Emergent Scale-Dependent Geometry from a Fractal Prime Network

This preprint presents a discrete arithmetic network of 360 000 primes with a modulo-15 edge rule. Distance is the shortest-path length. The network dimension from the scaling of N(ℓ) is non-integer and scale-dependent (1.76 / 1.34 / 1.13). The same points form a continuous spiral with box-counting dimension ≈ 1.13. The function π_eff(r) is an analytical construction satisfying π_eff(0) = 1 and π_eff → π. Circular velocity from mass proportional to N(ℓ) has near-zero slope. Comparison with 143 SPARC galaxies shows better shape agreement for slower galaxies. The density proxy has a slope of the same order as the NFW shape. The work states explicitly which structures are measurements and which are constructions. Code, figures and data are included.