6 · Slide a zero along the array

Seven microphones of the toy array, and one zero you can move. Each microphone's share changes; the array's total does not.

Try this. Park the zero on a microphone, then exactly between two. The shares redistribute completely, and the total printed below never moves. This fair sharing is why the table's diagonal counts every zero once, wherever it sits.

The window and spacing are the toy family's, the same ones the figures and tests use, and the identity is the paper's Lemma 2.2. The total drifts from one only when the zero approaches the ends of a finite array, which is the edge effect the paper bounds.