Resonant Touchdown of Fisher Zeros and Super-Kink Dynamical Quantum Phase Transitions via Lagrangian-Engineered Riemann Potentials

Dynamical quantum phase transitions (DQPT) provide a non-equilibrium framework for probing quantum criticality, where non-analytic kinks in the Loschmidt rate function lambda(t) signal the intersection of complex Fisher zeros with the physical real-time axis. In disordered many-body localized systems, spatial inhomogeneity typically induces multi-frequency dephasing that melts these non-analytic signatures into diffuse crossovers. Here, rather than attempting to enforce a global, multi-period isochronous clock across a strongly dephased spectrum, we present an augmented Lagrangian inverse engineering framework that strategically drives the complex Fisher-zero trajectory down to the physical real-time axis (tau = 0) at a targeted resonance time t*. By combining Riemann Gaussian Unitary Ensemble (GUE) level-repulsion kernels with variational penalty constraints, the engineered landscape drives the return probability down to the computational precision floor (L_min = 2.03e-13), establishing an ultra-sharp singular super-kink of peak amplitude lambda_max = 2.9219 in a transverse-field Ising chain (L=10, D=1024). We systematically demonstrate the scalability of this touchdown mechanism up to L=12 (D=4096, L_min ~ 5.40e-8) and establish a four-point finite-size scaling suite (L in {6, 8, 10, 12}). We distinguish the boundary-dominated reflection regime at small system sizes from the asymptotic linear critical-time scaling (t* proportional to L) governed by bounded quantum propagation, confirming the mathematical preservation of non-analytic criticality. This protocol establishes an operational variational quantum-control methodology for synthesizing on-demand non-equilibrium quantum criticality in programmable quantum processors.