Newtonian Circular Orbital Velocity

$$v(r) = \sqrt{\frac{GM_{enc}(r)}{r}}$$

Status: Proven.

This follows directly from setting Newtonian gravity equal to the centripetal force required for a circular orbit: $GM_{enc}m/r^2 = mv^2/r$, solved for $v$. $M_{enc}(r)$ is the mass enclosed within radius $r$ — for a point mass or any spherically symmetric distribution, only the enclosed mass matters (Newton’s shell theorem). This is the formula behind every basic orbital-velocity calculation, from satellites to planets to the naive (pre-dark-matter-problem) prediction for how fast stars should orbit within a galaxy.

Real, independent confirmation: confirmed continuously by satellite orbits, planetary motion, and binary star systems — one of the most thoroughly tested equations in physics for two-body, weak-field systems.

Used in: Paper 1, Paper 2, and Paper 4, as the un-modified half of the galactic rotation-curve formula $v(r)=\sqrt{G_{eff}(r)M_{enc}(r)/r}$ — this equation itself is correct and unmodified; what’s PBT-specific is replacing plain $G$ with a scale-dependent $G_{eff}(r)$, which is a separate, unproven claim — see Rotation-Curve Effective Gravity.