Quantum magic is necessary but not sufficient for wormhole-inspired teleportation

We investigate the dynamics of Quantum magic, formally known as non-stabilizerness, quantified by the stabilizer Rényi entropy (SRE), across the stages of the wormhole-inspired teleportation protocol (WITP) in the Sachdev-Ye-Kitaev (SYK) model. By tracking the SRE of the full pure state across scrambling, message insertion, left-right coupling, and right-side extraction, we uncover a regime-dependent relationship between magic accumulation and teleportation fidelity. In the gravitational (low temperature) regime, fidelity rises concurrently with magic from early times, whereas in the peaked-size (high temperature) regime, the magic saturates near the Haar-typical value before teleportation onset. A baseline-subtracted diagnostic comparing coupled and uncoupled protocols reveals that the double-trace coupling first suppresses and then channels non-stabilizer resources toward the teleportation signal, with the channel amplitude decreasing monotonically with inverse temperature. Comparison with a chaotic random two-local model, which generates near-maximal magic yet fails to teleport, demonstrates that structured magic redistribution, rather than raw non-stabilizerness, underlies successful wormhole traversal. Moreover, the magic transiently dips at the fidelity peak, marking the teleportation event in the time domain. Our results are robust across the three system sizes studied $N_maj=8,10,12$, and the fidelity-magic trajectories exhibit an approximate collapse across system sizes when the SRE is normalized by the Haar-typical prediction.