Using the conformal equivalence of translational KMS states on chiral theories with dilational KMS states obtained from restricting the vacuum state to an interval (the chiral inversion of the Unruh-effect) it was shown in a previous publications that the diverging volume (length) factor of the thermodynamic limit corresponds to the logarithmic increase in the attenuation length of the localization-caused vacuum polarization cloud near the causal boundary. This is not a coincidence but rather a structural consequence of the fact that both operator algebras are of the same unique von Neumann type which is completely different from that met in quantum mechanical algebras. Together with the technique of holographic projection this leads to the universal area proportionality. The main aim in this paper is to present a derivation which is more in the spirit of recent work on entanglement entropy in condensed matter physics, especially to that of the replica trick as used by Cardy and collaborators. The essential new ingredient is the use of the split property which already has shown its constructive power in securing the existence of models of factorizing theories.