/The Hamiltonian Constraint A First-Class Constraint and the Limits of the Zero-Energy ArgumentThe scalar Hamiltonian constraint C = 0 in general relatvity is read, in expositons of the canonical formalism, as a statement that the total energy of a closed universe vanishes. From this, popular accounts and some expository treatments infer that the universe could arise as a zero-energy quantum fuctuaton, a creaton from nothing. This paper argues that the inference rests on a category error. The constraint is a frst-class constraint that follows from the difeomorphism invariance of the acton, and its content is a restricton on phase space. The energy of the canonical formulaton sits in a boundary term, which a compact slice without boundary leaves absent, so the standard asymptotc and quasi-local defnitons of gravitatonal energy have nothing to atach to. A physical Hamiltonian can nevertheless arise on such a slice once a mater clock deparametrises the theory, and its value is generically nonzero. The vanishing of the unreduced constraint therefore fxes the value of no physical energy at all. Four further setngs sharpen the point: the free value of the cosmological constant, the comparison with electric charge, the foliaton dependence of any energy split, and canonical quantsaton with its boundary proposals. In each the constraint remains a statement of internal consistency within a pre-given formalism. It leaves the origin of that formalism outside its scope, and it licenses no talk of creaton ex nihilo. K
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