Four. (50 points) In a $101 \times 101$ grid, each cell is filled with a number from the set $\left\{1,2, \cdots, 101^{2}\right\}$, and each number in the set is used exactly once. The left and right boundaries of the grid are considered the same line, and the top and bottom boundaries are also considered the same line (i.e., it is a torus). If no matter how the numbers are filled, there always exist two adjacent cells (sharing a common edge) such that the difference between the numbers in these two cells is at least $M$, find the maximum value of $M$.

In a $101 \times 101$ grid, each cell is filled with a number from the set $\left\{1,2, \cdots, 101^{2}\right\}$, and each number in the set is used exactly once. The left and right boundaries of the grid are considered the same line, and the top and bottom boundaries are also considered the same line (i.e., it is a torus). If no matter how the numbers are filled, there always exist two adjacent cells (sharing a common edge) such that the difference between the numbers in these two cells is at least $M$, find the maximum value of $M$.
From left to right, number each column:
$$
-50,-49, \cdots,-1,0,1, \cdots, 49,50 \text {; }
$$

From bottom to top, number each row:
$$
-50,-49, \cdots,-1,0,1, \cdots, 49,50 \text {. }
$$

The cell at the intersection of the $i$-th column and the $j$-th row is denoted by the ordered pair $(i, j)$. The center cell of the grid is denoted by $(0,0)$. All cells are classified based on $|i|+|j|$ as follows:
$$
S_{k}=\{(i, j)|| i|+| j \mid=k\}(k=0,1, \cdots, 100) .
$$

It is easy to see that when $k=50$, $|S_{k}|$ is maximized at 200.
Since adjacent cells are either of the same type or belong to consecutive types (i.e., $S_{k}$ and $S_{k+1}$), the numbers $1,2, \cdots, 101^{2}$ can be sequentially filled into the cells of $S_{0}, S_{1}, \cdots, S_{100}$ such that the difference between the numbers in any two adjacent cells does not exceed 201 (as shown in Figure 6). Therefore, $M \leqslant 201$.