Ground-state degeneracy of the Fractal Ising model for arbitrary system sizes

Derive an analytical expression for the ground-state degeneracy of the three-dimensional Fractal Ising model for all lattice sizes L not restricted to the explicitly classified size families.

Background

The Fractal Ising model has a system-size-dependent symmetry group under periodic boundary conditions. The thesis derives the degeneracy for several families of sizes, including L=2n and other algebraically characterized sequences.

For general values of L outside those families, the symmetry generators and resulting degeneracy are not fully characterized. An analytical formula would complete the classification of the model's size-dependent ground-state structure.

References

While it is not clear whether an analytical expression exists to describe the GSD for all values of $L$, it is possible to identify several classes of system sizes that have well-defined ground-state degeneracy:

— Subsystem Symmetries and Fracton Models in Quantum Error Correction  (2608.18961 - Canossa, 19 Aug 2026) in Chapter 2, Section "Fractal Ising Model"