The Royal Ka
The man was mortal; scholars argue a divine 'kingship-soul' was passed down unbroken.
Also: royal ka · kꜣ · king's ka · divine office of kingship · L. Bell royal ka
Egyptians understood that each person had a ka, a vital life-force. Modern scholarship (notably L. Bell) has reconstructed a 'royal ka' — the divine essence of the kingship itself, imagined as passed from one king to the next, so that it is the office rather than the man that is truly divine. This is an interpretive reconstruction that helps explain how Egyptians could hold together the king's mortality and the kingship's divinity; it is not a doctrine the Egyptians spelled out in exactly these terms.
In short
An interpretive modern reconstruction (notably by L. Bell) of a divine 'royal ka' transmitted between kings as the locus of royal divinity — proposed to explain how the office, and its current holder, could be divine while the individual remained mortal; not a doctrine the Egyptians stated in these terms.
Where it’s found
Royal ritual texts and reliefs; rooted in the Old Kingdom ka concept; modern interpretation (L. Bell)
How the idea developed across Egypt
The royal-ka concept is rooted in the broader ka idea (Old Kingdom onward) and is supported by royal ritual and iconography (e.g. the king shown with his ka); the specific 'royal ka transmitted between kings' model is an influential modern interpretation (notably L. Bell), not an explicit Egyptian formulation.
Modern scholarship (notably L. Bell) has reconstructed a 'royal ka' transmitted between kings as the locus of royal divinity; this is an interpretive reconstruction, not a doctrine the Egyptians stated in these terms. The general 'ka' (life-force of any person) has its own entry under death-afterlife.