We extend Closure Theory (CT) to bound systems and atomic structure. In prior work,radiation was interpreted as closure imbalance relaxation, and quantisation as discrete closurelocking. In the present work, we show that atomic structure arises from synchronised closureconfigurations governed by recurrence.Using the closure recurrence relation mc2T0 = Nh, we demonstrate that admissible boundstates form a discrete ladder corresponding to stable synchronised closure configurations. Theprincipal quantum number n is reinterpreted as a recurrence index labelling these configurations.We recover the Bohr radius, atomic energy scale, and Rydberg spectral structure as directconsequences of closure geometry. Transitions between atomic states correspond to discreteclosure reconfiguration events, consistent with the emission law hν = ∆E derived in ClosureTheory II.In this framework, atomic structure is not imposed through wavefunction quantisation, butemerges from the synchronisation of curvature closure under recurrence constraints. Quantumnumbers are thus reinterpreted as closure indices, and the atomic spectrum as the observableprojection of a recurrence ladder.