Multi-Scale Tests of a Modified Gravity Theory Based on Fermion–Boson Duality: From Galactic Rotation Curves to Wide Binary Stars

This archive provides the preprint and Mathematica code for a modified-gravity framework based on energy-weighted Fermion–Boson Duality (FBD). The approach explains flat galaxy rotation curves without dark matter and proposes a clean, discriminating test using wide binary stars. The key ingredient is a distance-dependent statistical transition function, T (r) = 1 + exp ( (r − rfb) /λ) ^−1, which smoothly connects Newtonian gravity (inner regions) to an enhanced-gravity regime (outer regions). The transition radius is rfb = sqrt (GM/a0), and the same scaling is applied across multiple mass scales (galaxies to binaries). For galaxies, the framework naturally reproduces the baryonic Tully–Fisher relation (v⁴ ∝ M) and provides a statistical–mechanical interpretation of the MOND acceleration scale a0 ≈ 1. 2×10^−10 m/s². For wide binaries, FBD predicts a sharper (more Newton-consistent) behavior in the transition region: around s ≈ rfb, FBD remains nearly Newtonian (η ≈ 1. 0), whereas MOND predicts a significant enhancement (η ≈ 1. 13–1. 27), depending on the MOND interpolation function. The expected difference (about 11–21% in the transition region) is, in principle, testable with high-precision data and is discussed in the context of the ongoing Chae (2023) vs. Banik et al. (2024) debate. The included Mathematica notebook reproduces the tables/figures and exports the PDF figures used in the manuscript, including an explicit FBD–MOND comparison (Simple/Standard/RAR interpolation choices) and an observation-comparison figure. Version history Version 5: Restructured the theoretical foundation to make explicit the development lineage QED → QCD → Gravity, with QCD-FBD serving as the formal bridge that justifies the replacement of energy-dependence by distance-dependence via the natural inverse-proportional correspondence between energy and distance in QCD. Repositioned the extension from QCD to gravity as an expansion of the domain of applicability of an effective theory, rather than a direct derivation, in order to clearly delineate the logical status of each step. Added a boxed summary of the cross-scale development lineage (QED → QCD → Gravity) and a boxed contrast with MOND emphasizing that the FBD transition function is uniquely determined by two-state statistical mechanics, whereas MOND interpolation functions are chosen empirically. Expanded the discussion of alternative transition-function formulations (logarithmic-distance version Tₗog and acceleration-ratio version Tₐ), with a new appendix section on the properties of Tₗog. Added quantitative sensitivity analysis of the Newtonian recovery at r → 0 as a function of the width parameter α (Table: T (0) vs. α), and clarified the physical acceptance criterion (α ≲ 0. 15 for Solar System compatibility, α = 0. 2 adopted as the standard value). Strengthened the structural analogy between galactic-scale FBD and QCD confinement (both share the feature that the F-type mediating component dominates at long range), explicitly contrasting this with the Yukawa (nuclear-force) behavior. Expanded the Limitations and Future Challenges section to include: (i) the need to re-derive the energy–distance correspondence at gravitational scales from first principles, (ii) the general-relativistic (covariant) extension required for lensing/time-delay/cosmology, (iii) the galaxy-cluster problem, (iv) potential issues with cosmological extrapolation, and (v) the formulation of EFE within the FBD transition function argument. ・ Version 4: ・ Major revision: expanded from galaxy rotation curves to a multi-scale framework including wide binary stars. ・Added a dedicated FBD vs. MOND comparison (η in the transition region) and updated the discriminant range to ηMOND ≈ 1. 13–1. 27 (thus FBD–MOND difference ≈ 11–21%). ・Added wide-binary formulation/predictions, observational comparison (Chae 2023 vs. Banik et al. 2024), and an expanded discussion of external-field effects (EFE). ・ Reorganized the archive files accordingly (manuscript Ver4 + MOND-comparison code/output). ・ Version 3: ・Added §7. 2 “Qualitative Analogy with QCD” discussing structural similarity between QCD confinement/asymptotic freedom and galactic FBD transitions. ・ Added references: Gross & Wilczek (1973), Politzer (1973). ・ Minor figure improvements: adjusted plot ranges and legend positions in selected figures. ・Version 2: ・Ensured dynamical consistency by defining acceleration as the primary quantity and deriving potential by integration.