A Novel Approach to the Missing Mass Problem by Positing a Simple Spatial Density Law for Non-Baryonic Matter

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 approach to the missing mass problem that posits two properties of the observed non-baryonic mass component. First, it is real, gravitating mass — a claim already accepted by the dark matter community. Second, its local density obeys a simple spatial law: it is proportional to the local gravitational potential excess above the cosmic Machian baseline, ρ_loc = ξ φ_loc. No microphysical derivation of this proportionality is claimed. The result is conditional: if the non-baryonic mass density obeys this law, MOND-like phenomenology emerges as a structural consequence at galactic scales. The framework applies at cluster scales, but quantitative treatment is deferred to future work. Key results: Flat rotation curves emerge as a structural consequence of the Yukawa exterior solution, not an assumption. The framework implies a unique interpolating function ν_GPRF(x) = exp(−1/√x)(1 + 1/√x) — not a free functional choice, but the unique function implied by the Yukawa exterior solution under the BTFR normalization. Its value at the MOND transition ν(1) = 2e⁻¹ ≈ 0.74 gives the analytic origin of the ~0.4 dex RAR normalization gap. Closing this gap is the central quantitative target of subsequent work. Note: ν_GPRF is valid only for kr ≳ 1 (exterior regime); dwarf galaxy and LSB interior kinematics are governed by the full interior self-consistent solution, deferred to subsequent work. The framework predicts a specific radial acceleration relation (RAR) — a falsifiable, parameter-free prediction inviting direct confrontation with the McGaugh et al. (2016) observed correlation. Full quantitative comparison requires the interior self-consistent solution (subsequent work). For the Milky Way the normalization gives k⁻¹ ≈ 14 kpc, coinciding with the Keplerian decline onset reported by Gaia DR3 — derived from the BTFR alone, not fitted to the outer rotation curve. This three-way comparison is the paper's central empirical result: standard MOND cannot predict this decline (ν_MOND → x⁻¹/² gives asymptotically flat curves); standard CDM requires non-generic NFW halo truncation to accommodate it. Three galaxy regimes emerge from a single parameter-free prediction k⁻¹ ∝ v²_flat/a₀. Lensing mass peaks in merging cluster systems align with BCGs as a qualitative structural consequence of the co-distribution rule ρ_loc = ξ φ_loc, confirmed at sub-10 kpc resolution by JWST (Cha et al. 2025 ApJL). Quantitative treatment pending derivation of the relaxation timescale. Whether the non-baryonic component sources the full GR stress-energy tensor or only the Newtonian potential yields distinct lensing predictions testable with current cluster data — identified as a primary question for subsequent work. Newtonian gravity is recovered in the Solar System without a separate patch. Consistent with the Gallardo et al. (2026) kSZ force-law test at 30–230 Mpc scales. No fundamental principles were modified. A graphical abstract summarizing the framework and key results is included as a companion file. To be submitted to: Galaxies (MDPI, ISSN 2075-4434)Preprint DOI: https://doi.org/10.5281/zenodo.20450130