Theorem: P(C_4, k) expanded = k^4 - 4 k^3 + 6 k^2 - 3 k (explicit polynomial identity)

Theorem (C_4 chromatic-polynomial-expansion canonical): the chromatic polynomial of the 4-cycle C_4 at k colours is P(C_4, k) = (k-1)^4 + (k-1), which expands as k^4 - 4 k^3 + 6 k^2 - 3 k. Canonical sympy pins (symbolic k): k =…