Why it works: Ūrdhva-Tiryagbhyāṃ

For AB × CD: result = [$A\times C$] [$A\times D + B\times C$] [$B\times D$]. This is exactly the expansion of $(10A+B)(10C+D) = 10$$0AC + 10$(AD+BC) + BD. The crosswise products fill the middle. For 3-digit: the diamond has 5 nodes matching 5 partial-product positions.