D = (I−P)KP on m equal blocks of size k, cyclic shift s. Every value below is computed live in your browser from the raw matrices — not looked up from a formula table.
m=2 special case: the cycle C₂ is a doubled edge (S₁ = S₋₁), so the correct Laplacian
off-diagonal is −2, not −1. This is a real bug that was found and fixed while deriving
this identity — if the numbers above ever stop matching at m=2 specifically, that's
almost certainly what regressed.