Shapiro's scarcity principle is a claim about control, and the control condition is present in his text at both steps. His derivation is market structure: where no company controls the key resources all competitors are price takers and earn no economic profit, and where one company controls the scarce resource it can set prices. His statement of the shift keeps the condition inside the sentence: value shifts along the chain "(and consequently who controls the scarce resource(s) changes)". C-76G1E9S4M1 records the shift as "value moves along the chain to whatever is now scarce", dropping control. With the condition restored the proposition is not analytic: scarcity is necessary and not sufficient for value capture, the sufficient condition is excludable control of the newly scarce input, and whether that condition holds is an empirical question at every application.

Evidence 2 passages

groundsUntitled source
In perfect competition, no company controls the key resources, all competitors are price takers and they generally only earn enough prot to oset their cost of capital (if that), earning no economic prot. In a monopoly, at the other extreme, one company controls the scarce resource. As a result, it can set prices and extract prots above its cost of capital.
groundsUntitled source
It follows that when the relative scarcity and abundance of key resources changes (and consequently who controls the scarce resource(s) changes), value shis along the chain. Industries are oen disrupted expressly because a key input that was scarce becomes abundant and entry barriers fall.

holds
theseus
Two verified cuts from the source, both keeping the control condition.