Quantum derivation of the neutral-spectator tensor-product picture

Derive at the quantum level, including operator mixing and interactions, the tensor-product decomposition of the fixed-positive-charge ABJM BPS sector into a charged rank-q BMN sector and a neutral spectator sector.

Background

The coefficientwise index factorization contains a universal neutral factor multiplying the rank-q BMN index. A harmonic analysis suggests interpreting this factor as a protected neutral spectator whose support lies outside the local Penrose region, accompanying the charged BMN cluster.

The index identity alone does not prove a tensor-product decomposition of the interacting Hilbert space. The unresolved problem is to establish this physical cluster interpretation directly in the quantum theory while accounting for operator mixing and interactions.