{Paper 141: Filament Skeleton Fractal Dimension

Paper~139 established that box-counting the full galaxy distribution gives $d_H 1.89$ (all structures: walls, sheets, filaments combined), while the Davis \& Peebles~dp83 correlation dimension $D_2 = 1.23 0.04$ agrees with the VCML prediction $d_H = 2/ 1.236$ because the correlation function is pair-weighted and therefore filament-dominated. The definitive test is to isolate the filament skeleton directly. We apply multi-scale Hessian ridge detection (NEXUS$+$ style) to (i) 2.5\,M projected SDSS photometric galaxies and (ii) 126\,k volume-limited spectroscopic galaxies in 3D comoving space. After skeletonizing the filament network to a one-voxel-wide curve, we find $d_H(skeleton, 3D) = 1.22$--$1.25$ over $2$--$5$\,Mpc/$h$, with the 2/$ $ crossing consistently at $b 2.5$\,Mpc/$h$. This is within $1$--$2\%$ of $2/ = 1.2361$ and consistent with the D\&P value at $0.4 $. The three independent measurements---skeleton box-counting (3D), skeleton box-counting (2D projected), and correlation dimension---now all converge on $d_H 1.23$, confirming the VCML prediction to the precision available from current SDSS data.