The Derivative of Velocity Addition with Proper Time and What We Actually Measure in the Universe

This paper derives the time derivative of Einstein's velocity-addition formula with explicit distinction between coordinate time $t$ and proper time $t'$. A previous version of this paper used a common coordinate time $t$ for both the local velocity and the system velocity. That version is superseded here: it is a simplified model, not the standard special-relativistic transformation. The corrected formula is $$a^* = \frac{a_v'}{\gamma_u^3\left(1+vu/c^2\right)^3} + a_u\,\frac{1-v^2/c^2}{\left(1+vu/c^2\right)^2},$$ where $a_v'=dv/dt'$ is the local acceleration measured with proper time, $a_u=du/dt$ is the acceleration of the system measured with coordinate time, and $a^*=dv^*/dt$ is the acceleration measured from Earth. At low velocities the correction is negligible; at high velocities ( $v,u\sim 0.5$-- $0.9\,c$) the measured acceleration is 40--99% smaller than the Galilean sum. This affects the interpretation of the Hubble constant, the estimated age of the Universe, and the dark-energy explanation of accelerated expansion. The paper does not change any physical constants, does not introduce tired light, it only differentiates Einstein's 1905 formula with proper time taken into account.