Galaxy Rotation Curves from Baryonic Field Geometry

What if dark matter is not a substance but a shadow of the gravitational potential itself? We show that the discrepancy between observed and Newtonian rotation curves across 175 SPARC galaxies can be captured by a single formula involving only the baryonic gravitational potential 𝜙 and its derivatives — no dark matter, no modified force law, no free parameters per galaxy. The observed acceleration boost 𝑉obs2 /𝑉bar2 is empirically linear in 1/|𝜙| , with a slope that scales universally with the total baryonic mass encoded in (1/𝜙)′ . This yields a one-constant model, 𝑎 tot = 𝜙′ ⋅ (1 + 𝐶/√𝜙′) = 𝑎 𝑛 (1 + 𝐶/√𝑎 𝑛 ) with 𝐶 =√𝑎 0 ≈ 0.9 × 10 −5 , performs on par with MOND, with a marginal edge (54% vs 46%, median RMSE ratio 0.97) — and outperforms it at mass-to-light ratios independently favored by stellar dynamics. The baryonic Tully-Fisher relation 𝑉 4 ∝ 𝑀 emerges as an unimposed consequence. The numerical coincidence C² = a₀ ≈ cH₀ suggests these may be manifestations of the same underlying cosmological scale. We do not claim finality for the formula, but for the observation behind it: the Newtonian potential field of baryonic matter already contains the information that dark matter was invented to provide