Kaluza-Klein discreteness of the entropy: Symmetrical bath and CFT subsystem

We explore the entanglement entropy of CFT systems in contact with large bath system, such that the complete system lives on the boundary of $AdS_{d+1}$ spacetime. We are interested in finding the HEE of a bath (system-B) in contact with a central subsystem-A. We assume that the net size of systems A and B together remains fixed while allowing variation in individual sizes. This assumption is simply guided by the conservation laws. It is found that for large bath size the island entropy term are important. However other subleading (icebergs) terms do also contribute to bath entropy. The contributions are generally not separable from each other and all such contributions add together to give rise a fixed quantity. Further when accounted properly all such contributions will form part of higher entropy branch for the bath. Nevertheless the HEE of bath system should be subjected to minimality principle. The quantum minimality principle $ S_{quantum}[B]=\{S[A], S_{total}+S[A]\}_{min}$, is local in nature and gives rise to the Page curve. It is shown that the changes in bath entropy do capture Kaluza-Klein discreteness. The minimality principle would be applicable in finite temperature systems as well.