Design and mathematically simulate the operation of a quantum-powered spacecraft engine capable of interstellar travel. The engine should utilize theoretical quantum propulsion principles (e.g., quantum vacuum fluctuations or exotic matter) and be optimized for a multi-star system trajectory within 100 light-years. Part 1: Quantum Field Integration 1, Derive a system of differential equations describing the quantum fluctuations of the vacuum state around the spacecraft. Assume that spacetime curvature varies non-linearly within the interstellar medium. 2, Integrate these equations into a 12-dimensional tensor space accounting for: Relativistic effects Quantum decoherence over macroscopic distances Exotic matter generation efficiency Part 2: Energy Budget and Thermodynamics Calculate the total energy required to maintain quantum coherence in the engine's core. Assume energy extraction from a hypothetical 10^12 Tesla magnetic monopole field. Solve for the thermodynamic efficiency of the engine while accounting for: Entropy fluctuations in a 4-dimensional spacetime manifold Multi-phase quantum energy loss at Planck-scale interactions Derive a feasible (or infeasible) energy budget for a 10-light-year journey using real-time quantum entanglement energy transfer. Part 3: Multi-Star System Navigation Using a generalized N-body problem solver, simulate a 3-star gravitational slingshot maneuver where: The stars are modeled as rapidly rotating neutron stars emitting bursts of gamma radiation. Gravitational anomalies near each star introduce unpredicted trajectory perturbations. Optimize the trajectory using Lagrangian mechanics extended to a 6-body system with quantum drag forces. Present the system of equations and solve numerically using at least fourth-order Runge-Kutta methods. Part 4: Engine Material Engineering Design a material capable of withstanding continuous interaction with exotic matter and extreme quantum effects. Utilize Schrödinger's wave equations to define the electron density and resilience under quantum tunneling bombardments. Solve for a stable material composition using partial differential equations in 7-variable Hilbert space. Part 5: Unification with Hypothetical Physics Integrate your propulsion model with unsolved problems in physics, such as: Unifying General Relativity and Quantum Mechanics Validating a dynamic Alcubierre metric without the need for negative energy Propose modifications to the Standard Model to allow stable exotic matter production and mathematically prove your hypotheses. Deliverables: A 50 to 100-page mathematical dissertation detailing your findings. 3D visualizations and simulations of the quantum field effects and trajectory optimizations. Proof of computational feasibility, including source code for the simulations. Marks will be awarded based on: The novelty of your hypotheses. Accuracy in applying known physical and mathematical principles. Innovation in merging theoretical physics with practical engineering constraints. Hint: There are no complete solutions; strive to innovate beyond known science.