A Simple Explanation for Why Anomalous Galaxy Rotation Curves Do Not Break Newton's Law of Gravitation

The Gravitational Potential Relationship Framework (GPRF) James Richard Marsen | Independent Researcher | ORCID: 0009-0004-5112-0181 This paper presents the Gravitational Potential Relationship Framework (GPRF), a structural explanation for the anomalous rotation curves of nearby galaxies. The framework posits a single constitutive relation: non-baryonic matter density is proportional to the gravitational potential, ρₙb ∝ φ. No microphysical derivation of this proportionality is claimed. The paper is conditional: if non-baryonic density obeys this law, the following consequences follow by algebra, without modifying gravity, without a freely chosen interpolating function, and without a free parameter per galaxy. Key results established in this paper: Flat rotation curves emerge as a structural consequence of the Yukawa exterior solution of the augmented Poisson equation — not an assumption. The Baryonic Tully–Fisher relation is recovered: the tight empirical scaling v⁴flat ∝ GMb a₀ follows from a single normalization condition on the coupling constant ξ. A unique parameter-free ν-function νGPRF (x) = exp (−1/√x) (1 + 1/√x) is derived from the Yukawa exterior solution — not chosen. Unlike standard MOND interpolating functions, which are selected phenomenologically from infinitely many candidates, νGPRF is what the constitutive relation implies. Its value at the MOND transition, νₗocal (1) = 2e⁻¹ ≈ 0. 74, gives the analytic origin of the RAR normalization gap. Key result: the enclosed mass integral with the BG2016 disk+bulge model closes most of this gap — νₑncBG16 (1) ≈ 1. 358 is within the 1σ scatter of the observed RAR (McGaugh+2016, central value ≈ 1. 58), reducing the normalization discrepancy from a factor of ~2 to ~14% without any new free parameters. The remaining gap reflects the fixed-source and spherical-symmetry approximations; closing it is the central quantitative target of GPRF II. The Milky Way outer halo cutoff scale k⁻¹ ≈ 14 kpc is predicted from the BTFR normalization alone, without fitting to the outer rotation curve. This coincides with the onset of the outer rotation curve decline reported by Gaia DR3. Standard MOND cannot predict this decline; standard CDM requires non-generic halo truncation to accommodate it. The full enclosed mass solution gives vc ≈ 187 km/s at 27. 3 kpc against the observed 173 ± 17 km/s — within ~1σ, with the residual gap having identified physical origins. The decline is standard Newtonian dynamics applied to the full enclosed mass: the Yukawa profile of Mₙb asymptotes to a finite constant, and the rotation curve declines because Mₙb stops growing — not because gravity weakens. A three-regime galaxy structure — Before-MOND (kr ≪ 1, inner disk), MOND-like (kr ~ 1, flat curves), and Beyond-MOND (kr ≫ 1, outer decline) — emerges parameter-free from a single coupling constant. The Gaia DR3 outer decline is the first observational detection of the Beyond-MOND regime. Halo cutoff scales vary with galaxy mass as k⁻¹ ∝ v²flat/a₀, consistent with 10 SPARC galaxies spanning three orders of magnitude in rotation speed. This is a falsifiable prediction: dwarf galaxies are interior-dominated with no visible cutoff; massive spirals have their cutoff beyond current survey limits; SKA will test this systematically. The GPRF environmental effect is opposite in sign to the MOND external field effect: in denser environments, the higher environmental potential floor amplifies the non-baryonic component rather than suppressing it. This is a direct discriminant between GPRF and MOND+EFE. It is argued qualitatively that GPRF applies to cluster-scale dynamics. In merging systems such as the Bullet Cluster, the co-distribution rule ρₙb = ξφ (x) naturally produces lensing mass peaks at the BCGs — where potential peaks — while X-ray emission concentrates in the flatter interior gas. This structural account is consistent with sub-10 kpc resolution JWST lensing data (Cha et al. 2025). Quantitative cluster-scale integration is deferred to subsequent work. Newtonian gravity is recovered in the Solar System to first-order precision: the non-baryonic correction to orbital dynamics is negligible at kr ~ 10⁻⁹ for k⁻¹ ≈ 14 kpc, consistent with all planetary ephemeris constraints. Second-order effects (gravitational light deflection, Shapiro delay) depend on the total refractive index n = 1 + 2φₜot/c², which includes the non-baryonic contribution to φₜot; this GR limit is deferred to GPRF II. The framework predicts Newton’s gravitational law holds at cosmological scales (kr ≫ 1, ρₙb → 0), consistent with the Gallardo et al. (2026) kSZ gravitational force-law test at 30–230 Mpc. The Gallardo result is a consistency check, not an independent test of the non-baryonic sector: GPRF predicts standard Newtonian gravity there by construction. No fundamental principles were modified. Explicitly deferred to subsequent work: full interior self-consistent solution; quantitative RAR comparison with the full SPARC sample; quantitative cluster-scale integration; first-principles derivation of ξ from cosmological boundary conditions; connection a₀ ≈ cH₀/2π as a consequence of the causal horizon. Graphical Abstract, Extended FAQ, Technical Summary, and General Audience Overview are included as companion files. Submitted to: Galaxies (MDPI, ISSN 2075-4434) Preprint DOI: https: //doi. org/10. 5281/zenodo. 20450130