The European Union presents a puzzle to political systems scholars: how can a developing polity, with all its attendant functional weaknesses, be rendered politically stable even through moments of a policy crisis? Building on insights from the literature on fiscal federalism, this article challenges much conventional wisdom on Europe’s incompleteness. This is based on the corollary of Jonathan Rodden’s concept of Hamilton’s Paradox: whereas a strong centre cannot resist exploitation by states because it has the means to rescue them, a weak centre’s lack of exploitable capacity may induce states to support, and even empower, it in a crisis. This article argues that in providing a contemporaneous stress-test, Covid-19 serves to expose both the pathologies of a strong-centred federation and the surprising resilience of a weak one. It highlights three polity features—powers, decision-making modes and integrity—and charts their political implications during an acute crisis. The article argues that in the EU these features incentivise cooperative ‘polity maintenance’ between polarised states, a feature absent in an American polity marked by rivalry between polarised parties. The article thus challenges notions that the EU’s incompleteness necessarily leads it to dysfunction or that it should strive to emulate established federations.