X-ray characteristic spectra constitute one of the most sensitive probes for testing relativistic quantum mechanics and quantum electrodynamics, playing a key role in both atomic physics research and industrial applications. Advanced relativistic theory has, however, been unable to predict such spectra, calling several foundational ideas into question. A core problem lies in the process to convergence and the lack of sensitive metrics for convergence of many-body systems. We present a convergence metric for many-body systems: the radiative-lifetime-invariance metric ($\mathrm{\Sigma }$ metric) which can be used to identify cases of false convergence that go undetected using previous criteria. This new metric suggests that the radiative lifetime of all upper states of the $K\alpha$ transition are the same to within 0.347% under appropriate circumstances. We develop a suite of convergence criteria innovating best practice in many-body theory. We apply the suite to develop a theoretical model for the copper $K\alpha$ satellite spectrum which, despite being the best-tested system experimentally and theoretically in x-ray physics, is yet to be adequately explained theoretically. We predict the dominant $\left[1s\right]\to \left[2p\right]$ transition spectrum, and all major satellite transition spectra. The resulting wave functions are highly convergent, according to the range of convergence metrics developed herein. The theoretical Cu $K\alpha$ spectrum is in strong agreement with the world-leading experiments. We confirm the necessity and accuracy of the diagram spectrum and the $\left[4s\right]$ satellite spectrum; define and demonstrate the necessity of a convergent $\left[3d\right]$ satellite spectrum; and calculate and highlight the importance of the $\left[3p\right]$, $\left[3s\right]$, $\left[3d4s\right]$, and $\left[3d^2\right]$ shakeoff satellite processes. All are real and observed in the experimental spectra.

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• Atomic spectra
• Electronic excitation & ionization
• Photoemission
• Relativistic & quantum electrodynamic effects in atoms, molecules,& ions
• Single- and few-photon ionization & excitation
• Spontaneous emission
• Adiabatic approximation
• Hartree-Fock methods