Baran Complex Energy Formalism: General Gamma, Axioms, and the Complex Decomposition of Relativistic Energy

This work introduces the Baran Complex Energy Formalism, built on one axiom and one definition. Axiom: E := γ_general · mc² Definition: γ_gen := γ_ξ + i·γ_U where γ_ξ (external gamma) comes from the spacetime metric and γ_U (intrinsic gamma) is a property of matter itself, independent of the metric. In the standard case γ_ξ = γ − 1 and γ_U = 1, yielding the Baran Law: E = (γ − 1 + i)mc². The real part carries external energy (kinetic, gravitational); the imaginary part carries intrinsic rest energy. Four theorems are proven: the total energy relation Re(E) + Im(E) = γ·mc², Einstein's formula as a special case |E| = mc² at rest, the general modulus, and the conjugate product. The Newtonian limit naturally recovers (1/2)mv² + GMm/r + i·mc². The formalism covers Lorentz, Schwarzschild, and Kerr metrics, as well as mass-energy conversion, annihilation, and pair production via the choice of γ_ξ and γ_U.