Every architecture of six two-qubit gates is locally universal on three qubits

We present the first analytical determination of the exact accessible dimension for every fixed architecture of arbitrary two-qubit gates on three qubits. A support word represents which pair of qubits each two-qubit gate acts on, and its reduced length is obtained by merging consecutive gates on the same pair. If the reduced length is r, then the set of implementable three-qubit unitaries has accessible dimension d(w)=min\{63,9r+9\}. Consequently, six arbitrary two-qubit gates are necessary and sufficient for local universality: every fixed architecture reaches a nonempty open subset of SU(8) when its reduced length is at least six. The result is stronger than existence of one favorable architecture. Every reduced six-slot support word is locally universal, including the alternating nearest-neighbor line AB,BC,AB,BC,AB,BC. The upper bound follows from a standard parameter-counting argument. The matching lower bounds are proved by Jacobian certificates in the Pauli basis. Up to qubit relabeling and reversal, there are 22 reduced architectures of lengths two through six. For each one, a product of rational Pauli rotations yields a nonzero maximal minor modulo the prime 1{,}000{,}003. Thus, any obstruction to global universality with six two-qubit gates must go beyond parameter counting, connectivity, and differential rank.