Broadband lasers suppress the absolute two-plasmon-decay instability that preheats inertial-fusion targets. The usual scaling relates this suppression to the drive's coherence time, a power-spectrum statistic insensitive to spectral phase. By gradient descent through a differentiable enveloped wave solver, we show that at fixed power spectrum phase optimization raises the threshold from the random-phase median of 2.8 Imono to 4.1 Imono, but produces a transform-limited (TL) pulse train with peak-to-average ratio R=32. Joint amplitude–phase optimization reaches 5.6 Imono at R=3.6, above the best tested TL line-count result of 4.5 Imono at R=16; relaxing the peak constraint raises the joint threshold to 6.4 Imono. An exact growth-rate budget evaluated inside the simulation separates the two mechanisms. TL suppresses coupling mainly by concentrating the pump field into short bursts, thereby lowering its time-averaged magnitude at fixed average intensity. The joint optima retain near-random values of this amplitude measure while suppressing both the available coupling and the fraction realized through pump–daughter alignment. The advantage of joint optimization persists from 1% to 4% bandwidth and from OMEGA-scale to ignition-scale conditions.