Stabilizer Quantum Gravity IV: Defect Sectors, Chirality, and the Constructive Matter Problem

This paper forms Part IV of the Stabilizer Quantum Gravity (SQG) program, formalizing the constructive matter problem within an emergent operator-algebraic framework. In SQG, localized stabilization deficits naturally act as effective geometric sources. However, source-like behavior alone is insufficient to recover the phenomenology of the Standard Model. This work isolates the exact structural step at which the matter problem becomes mathematically meaningful: the transition from localized sourcing defects to persistent, chiral transport sectors. We define recoverable defect sectors, topological persistence, effective transport, and an effective chirality index. We establish the minimal constructive target for matter emergence: the existence of a recoverable defect sector that is topologically persistent, transport-capable, and carries a stable nonzero chirality index. Furthermore, we explicitly outline the obstruction pathways, such as chiral triviality (the mirror obstruction), instability under recovery, and transport failure, that could block the realization of realistic matter. This transitions the SQG matter program from heuristic interpretation to a formal theorem-engineering target.