Closure Theory III: Atomic Structure as Closure Synchronisation Recurrence, Spectra, and the Emergence of Quantum Numbers

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 closure
locking. In the present work, we show that atomic structure arises from synchronised closure
configurations governed by recurrence.
Using the closure recurrence relation mc2T0 = Nh, we demonstrate that admissible bound
states form a discrete ladder corresponding to stable synchronised closure configurations. The
principal 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 direct
consequences of closure geometry. Transitions between atomic states correspond to discrete
closure reconfiguration events, consistent with the emission law hν = ∆E derived in Closure
Theory II.
In this framework, atomic structure is not imposed through wavefunction quantisation, but
emerges from the synchronisation of curvature closure under recurrence constraints. Quantum
numbers are thus reinterpreted as closure indices, and the atomic spectrum as the observable
projection of a recurrence ladder.