The y > x, x = y problem

[u/Parking-War-2982]

Mathematical Problem: Minimum Number of Vectors in an \(N\times N\) Grid
Consider an \(N\times N\) square grid, where \(N\) is any positive integer.
Rules and Definitions
• Empty boxes: Exactly one box in each row and exactly one box in each column must be empty. Therefore, the total number of empty boxes is \(x=N\).
• Vectors: Every non-empty box belongs to a horizontal or vertical vector. A vector may occupy a single box or multiple consecutive boxes, but it must be straight: it cannot turn or be both horizontal and vertical.
• Counting vectors: A vector is a maximal uninterrupted straight run of non-empty boxes. Consecutive non-empty boxes in the same row or column may belong to one vector; separate runs count as separate vectors.
• Objective: The positions of the empty boxes may vary, provided the rules above are satisfied. Vectors may be horizontal, vertical, or a mixture of both.
The Question
Let \(y\) denote the total number of vectors and \(x=N\) the total number of empty boxes.
Is it true that (y>x) for every valid configuration whenever ( N > 3 ), regardless of the grid size or the arrangement of the empty boxes?