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 work 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 total gravitational potential at that point — the summed gravitational contribution of every mass in the universe. 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. It is argued that the framework also 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 baryonic Tully-Fisher relation slope is recovered under one normalization condition with one empirical input. The framework implies a unique interpolating function ν_GPRF(x) = exp(-1/√x)(1 + 1/√x) — derived, not postulated — whose value at the MOND transition ν(1) = 2e⁻¹ ≈ 0.74 gives the analytic origin of the RAR normalization gap. 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 (GPRF II). 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 is a self-consistency check, not an independent prediction. 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 naturally expected structural consequence — neither CDM nor MOND recovers this from structure alone. 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. v1.0.9 changes: Framework renamed "Gravitational Potential Response Framework" → "Gravitational Potential Relationship Framework." "Response" carries an unintended mechanistic connotation — the constitutive relation ρ_nb = ξ δφ is a statement about the spatial density relationship between baryonic and non-baryonic matter, not a causal response to the potential. Bullet Cluster schematic figure added (uniform gas, lensing peaks aligned with BCGs). Three-regime enclosed mass figure (Fig. 7) and rotation curve figure (Fig. 8) added as separate figures, replacing the earlier combined two-panel figure. Table 2 added to the Discussion: a structured paradigm comparison across scales (Newtonian / MOND / GPRF) for the inner galaxy, flat rotation curve regime, outer decline, and Bullet Cluster. Beyond-MOND regime introduced with correct physical framing: exponential Yukawa augmentation e^{-kr} → 0 beyond the baryonic extent, δρ_nb → 0 — Keplerian re-emergence. Unique structural prediction with no MOND or CDM equivalent; first observational identification is the Gaia DR3 outer halo decline. Deep-MOND (x ≪ 1) and Beyond-MOND (kr ≫ 1) shown to be the same physical regime — ν_GPRF → 0 is the prediction associated with the Gaia DR3 outer halo decline. Gaia DR3 comparison corrected and tightened following a citation audit: the point-mass exterior solution reproduces the onset radius (k⁻¹ ≈ 14 kpc) and the qualitative decline mechanism, but does not yet reproduce the observed decline magnitude. Against Jiao et al. (2023) and Ou et al. (2024), the bias-corrected point-mass solution predicts ~100–116 km/s at 27.3 kpc versus an observed 173 ± 17 km/s. This shortfall is attributed to the point-mass approximation, which omits the enclosed non-baryonic mass contribution at intermediate radii, and is identified as the central quantitative target of GPRF II. Earlier language describing this as a quantitative match has been corrected. Order-of-magnitude estimate of ξ ≈ 6×10⁻³³ kg s² m⁻⁵ added with halo density consistency check (~0.09 M☉ pc⁻³ at r ~ k⁻¹, consistent with local halo estimates). k universality clarified — apparent galaxy-to-galaxy variation in k⁻¹ ∝ v²_flat/a₀ reflects the BTFR, not a variable coupling constant. Definitive test: single-ξ fit to SPARC sample (GPRF II). 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