arXiv:2611.14429v1 Announce Type: new Abstract: The median $\nu_\Gamma(\alpha)$ of the gamma distribution has no closed form, but its behavior at both ends is known: it is asymptotic to $2^{-1/\alpha}e^{-\gamma}$ (with $\gamma$ being the Euler-Mascheroni constant) as the shape parameter $\alpha \to 0$, and approaches $\alpha - 1/3$ as $\alpha \to \infty$. We write approximations as interpolations $(1 - g)\nu_0 + g\nu_\infty$ between two elementary-function bounds or approximations, each exact in one limit, with an interpolator $g(\alpha)$ going from 0 to 1, and compare three such forms: a first one between previously proven bounds, a second ending in an approximation from the beta median that is exact at $\alpha = 1$ but does not quite approach $\alpha - 1/3$ as $\alpha \to \infty$, and a third one, between bounds proved here, that corrects the large-$\alpha$ end of the second. For each form, the ideal interpolator's slope at 0, approach to 1, and value at $\alpha = 1$ are derived in closed form. Rational interpolators with every coefficient set in closed form from these properties, with no numerical fitting, reach a worst relative error of 0.0011 over all $\alpha > 0$, with absolute error below 0.0002. Fitting some of these or other coefficients numerically to minimize the worst absolute-or-relative error then trades accuracy against complexity: the Pareto frontier reaches 0.000087 with two fitted constants and 0.000041 with five, and from two on uses degree-4 rationals of the second form with no $\alpha^2$ term in the numerator. The proofs that the end functions of the first and third forms are bounds are machine-checked in Lean 4.