Suppose

$$ P \subseteq \{1,2,\dots,N\},\quad |P| = K $$

We calculate the differences as:

$$ d=p_i-p_j\mod N,\quad i\ne j $$

Now let $a_d$ denote the number of occurrence of $d$ (for $d = 0, 1, 2, \dots , N − 1$), then we have a set

$$ D=\{a_0,a_1,a_2,...,a_{N-1} \} $$

Let's define a valid $D$ as a $D$ which is resulted from a points set $P$ (a $D$ which there exists some $P$ leading to it).

Reconstructing $P$ from $D$ is called beltway reconstruction and there are some algorithms for it, but I'm looking for some criteria that indicate the given $D$ is not a valid one.

One is $\sum_{d=1}^{N-1} a_d = K(K-1)$, i.e. if this equality doesn't hold for the given $D$ that $D$ is not a valid one.

The other is $a_i = a_{N-i}$ for $i=1,2,\cdots,N-1$

**EDIT**:
consider these two sets of $D$'s ($a_0=K$ is included, too)

$(N,K) = (10,5)$:

$$\color{green}{\{5, 3, 2, 2, 2, 2, 2, 2, 2, 3\}}$$

$$\{5,2, 3, 2, 2, 2, 2, 2, 3, 2\}$$

$$\color{green}{\{5, 2, 2, 3, 2, 2, 2, 3, 2, 2\}}$$

$$\{5,2, 2, 2, 3, 2, 3, 2, 2, 2\}$$

$(N,K) = (11,8)$ :

$$\color{green}{ \{ 8, 6 , 6, 6 , 5, 5 , 5, 5 , 6, 6 , 6 \}}$$

$$\{ 8 , 6 , 6 , 5 , 6 , 5 , 5 , 6 , 5 , 6 , 6\}$$

$$\{ 8 , 6 , 6 , 5 , 5 , 6 , 6 , 5 , 5 , 6 , 6\}$$

$$\color{green}{\{ 8 , 6, 5 , 6, 6 , 5, 5 , 6, 6 , 5, 6 \}}$$

$$\{ 8, 6 ,5, 6 ,5, 6 ,6, 5 ,6, 5 ,6\}$$

$$\color{green}{\{ 8 , 6 , 5 , 5 , 6 , 6 , 6 , 6 , 5 , 5 , 6 \}}$$

$$\{ 8 , 5 , 6 , 6 , 6 , 5 , 5 , 6 , 6 , 6 , 5\}$$

$$\color{green}{\{ 8 , 5 , 6 , 6 , 5 , 6 , 6 , 5 , 6 , 6 , 5 \}}$$

$$\color{green}{\{ 8 , 5 , 6 , 5 , 6 , 6 , 6 , 6 , 5 , 6 , 5 \}}$$

$$\{ 8 ,5, 5 ,6, 6 ,6, 6 ,6, 6 ,5, 5\}$$

Only green ones are permissible and the others are not, while they satisfy symmetric criteria suggested thanks to Thomas. So there must be some asymmetric criteria.

**Are there any other criteria?**

This question is related, if helps.