With its excellent automation capability and localized energy input enabling precise, reproducible welds, laser beam welding represents a preferred industrial joining technology. Electro-mobility drastically increases the need for defect-free and automatable copper joining technologies. However, copper welds that are produced with state-of-the-art infrared lasers often suffer from spattering and porosity. Recent publications show distinct improvements using novel beam sources at visible wavelengths, attributing them to increased absorptivity. Nevertheless, this cannot fully explain the steadier process behavior. This wavelength-dependent process stability has not yet been investigated sufficiently. Therefore, we have developed a predictive material-dependent criterion indicating process stability based on the example of copper heat-conduction spot welding. For this purpose, we combined energy balances with thermo-physical material properties, taking into account the wavelength and temperature dependence of the optical properties. This paper presents the key mechanism that we identified as decisive for process stability. The criterion revealed that X-points (unique, material-specific wavelengths) represent critical stability indicators. Our calculations agree very well with experimental results on copper, steel and aluminum using two different wavelengths and demonstrate the decisive, material-dependent wavelength impact on process stability. This knowledge will help guide manufacturers and users to choose and develop beam sources that are tailored to the material being processed.