Rotation-Curve Effective Gravity G_eff(r)

Current formula (adopted 2026-09-30, Paper 1 Rev 5)

$$g_{obs} = \frac{g_{bar}}{1-e^{-\sqrt{g_{bar}/a_0}}}, \qquad a_0 = 1.20\times10^{-10}\ \text{m/s}^2, \qquad g_{bar}=\frac{GM_{bar}}{r^2}$$

with Milgrom’s External Field Effect applied for any system embedded in a larger ambient field (see below).

Status: Real, independently-published, precision-tested astrophysics — not a PBT derivation. This is the radial acceleration relation (McGaugh, Lelli & Schombert, Phys. Rev. Lett. 117, 201101, 2016), fit to 2693 points across 153 real galaxies, adopted here to replace the badly-miscalibrated formula below. At the Milky Way single point ($r=10$ kpc, $M_{enc}\approx6\times10^{10},M_\odot$): $v\approx213.5$ km/s against the real $\approx220$ km/s target — not the old formula’s 7005 km/s. Root cause of the old formula’s failure, not just a bad parameter choice: $k(r/r_0)^\gamma$ grows linearly with raw radius off a coefficient of ~1900, so it was only ever going to look reasonable in the narrow window around the one radius it was fit to.

External Field Effect (Famaey & McGaugh, Living Rev. Relativity 15, 10, 2012, Sec. 6.3). A system embedded in a larger ambient gravitational field has its own internal boost suppressed by that field — standard MOND phenomenology, not invented for PBT. Milgrom’s real quasi-Newtonian EFE approximation is direction-dependent (an anisotropic dilation along the external field’s own direction); the check here uses the same isotropic interpolation function as an order-of-magnitude stand-in, not a claim of matching the full vector theory. The Milky Way’s own field at the solar neighborhood ($V\approx220$ km/s, $R\approx8$ kpc) gives $g_{ext}=V^2/R\approx1.96\times10^{-10}$ m/s², comparable to $a_0$. Checked at real planet-formation scales (molecular cloud, pre-stellar core, protoplanetary disk, planetesimal orbit — all embedded in this same field): disk and planetesimal scale need no correction at all, since their own internal gravity already vastly exceeds both $a_0$ and $g_{ext}$. At cloud/core scale, two different failures both come down to the same shared residual: the old formula’s own failure was 10.5x at giant-molecular-cloud scale (only 1.9% at pre-stellar-core scale — a raw-radius formula, so the two scales fail by very different amounts); the new formula (RAR alone, without EFE) would itself fail by 246% at pre-stellar-core scale, since that scale’s own $g_{bar}$ sits well below $a_0$. With EFE applied, both land at the same ~39% residual, stated plainly here rather than rounded away — real progress, not a fully closed case. That residual sits in the same regime ($g_{ext}/a_0$ of order 1) that real, current wide-binary-star tests of MOND/EFE are actively contested over (e.g. Chae 2023/2024 and Hernandez et al. report MOND-favoring anomalies from the same Gaia wide-binary catalogue that Banik et al. 2024 and Pittordis & Sutherland report excludes MOND at high significance — a real, live methodological dispute, not settled either way). Full computation: static/models/pbt-gravity-rar-efe-fix-v1.py.

What this is not: a claim that PBT has derived or discovered the radial acceleration relation. It hasn’t — this is importing a real, established empirical law, the same “agreement is the pass condition, not a weakness” move already used for magnetism and electromagnetism elsewhere in this project. Deriving why the shadow/flux mechanism should produce this relation from its own primitives remains open.

Superseded formula (Papers 1/2/4, until 2026-09-30)

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

Kept below for the historical record, not deleted — this is no longer this project’s working formula.

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.