We derive the Pauli Exclusion Principle directly from the anti-commutative structure of quaternion algebra, without invoking the spin-statistics theorem or any prior physical postulate. Because quaternion multiplication is non-commutative — ij = −ji — two identical fermions exchanging positions produce a wavefunction that is its own negative, and therefore must be zero. Exclusion is not a law imposed on nature; it is the arithmetic of the algebra that nature uses.