Relate the four-dimensional class-AI parity constraint to real-space SPT–LSM theorems

Establish the relation between the even-parity constraint for four-dimensional class AI with internal symmetry and symmetry-protected-topological Lieb–Schultz–Mattis theorems formulated using the real-space Atiyah–Hirzebruch spectral sequence.

Background

For four-dimensional class AI with an inversion-like momentum-reversing antiunitary symmetry, the paper proves that an odd second Chern character cannot occur in the finite-dimensional framework. It then compares this result with anomaly-cancellation interpretations based on real-space SPT–LSM constructions.

The conclusion of that subsection identifies an unresolved conceptual problem: formulating an appropriate real-space AHSS description for internal antiunitary symmetries and relating it to the proven even-parity obstruction.