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).

Tested directly against real, independently-known pressures at scales it was never calibrated for (2026-07-24/28 audit) — it was only ever fit at galactic scale. Evaluated unmodified at the Bohr radius against the real atomic unit of pressure: 7 to 36.5 orders of magnitude too small, depending on $\gamma$. Evaluated at roughly one femtometer against the MIT bag-model nucleon confinement pressure: 18 to 38.2 orders too small. A separate, more direct test — fitting this same formula to real nuclear binding energy specifically — misses by 18 to 30-plus orders of magnitude for every $\gamma$ in the stated range. The failure grows monotonically worse the further the evaluation scale sits from $l_0$ in either direction (see Paper 14’s cross-scale audit) — not an isolated nuclear-scale problem, but the same formula failing catastrophically everywhere outside the one regime it was built for.

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 — and the direct tests above found the opposite of confirmation everywhere they were tried. $\varepsilon_0$ 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; audited directly in Paper 14.