Simulations based on fractal initial conditions feeding angular momentum growth via the Zel'dovich approximation and torque alignment.
B.5. Ray-Tracing Setup for CMB Anisotropy Mapping
Tracing null geodesics through layered topological geometries to compute temperature fluctuations.
B.6. Data Availability and Format
All simulation outputs are stored in:
HDF5 format for large particle datasets.
JSON format for configuration files and metadata.
FITS format for synthetic CMB sky maps and galaxy spin alignment catalogs.
To request full datasets and custom visualization scripts, contact the corresponding author or access the linked Zenodo repository (will be made public upon peer review acceptance).
Appendix C: Analytical Derivation of Fractal Spin Alignment in Large-Scale Structure
This appendix presents the full derivation of spin--filament alignment arising from fractal initial conditions, based on tidal torque theory (TTT) extended to a hierarchical density field with non-integer Hausdorff dimension DHD_H.
C.1. Background: Tidal Torque Theory (TTT) Refresher
In standard TTT, the angular momentum L\mathbf{L} of a protohalo forming in the linear regime is approximated as:
Li(t)a2(t)ijkTjlIlkL_i(t) \propto a^2(t) \, \epsilon_{ijk} T_{jl} I_{lk}
