Quantum Circuit Topology Reduces to Classical Transfer Matrices: 24 Theorems on Controlled-Unitary Networks

We prove that quantum circuit topology and classical network topology are the same mathematical object, connected by a universal 4x4 transfer matrix with bond dimension 4 = 2x2 (tensor product of hub-state and relay-action qubits). Twenty-four theorems establish: (T1-T2) non-transitivity of controlled-unitary topology with closed-form fidelity F=cos^4(alpha); (T15-T16) a universal transfer matrix valid for ALL controlled-unitary gates; (T17) the projector angle where the transfer matrix becomes a classical stochastic matrix; (T19) relay states are eigenvectors of the gate at the projector; (T20-T21) tree factorization and back-edge rules for general graph topologies; (T22) self-referential structure: the transfer matrix IS itself a controlled-unitary gate and the RG flow always converges to the projector; (T23) a novelty formula N=|sum A_j v_j| with conservation law sum A_j=0 connecting attention mechanisms to transfer matrices; (T24) deflection D^2=1-Z^2 with a breathing-mode ratchet driven by nonlinear self-amplification. Includes a topology-aware quantum compiler that outperforms SWAP-count heuristics by up to 35% using per-qubit IBM calibration data. All results verified: 42,000+ computational tests, zero failures.