Dark Matter Induced by Gravitational Potential: MOND-like Phenomenology at Galactic Scales

The Gravitational Potential Response Framework (GPRF) Dark Matter Induced by Gravitational Potential: MOND-like Phenomenology at Galactic Scales Rotation Curves, the Baryonic Tully-Fisher Relation, and the Outer Halo Cutoff James Richard Marsen | Independent Researcher | ORCID: 0009-0004-5112-0181 This work presents the Gravitational Potential Response Framework (GPRF), a structural approach to galactic dynamics that recovers the empirical successes of both dark matter and Modified Newtonian Dynamics — flat rotation curves, the tight mass-velocity correlation observed across all spiral galaxies, and the finite boundary of the galactic halo — from a single linear constitutive relation and one coupling parameter, without new particles, modified gravity, or ad hoc assumptions. The MOND formula emerges as a mathematical consequence rather than a starting assumption. The constitutive relation ρₗoc = ξ φₗoc combined with the standard Newtonian operator and classical potential theory yields an augmented Poisson equation whose Yukawa exterior solution produces a broad velocity plateau and a structural halo cutoff. The −k² operator follows conditionally from the environmental decomposition φₗoc = φₜot − Φbg, the Sciama–Brans-Dicke requirement that a physically real substrate density exists, and the augmentation hypothesis ξ > 0; the sign is not an independent postulate (Appendix D). The induction parameter is normalized per galaxy to the baryonic Tully-Fisher relation via k² = a₀/ (GMb), taking one empirical scale a₀ through one normalization condition; this recovers the correct BTFR slope and a specific radial acceleration relation. For the Milky Way this normalization gives k⁻¹ ≈ 14 kpc, consistent with the Keplerian decline recently revealed by Gaia DR3 as a self-consistency check rather than an independent prediction. Because the cutoff scale varies with galaxy mass as k⁻¹ ∝ v²flat/a₀, the framework predicts a specific, falsifiable pattern: halo cutoffs appear at progressively larger radii in lower-mass galaxies. Key results: The constitutive relation ρₗoc = ξ φₗoc combined with the standard Newtonian operator yields an augmented Poisson equation whose −k² operator follows conditionally from three named assumptions: local validity of the Poisson equation on galactic scales, the Sciama–Brans-Dicke requirement that a physically real background substrate density exists, and the augmentation hypothesis ξ > 0. The sign is not an independent postulate. The Yukawa exterior solution produces MOND-like 1/r scaling at intermediate radii (kr ∼ 1) as a structural consequence, not an assumption. The baryonic Tully-Fisher relation slope is recovered under the normalization k² = a₀/ (GMb) — one empirical input, one condition. 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. Three galaxy regimes emerge from a single parameter-free prediction k⁻¹ ∝ v²flat/a₀: dwarf galaxies are interior-dominated throughout; MW-mass systems show the cutoff within the observable range; massive spirals have their cutoff beyond current survey reach. Newtonian gravity is recovered in the Solar System and wide-binary regime (kr ∼ 10⁻⁹) without a separate patch. The framework is consistent with the Gallardo et al. (2026) kSZ force-law test at 30–230 Mpc scales. No fundamental principles were modified. The Newtonian gravitational operator is unchanged. The framework adds one constitutive relation. Key changes in v1. 0. 6: The notation δφ is replaced by φₗoc ≡ φₜot − Φbg throughout the framework, making explicit that the dynamical variable is a local potential excess above the environmental baseline — not a deviation from zero. This dissolves the empty-universe assumption implicit in the earlier notation. The −k² operator in the augmented Poisson equation is now presented as a conditional consequence of three named assumptions: The local validity of the Poisson equation for Φbg on galactic scales (standard cosmological perturbation theory) ; The Sciama–Brans-Dicke postulate that a physically real background substrate density ρₙb, env exists; The augmentation hypothesis ξ > 0. Appendix D is reframed as a consistency check rather than a derivation. A new logical structure paragraph in Section 2 explicitly lists what is assumed, what is conditionally derived, and what remains open. Submitted to European Physical Journal Plus (EPJ Plus). Preprint DOI: https: //doi. org/10. 5281/zenodo. 20450130