad (201)$$

We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (i, 0.5, 0, \dots, 0), x_3 = (i, -0.5, 0, \dots, 0), x_4 = (2i, 0, \dots, 0), \quad (202)$$

and get $z = \frac{1}{9}$, $\bar{z} = 9$, which is in the range corresponding to class T. So we conclude that (1234) is in class T. The results of OPE convergence properties in the three channels are shown in table 8. Here we use the fact that $(i_1i_2i_3i_4)$ and $(i_1i_3i_2i_4)$ are the same causal ordering (the little group is $\mathbb{Z}_2$).

### C.5 Type 5

The type 5 causal ordering is given by

$$a \underset{d}{\overset{b}{\rightleftharpoons}} c, \quad b \underset{d}{\overset{c}{\rightleftharpoons}} a. \qquad (203)$$
<table><caption>Table 8: OPE convergence properties of type 4 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td>(1234), (4321)</td><td>T</td><td>✗</td><td>✓</td><td>✗</td></tr><tr><td>(1243), (4312), (2134), (3421)</td><td>U</td><td>✗</td><td>✗</td><td>✗</td></tr><tr><td>(1342), (4213), (2341), (3214)</td><td>S</td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>(2143), (3412)</td><td>T</td><td>✗</td><td>✗</td><td>✗</td></tr></tbody></table>

We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (i, 0.5, 0, \dots, 0), x_3 = (i, 0, \dots, 0), x_4 = (i, -0.5, \dots, 0), \quad (204)$$

and get $z = \frac{1}{4}$, $\bar{z} = \frac{3}{4}$, which is in the range corresponding to subclass E<sub>st</sub>. So (1234) is in class E. We would like to find a particular configuration in each subclass of class E. The little group of this causal type is S<sub>3</sub>, which corresponds to permutations among b, c, d in (203). By looking up table 3, we see that permuting x<sub>2</sub>, x<sub>3</sub> in (204) gives E<sub>tu</sub> and permuting x<sub>3</sub>, x<sub>4</sub> gives E<sub>su</sub>. To realize E<sub>stu</sub> we choose the following configuration in (1234):

$$x_1 = 0, x_2 = (i, 0.5, 0, \dots, 0), x_3 = (i, -0.5, 0, \dots, 0), x_4 = (i, 0, 0.5, 0, \dots, 0), \quad (205)$$

and get $z = i$, $\bar{z} = -i$, which is indeed in the range corresponding to subclass E<sub>stu</sub>. So we conclude that the configurations of (1234) do appear in all subclasses of class E in $d \ge 3$, while they only appear in subclasses E<sub>st</sub>, E<sub>su</sub>, E<sub>tu</sub> in 2d.<sup>42</sup>

The results of OPE convergence properties in the three channels are shown in table 9. Here we use the fact that for $(i_1 i_2 i_3 i_4)$, any permutation of 2, 3, 4 does not change the causal ordering (the little group is S<sub>3</sub>).

<table><caption>Table 9: OPE convergence properties of type 5 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td rowspan="4">(1234), (4321)</td><td>E<sub>st</sub></td><td>✓</td><td>✓</td><td>✗</td></tr><tr><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>E<sub>tu</sub></td><td>✗</td><td>✓</td><td>✓</td></tr><tr><td>E<sub>stu</sub></td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td rowspan="4">(2134), (3124)</td><td>E<sub>st</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>tu</sub></td><td>✗</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>stu</sub></td><td>✓</td><td>✗</td><td>✗</td></tr></tbody></table>

### C.6 Type 6

The type 6 causal ordering is given by

$$a \rightarrow b \rightarrow c \\ d \quad . \quad (206)$$

<footnote n="42"><sup>42</sup>We used two dimensions in (204) and three dimensions in (205). On the other hand, as mentioned at the end of section 5.3.2, subclass E<sub>stu</sub> does not exist in 2d because z, $\bar{z}$ can only be real.</footnote>
We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (i, 0, \dots, 0), x_3 = (2i, 0, \dots, 0), x_4 = (i, 2, \dots, 0), \quad (207)$$

and get $z = \frac{1}{4}$, $\bar{z} = \frac{3}{4}$, which is in the range corresponding to subclass E<sub>st</sub>. By lemma C.1, the whole (1234) is in subclass E<sub>st</sub>. The results of OPE convergence properties in the three channels are shown in table 10.

<table><caption>Table 10: OPE convergence properties of type 6 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td>(1234), (4321), (3214), (2341)</td><td>E<sub>st</sub></td><td>✓</td><td>✓</td><td>✗</td></tr><tr><td>(1243), (4312), (4213), (1342)</td><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>(1324), (4231), (2314), (3241)</td><td>E<sub>tu</sub></td><td>✗</td><td>✓</td><td>✗</td></tr><tr><td>(1423), (4132), (2413), (3142)</td><td>E<sub>tu</sub></td><td>✗</td><td>✗</td><td>✓</td></tr><tr><td>(1432), (4123), (3412), (2143)</td><td>E<sub>st</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>(2134), (3421), (3124), (2431)</td><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✗</td></tr></tbody></table>

### C.7 Type 7

The type 7 causal ordering is given by

$$ \begin{array}{ccc} a & b & b \\ \multicolumn{3}{c}{c,} \\ d & & d \end{array} \quad \begin{array}{c} a \\ b \\ c \\ d \end{array}. \quad (208) $$

We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (i, 0.5, 0, \dots, 0), x_3 = (i, -0.5, 0, \dots, 0), x_4 = (0, 2, 0, \dots, 0), \quad (209)$$

and get $z = \frac{7}{15}$, $\bar{z} = 9$, which is in the range corresponding to class T. So we conclude that (1234) is in class T. The results of OPE convergence properties in the three channels are shown in table 11. Here we use the fact that $(i_1 i_2 i_3 i_4)$ and $(i_1 i_3 i_2 i_4)$ are the same causal ordering (the little group is $\mathbb{Z}_2$).

<table><caption>Table 11: OPE convergence properties of type 7 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td>(1234), (4321)</td><td>T</td><td>✗</td><td>✓</td><td>✗</td></tr><tr><td>(1243), (4312)</td><td>U</td><td>✗</td><td>✗</td><td>✓</td></tr><tr><td>(1342), (4213)</td><td>S</td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>(2134), (3421)</td><td>U</td><td>✗</td><td>✗</td><td>✗</td></tr><tr><td>(2143), (3412)</td><td>T</td><td>✗</td><td>✗</td><td>✗</td></tr><tr><td>(2341), (3124)</td><td>S</td><td>✓</td><td>✗</td><td>✗</td></tr></tbody></table>
### C.8 Type 8

The type 8 causal ordering is given by

<figure><img src="image_25.png" /></figure>

We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (i, -0.5, 0, \dots, 0), x_3 = (0, 1, 0, \dots, 0), x_4 = (i, 0.5, 0, \dots, 0), \quad (211)$$

and get $z = \frac{1}{4}$, $\bar{z} = \frac{9}{4}$, which is in the range corresponding to class T. So we conclude that (1234) is in class T. The results of OPE convergence properties in the three channels are shown in table 12.

<table><caption>Table 12: OPE convergence properties of type 8 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td>(1234), (4321)</td><td>T</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(1243), (4312), (3421), (2134)</td><td>U</td><td>X</td><td>X</td><td>✓</td></tr><tr><td>(1342), (4213), (2431), (3124)</td><td>S</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>(2143), (3412)</td><td>T</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(1324), (4231)</td><td>T</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(1432), (4123), (2341), (3214)</td><td>S</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>(1423), (4132), (3241), (2314)</td><td>U</td><td>X</td><td>X</td><td>X</td></tr><tr><td>(2413), (3142)</td><td>T</td><td>X</td><td>X</td><td>X</td></tr></tbody></table>

### C.9 Type 9

The type 9 causal ordering is given by

<figure><img src="image_26.png" /></figure>

We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (i, 0, \dots, 0), x_3 = (0, 2, 0, \dots, 0), x_4 = (0, 3, 0, \dots, 0), \quad (213)$$

and get $z = -\frac{1}{8}$, $\bar{z} = \frac{1}{4}$, which is in the range corresponding to class S. So we conclude that (1234) is in class S. The results of OPE convergence properties in the three channels are shown in table 13. Here we use the fact that $(i_1 i_2 i_3 i_4)$ and $(i_1 i_2 i_4 i_3)$ are the same causal ordering (the little group is $\mathbb{Z}_2$).

### C.10 Type 10

The type 10 causal ordering is given by

<figure><img src="image_27.png" /></figure>
<table><caption>Table 13: OPE convergence properties of type 9 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td>(1234), (4312), (2134), (3412)</td><td>S</td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>(1324), (4213), (3124), (2413)</td><td>U</td><td>✗</td><td>✗</td><td>✓</td></tr><tr><td>(1423), (4123)</td><td>T</td><td>✗</td><td>✓</td><td>✗</td></tr><tr><td>(2314), (3214)</td><td>T</td><td>✗</td><td>✓</td><td>✗</td></tr></tbody></table>

We choose a particular configuration for (1234):

$$x_1 = 0, x_2 = (0, 1, 0, \dots, 0), x_3 = (2i, 0, \dots, 0), x_4 = (2i, 1, \dots, 0), \quad (215)$$

and get $z = \bar{z} = \frac{1}{4}$, which is in the range corresponding to subclass $E_{st}$. So (1234) is in class E. We would like to find a particular configuration in each subclass of class E. The little group of this causal type is $\mathbb{Z}_2 \times \mathbb{Z}_2$, which is generated by $a \leftrightarrow b$ and $c \leftrightarrow d$ in (214). By looking up table 3, we see that by acting with the little group on configuration (215), we can get $E_{su}$, but we cannot get $E_{tu}$. The underlying fact is that the 2d configurations of (1234) do not appear in $E_{tu}$ (it is obvious that 2d configurations do not appear in $E_{stu}$). Let us show this fact. In 2d Minkowski space we can use the light-cone coordinates:

$$z_k = t_k + \mathbf{x}_k, \quad \bar{z}_k = t_k - \mathbf{x}_k, \quad (x_k = (it_k, \mathbf{x}_k)). \quad (216)$$

The causal ordering (214) implies

$$z_3, z_4 > z_1, z_2, \quad \bar{z}_3, \bar{z}_4 > \bar{z}_1, \bar{z}_2, \\ (z_1 - z_2)(\bar{z}_1 - \bar{z}_2) < 0, \quad (z_3 - z_4)(\bar{z}_3 - \bar{z}_4) < 0. \quad (217)$$

Since the little group $\mathbb{Z}_2 \times \mathbb{Z}_2$ of (1234) preserves $E_{tu}$ (see table 3), by the $\mathbb{Z}_2 \times \mathbb{Z}_2$-action, it suffices to show that $E_{tu}$ configurations do not exist when

$$z_3, z_4 > z_1, z_2, \quad \bar{z}_3, \bar{z}_4 > \bar{z}_1, \bar{z}_2, \\ z_1 - z_2 < 0, \quad \bar{z}_1 - \bar{z}_2 > 0, \\ z_3 - z_4 < 0, \quad \bar{z}_3 - \bar{z}_4 > 0. \quad (218)$$

In this case the computation is straightforward:

$$z = \frac{(z_2 - z_1)(z_4 - z_3)}{(z_3 - z_1)(z_4 - z_2)} < \frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_1)(z_4 - z_2)} = 1, \\ \bar{z} = \frac{(\bar{z}_1 - \bar{z}_2)(\bar{z}_3 - \bar{z}_4)}{(\bar{z}_3 - \bar{z}_1)(\bar{z}_4 - \bar{z}_2)} < \frac{(\bar{z}_4 - \bar{z}_2)(\bar{z}_3 - \bar{z}_1)}{(\bar{z}_3 - \bar{z}_1)(\bar{z}_4 - \bar{z}_2)} = 1. \quad (219)$$

So we conclude that the 2d configurations in (1234) have $z, \bar{z} < 1$, i.e. (1234) does not intersect with subclass $E_{tu}$ in 2d. To find a $E_{tu}$ configuration in (1234) we need to construct it in $d \ge 3$. We choose the 3d configuration (114) and get $z \approx 1.1, \bar{z} \approx 6.3$, which is in the range corresponding to subclass $E_{tu}$.

To realize $E_{stu}$ we choose the following configuration in (1234):

$$x_1 = 0, x_2 = (0, 0.5, 0, \dots, 0), x_3 = (2i, 0, 0.5, 0, \dots, 0), x_4 = (i, 0.5, 0, \dots, 0), \quad (220)$$

and get $z \approx 0.33 + 0.24i, \bar{z} = 0.33 - 0.24i$, which is in the range corresponding to subclass $E_{stu}$. So we conclude that the configurations of (1234) do appear in all subclasses of class E in $d \ge 3$, while they only appear in subclasses $E_{st}, E_{su}$ in 2d.

The results of OPE convergence properties in the three channels are shown in table 14: Here we use the fact $(i_1i_2i_3i_4), (i_2i_1i_3i_4), (i_1i_2i_4i_3)$ and $(i_2i_1i_4i_3)$ are the same causal ordering (the little group is $\mathbb{Z}_2 \times \mathbb{Z}_2$).
<table><caption>Table 14: OPE convergence properties of type 10 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td rowspan="4">(1234), (3412)</td><td>E<sub>st</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>tu</sub></td><td>✗</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>stu</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td rowspan="4">(1324), (2413)</td><td>E<sub>st</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>tu</sub></td><td>✗</td><td>✗</td><td>✗</td></tr><tr><td>E<sub>stu</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td rowspan="4">(1423), (2314)</td><td>E<sub>st</sub></td><td>✓</td><td>✓</td><td>✗</td></tr><tr><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>E<sub>tu</sub></td><td>✗</td><td>✓</td><td>✓</td></tr><tr><td>E<sub>stu</sub></td><td>✓</td><td>✓</td><td>✓</td></tr></tbody></table>

### C.11 Type 11

The type 11 causal ordering is given by

$$ 
\begin{array}{l}
a \rightarrow b \\
c \rightarrow d
\end{array}
\qquad (221) $$

We choose a particular configuration for (1234):

$$ x_1 = 0, \quad x_2 = (i, 0, \dots, 0), \quad x_3 = (0, 2, \dots, 0), \quad x_4 = (i, 2, \dots, 0), \qquad (222) $$

and get $z = \bar{z} = \frac{1}{4}$, which is in the range corresponding to subclass E<sub>st</sub>. By lemma C.1, the whole (1234) is in subclass E<sub>st</sub>. The results of OPE convergence properties in the three channels are shown in table 15. Here we use the fact that (i<sub>1</sub>i<sub>2</sub>i<sub>3</sub>i<sub>4</sub>) and (i<sub>3</sub>i<sub>4</sub>i<sub>1</sub>i<sub>2</sub>) are the same causal

<table><caption>Table 15: OPE convergence properties of type 11 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td>(1234), (2143)</td><td>E<sub>st</sub></td><td>✓</td><td>✗</td><td>✗</td></tr><tr><td>(1243), (2134)</td><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>(1324), (3142)</td><td>E<sub>tu</sub></td><td>✗</td><td>✗</td><td>✓</td></tr><tr><td>(1342), (3124)</td><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>(1423), (3241)</td><td>E<sub>tu</sub></td><td>✗</td><td>✓</td><td>✗</td></tr><tr><td>(1432), (2341)</td><td>E<sub>st</sub></td><td>✓</td><td>✓</td><td>✗</td></tr></tbody></table>

ordering (the little group is $\mathbb{Z}_2$).

### C.12 Type 12

The type 12 causal ordering is given by

$$ a \ b \ c \ d. \qquad (223) $$
One can show that this causal type belongs to class E. In fact this is the "Euclidean" case (that's why this class is called class E), and there is no need to check numerically for this type. Suppose we have a totally space-like separated configuration $(x_1, x_2, x_3, x_4)$, where $x_k = (it_k, \mathbf{x}_k)$. We can reach this configuration via the path

$$x_k(s) = \begin{cases} ((1-2s)\epsilon_k, 2s\mathbf{x}_k) & s \in [0, 1/2], \\ ((2s-1)it_k, \mathbf{x}_k) & s \in [1/2, 1]. \end{cases} \qquad (224)$$

Along the path all the $x_i, x_j$ pairs are space-like separated. As a result, the totally space-like separated configurations always have $N_t = N_u = 0$ (as long as $N_t, N_u$ are well-defined). On the other hand, there is no doubt that all subclasses of class E can be realized in $d \ge 3$.<fnref n="43" /> We summarize the OPE convergence properties of this type in table 16. Here we use the fact that there is only 1 causal ordering in this type (the little group is $S_4$).

<table><caption>Table 16: OPE convergence properties of type 12 causal orderings</caption><thead><tr><th>causal ordering</th><th>class/subclass</th><th>s-channel</th><th>t-channel</th><th>u-channel</th></tr></thead><tbody><tr><td rowspan="4">(1234)</td><td>E<sub>st</sub></td><td>✓</td><td>✓</td><td>✗</td></tr><tr><td>E<sub>su</sub></td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>E<sub>tu</sub></td><td>✗</td><td>✓</td><td>✓</td></tr><tr><td>E<sub>stu</sub></td><td>✓</td><td>✓</td><td>✓</td></tr></tbody></table>

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<footnote n="43">⁴³One can realize one of the three subclasses $E_{st}, E_{su}, E_{tu}$ in 2d, and then realize the other two subclasses by $S_4$-action. For subclass $E_{stu}$, one can put four points in the hyperplane with $t = 0$.</footnote>
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