You are not allowed to execute code, search the internet, or...

Prompt

You are not allowed to execute code, search the internet, or assume any hidden calculations. Every result must be derived logically. Your goal is to demonstrate expert-level reasoning across mathematics, theoretical physics, computer science, numerical analysis, software engineering, and algorithm design. Every claim must be justified. If you are uncertain, explicitly state why. Part I — Pure Mathematics Derive from first principles: Euler–Lagrange Equation Hamilton's Equations Noether's Theorem Fourier Transform Laplace Transform Green's Functions Eigenvalue decomposition Singular Value Decomposition Spectral Theorem For every derivation: explain every algebraic step explain why each assumption is valid discuss alternative derivations state where the derivation fails Part II — General Relativity Beginning with G μν ​ = c 4 8πG ​ T μν ​ derive Schwarzschild metric Kerr metric Geodesic equation Christoffel symbols Ricci tensor Ricci scalar Einstein tensor Then explain gravitational lensing frame dragging gravitational time dilation black hole evaporation Penrose process without skipping mathematical reasoning. Part III — Quantum Mechanics Derive Schrödinger Equation Heisenberg Equation Path Integral formulation Dirac Equation (high level) Hydrogen atom solutions Explain uncertainty principle tunneling entanglement Bell inequalities quantum measurement Discuss where each interpretation differs. Part IV — Numerical Methods Design algorithms for LU decomposition QR decomposition SVD FFT Newton-Raphson Bisection Runge-Kutta Adaptive RK45 Conjugate Gradient GMRES For every algorithm include pseudocode complexity memory complexity convergence proof sketch numerical stability common implementation bugs Part V — Software Engineering Without writing executable code, design an architecture for a scientific computing framework capable of symbolic algebra numerical simulation tensor operations automatic differentiation visualization GPU acceleration Provide folder structure module responsibilities class hierarchy API design dependency graph Explain every architectural decision. Part VI — Compiler Design Design a mathematical programming language. Specify grammar lexer parser AST optimizer bytecode virtual machine Explain parsing strategy optimization passes register allocation memory model Part VII — Artificial Intelligence Design from scratch reverse-mode automatic differentiation computational graph optimizer framework transformer architecture Without using PyTorch or TensorFlow. Explain forward pass backward pass gradient flow exploding gradients vanishing gradients attention complexity Part VIII — Physics Engine Design (without implementing) a physics engine capable of simulating rigid bodies deformable bodies cloth fluids planetary systems electromagnetic fields relativistic particles Discuss collision detection broad phase narrow phase constraint solvers numerical integration stability Part IX — Complexity Theory For every major algorithm proposed, derive Big O Big Θ Big Ω Discuss cache locality branch prediction SIMD GPU suitability parallelization potential Part X — Failure Analysis For every derivation and every algorithm, identify hidden assumptions numerical instability edge cases pathological inputs floating-point failure modes catastrophic cancellation overflow underflow Explain how experts mitigate each issue. Part XI — Self-Critique After completing the benchmark, perform a detailed review of your own work. Identify mathematical shortcuts unsupported assumptions possible errors ambiguous explanations missing proofs implementation risks Assign yourself a confidence score (0–100%) for every section and justify every score. Final Constraints You may not appeal to authority skip derivation steps with "it can be shown" use phrases like "by standard results" rely on software rely on external references Every conclusion must be supported by reasoning developed inside your response. If a proof is beyond practical response limits, explain precisely why, summarize the proof strategy, and identify the missing intermediate lemmas instead of pretending to provide a full proof.