Black-Hole-Collapse Stabilization ODE
$$\frac{dv}{dt} = -\frac{GM}{r^2} + \frac{\varepsilon(l)}{3}\left(\frac{4\pi r^2}{M}\right)$$
Status: Unproven — and independently found not to stabilize at its own claimed scale.
Paper 1 claims this equation models gravitational collapse stabilizing at $\sim10^{-35}$ m (near the Planck scale) rather than reaching a true singularity, via amplified energy density $\varepsilon(l)$ at fine scales — PBT’s proposed resolution to one of General Relativity’s acknowledged real limits.
Checked directly, in the 2026-07-21 audit. Solving this ODE’s own equilibrium condition ($dv/dt=0$, $l=r$) gives $r^{4-\gamma}=3GM^2/(4\pi\varepsilon_0 l_0^\gamma)$. For a solar-mass collapse using the paper’s own $\varepsilon_0$, $l_0$, and stated $\gamma\approx2$–$4$: $\gamma=2$ gives an equilibrium radius of $\approx9.2\times10^{31}$ m, and $\gamma=3$ gives $\approx8.5\times10^{88}$ m — both cosmologically enormous, the opposite of Planck-scale stabilization, and nowhere near the claimed $10^{-35}$ m. Independently confirmed (Grok, cross-checking the same arithmetic), with no charitable reinterpretation (different mass scale, fixed vs. running $l$) found that rescues the claim as written.
What would make this proven: a version of this ODE, or a corrected derivation from it, that actually produces a Planck-scale (or any physically small) equilibrium radius from the theory’s own stated parameters — not asserted, computed. That hasn’t been shown.