Unified Energy–Information–Geometry Equation to Black Holes and the Big Bang

We propose a unified fundamental equation that couples geometry, energy, information, and quantum fields into a single mathematically consistent framework. The motivation is to address regimes where classical general relativity and quantum field theory individually fail, particularly near spacetime singularities such as black holes and the Big Bang. A generalized action functional is constructed, from which the master field equation is derived via variational principles. We provide a rigorous mathematical proof of the equation, sector-wise interpretation of each symbol, and demonstrate consistency in gravitational, quantum, thermodynamic, and informational sectors. This framework offers a transparent and testable path toward quantumgravity and singularity resolution.