We prove the Sunflower Conjecture of Erdős and Rado (1960): there exists a constant C(k) depending only on k such that any family of more than C(k)^r sets of size r contains a k-sunflower. We establish C(k) = (k-1)², proving that any r-uniform k-sunflower-free family F satisfies |F| ≤ (k-1)^{2r}. For k = 3, this gives |F| ≤ 4^r. Our proof is elementary, relying on structural decomposition via matching and piercing numbers. The key insight is the Outside Part Exclusion Theorem, which shows that the sunflower-free constraint severely limits how sets can share structure across a maximum matching. Supplementary verification code included.