Rotation-Curve Effective Gravity G_eff(r)
$$G_{eff}(r) = G\left[1 + k\left(\frac{r}{r_0}\right)^\gamma\right]$$
Status: Unproven — and independently found not to reproduce its own claimed results.
This is PBT’s flagship dark-matter-free explanation for flat galactic rotation curves, feeding into $v(r)=\sqrt{G_{eff}(r)M_{enc}(r)/r}$ (the proven part of that equation).
Paper 1’s version, checked directly. Using the paper’s own stated $k\approx1900$, $\gamma=1$, $r_0=10$ kpc, and $M_{enc}\approx6\times10^{10}\,M_\odot$: plugging into the formula at $r=r_0$ gives $v\approx7005$ km/s — not the claimed ~220 km/s, off by roughly 32x. Plain Newtonian gravity alone ($k=0$, no PBT modification at all) gives a far more realistic $\approx161$ km/s with the same inputs. The $k$ value that would actually produce ~220 km/s is about 0.875, not 1900. Paper 2 repeats the identical parameters for the same claim.
Paper 4’s independently-parametrized version, also checked. Using its own stated $k=0.1$, $\gamma=2$: the same formula gives $\approx128$ km/s at 30 kpc, not the claimed ~250 km/s — about half. The curve does genuinely flatten in shape (a real qualitative feature of this formula), just not at the claimed magnitude.
Neither of the two independently-derived parameter sets in this paper series currently reproduces its own claimed output when the formula is computed directly from its own stated inputs — both findings from this session’s own 2026-07-22 audit, not previously caught in the 2026-07-21 corrigendum pass.
What would make this proven: a single, consistent parameter set that both matches real galactic rotation-curve data and is derived from (or at least consistent with) the same $\varepsilon(l)$/$G_{eff}(l)$ formula used elsewhere in the theory, rather than fit freely and separately for this one purpose. See Unproven Systems At Large for MOND, a real competing modified-gravity explanation for the same rotation-curve data, not yet cross-compared on this site.