Hierarchical Energy Density Scaling

$$\varepsilon(l) = \varepsilon_0 \left(\frac{l_0}{l}\right)^\gamma$$

Status: Unproven.

This is PBT’s own central postulate, not a result derived from anything more fundamental: energy density in the theory’s proposed subatomic medium is claimed to scale as a power law with the observation scale $l$, using free parameters $\varepsilon_0\approx7.4\times10^{35}$ J/m³, $l_0\approx10^{-25}$ m, and $\gamma\approx2$–$4$. This is the formula the entire “one mechanism unifies gravity, nuclear binding, and atomic bonding” claim is built on top of (see Effective Gravitational Coupling, which is derived from this one).

What would make this proven: an independent measurement of energy density at two or more of these vastly different scales (nuclear, atomic, planetary) that confirms the same power law with the same parameters — not just a formula that can be tuned to match known constants after the fact at each scale separately (see the note on Effective Gravitational Coupling below, where this session’s own audit found the “match” to the real gravitational constant $G$ is a normalization, not a prediction). No such independent, cross-scale measurement exists. $\varepsilon_0$ and $l_0$ are not independently measured constants of nature the way $G$ or $\hbar$ are — they’re fit to reproduce known results, which is a legitimate way to build a model but not the same thing as confirming one.

Used in: Paper 1 (introduced here) and referenced throughout nearly every other PBT paper as the theory’s core scaling relation.