Rotation-Curve Effective Gravity G_eff(r)

A spiral galaxy rendered in glowing particle streams, its outer disk glowing evenly bright all the way to the edge rather than fading, hinting at an unusually flat velocity profile.

$$G_{eff}(r) = G\left[1 + k\left(\frac{r}{r_0}\right)^\gamma\right]$$

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.

Checked against real data directly, not just against the paper’s own claim (2026-07-21/28 audit). Fit against the real SPARC database (Lelli, McGaugh, Schombert 2016 — 171 disk galaxies, 3375 rotation-curve points with real measurement uncertainties), scored as $\chi^2/\text{dof}$ against each point’s own measurement error ($\chi^2/\text{dof}\approx1$ means a model matches the data to its own precision). The published radius-based form ($k(r/r_0)^\gamma$, refit) reaches only $\chi^2/\text{dof}\approx246$ — residuals ~16× the data’s own precision. A reformulation in local baryonic acceleration instead of raw radius ($g_{bar}=V_{bar}^2/r$; fitted $k=0.99$, $\gamma=0.537$, $a_\dagger=6.68\times10^{-11}$ m/s²) does substantially better, $\chi^2/\text{dof}\approx82.5$ (~9× the data’s precision) — a real improvement, still roughly 16–80× short of MOND/NFW dark-matter-halo fits on the same sample ($\chi^2/\text{dof}\approx1$–5). See Paper 14 for the full cross-scale audit this feeds into.

What would make this proven: a single, consistent parameter set that both matches real galactic rotation-curve data to within its own measurement precision 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. Neither exists yet — the best real-data fit found so far ($\chi^2/\text{dof}\approx82.5$) is a genuine improvement over the published form but still falls well short of this bar.

Used in: Paper 1, Paper 2, Paper 4, and audited directly in Paper 14.