Diff(M) ⇒ Noether II ⇒ Jξ ≈ dQξ ⇒ Hξ = ∫∂Σ(Qξ − ξ·B) ⇒ ∂Σ = ∅ ⇒ Hξ = 0 Diffeomorphism invariance of the action, Diff(M), implies Noether’s Second Theorem, which establishes structural identities between the equations of motion. In General Relativity, these are the Bianchi identities, ∇μGμν = 0, which are structural and not dynamical equations. As a consequence, the associated Noether current satisfies Jξ ≈ dQξ, meaning the current is an exact differential form on shell. This implies that the Hamiltonian generator of diffeomorphisms is given by a boundary integral, Hξ = ∫∂Σ(Qξ − ξ·B), rather than a volume integral. If the spatial manifold is compact and without boundary, ∂Σ = ∅, for example Σ ≃ S³, then it follows strictly that Hξ = 0. This result is exact and follows directly from topology and Stokes’ theorem. It does not represent a violation of conservation law, but a stronger statement: in a closed universe without boundary, a global Noether generator of time cannot exist as an independent quantity, because there is no boundary on which it could be defined. The system is fully closed, with no external time and no external energy source; all dynamics represent internal transformations of the structure. We constructed a rigorous application of Noether’s Second Theorem to cosmology with compact boundaryless topology Σ ≃ S³, not as a special case but as a fundamental condition. We showed that the apparent tension between local conservation (Noether II) and absence of global Noether charge in General Relativity is not a contradiction but a topological consequence. We further connected this variational structure to a concrete physical framework, deriving mode spectra, CMB structure, an embedding variable, and falsifiable predictions. We therefore proved a rigorous physical theorem within the FBS³R cosmological model: in compact boundaryless cosmology with diffeomorphism-invariant action, the global Noether generator of time is absent, and this follows strictly from topology and Noether’s Second Theorem. This result establishes a fully closed variational and topological structure in which global consistency follows from compact geometry rather than postulated external parameters.
Batenin et al. (Tue,) studied this question.