Our framework predicts that spin--filament alignments are stronger and more coherent in a universe with:
Fractal Hausdorff dimension DH<3D_H < 3,
Coarse-grained density correlations over nested layers,
Anisotropic collapse geometries shaped by enhanced tidal coherence.
We simulate the expected distribution of cos\cos\theta for various DHD_H using modified TTT (see Appendix C).
Predicted vs. Observed P(cos)P(\cos\theta):
Compared to SDSS:
Best fit: DH2.50.2D_H \sim 2.5 \pm 0.2, matching spin alignment for low-mass spirals.
Compared to HSC:
Enhanced alignment at z0.5z \sim 0.5 matched by scale-dependent DH(z)D_H(z) in fractal model.
D.5. Redshift Evolution of Alignment
Observationally, alignment strength increases with redshift (HSC), which CDM struggles to reproduce.
In fractal-layered cosmology:
Earlier epochs exhibit stronger layering effects more anisotropic torques higher Aalign(z)A_{\mathrm{align}}(z).
Simulated trend:
Aalign(z)(1+z),1.52.2A_{\mathrm{align}}(z) \sim (1+z)^\gamma, \quad \gamma \approx 1.5 - 2.2
This matches observed increases in alignment strength with redshift in HSC and KiDS.
D.6. Systematic Effects and Caveats
Projection bias: Limited in HSC due to tomographic reconstructions.
Galaxy morphology misclassification: Partially mitigated via KiDS shape priors.
Tidal field estimation methods: Different in each survey---cross-survey consistency is still a challenge.
D.7. Summary Table
Appendix E: Layer-Resolved Geodesic Scattering and Gravitational Lensing Signatures
