Effective Gravitational Coupling G_eff(l)

$$G_{eff}(l) \approx \frac{\varepsilon(l)\, \sigma(l)^2}{4\pi\, m(l)^2}, \quad \sigma(l) \approx l^2, \quad m(l) \approx \frac{\hbar}{lc}$$

Status: Unproven.

This is PBT’s single central claim: one formula, built on hierarchical energy density scaling, meant to reproduce ordinary Newtonian gravity, the strong nuclear force, and atomic/chemical bonding as the same effective coupling evaluated at different scales $l$.

What’s actually been checked, and what was found. Paper 1 claims plugging in $l\approx10^{-15}$ m (nuclear scale) gives $G_{strong}\approx10^{29}$ m³kg⁻¹s⁻². This session’s 2026-07-21 audit recomputed this directly from the paper’s own stated $\varepsilon_0$, $l_0$, and $\gamma$ range and found it doesn’t follow — the computed values come out many orders of magnitude off in every case checked (solving for the $\gamma$ that would produce $10^{29}$ gives $\gamma\approx-0.07$, outside the paper’s own stated range and the wrong sign relative to its own claim that finer scales amplify $\varepsilon$).

A second, subtler problem found in the same audit. The paper’s “Gravitational Level” calculation — plugging in $\sigma\approx5.6\times10^{-50}$ m² and the proton mass to recover the real $G=6.6743\times10^{-11}$ — was found to be circular rather than an independent check: solving the same equation for $\varepsilon$ using those inputs reproduces $\varepsilon_0$ almost exactly, meaning $\sigma$ appears to have been chosen specifically to make this happen. It’s a normalization of the model’s own free parameters, not a prediction of a real, independently-measured constant from anything more fundamental.

What would make this proven: a version of this formula that reproduces $G$, nuclear binding, and atomic bonding simultaneously, from one consistent, non-circularly-chosen set of parameters — not tuned separately at each scale. That hasn’t been shown to work even once, let alone across all three scales at once.

Used in: Paper 1, Paper 2, and referenced throughout the series as the theory’s central unifying formula.