Emergent Hydrogenic Spectra from Finite-Response ECSM Potentials

This work investigates whether hydrogen-like atomic spectral structure can emerge naturally within the Emergent Condensate Superfluid Medium (ECSM) framework using a finite-response coherent potential. A family of ECSM radial response potentials is constructed and solved numerically using finite-difference eigenmode methods. The resulting spectrum exhibits: • discrete hydrogen-like shell structure, • approximate 1/n² energy scaling, • Lyman-, Balmer-, and Paschen-like transition hierarchies, • dipole-dominated transition behaviour, • and stable robustness convergence. The work does not attempt full precision QED spectroscopy, but instead demonstrates that leading atomic spectral organization may emerge naturally from coherent finite-response medium dynamics with minimal assumptions and without independently imposed microscopic force sectors. The paper forms part of the broader ECSM programme investigating emergent gravity, quantum structure, and cosmological dynamics from finite-response coherent-medium physics.