Explain the extensive bulk contribution to the fuzzy-sphere interaction correction

Explain the mechanism responsible for the additional bulk contribution proportional to the radius of the fuzzy sphere in the first-order quartic-interaction correction to entanglement entropy.

Background

For the fuzzy sphere, the first-order correction to the entanglement entropy violates the free-theory area law and scales extensively with the bulk degrees of freedom, while retaining the same degree of ultraviolet divergence as the free contribution. The numerical analysis shows that degrees of freedom throughout both hemispheres contribute significantly, rather than only those near the entangling circle.

The authors explicitly state that the origin of the additional bulk contribution proportional to the radius remains unresolved. Establishing this mechanism is important for determining whether the extensive behavior is an intrinsic feature of interacting fuzzy-space theories or a consequence of the perturbative treatment.