Rectifiability of every finite order for stationary integral varifolds

We give an alternative proof of a theorem of Brena, De Lellis and Franceschini (arXiv:2503.00649): the support of a stationary integral $d$ dimensional varifold in an open subset of ${\mathbf{R}}^n$ is $({\mathcal{H}}^d,d)$ rectifiable of class $(k,\alpha )$ for every positive integer $k$ and every $0<\alpha <1$, and in particular of class ${\mathcal{C}}^{\mathrm{\infty }}$. Our proof is independent of theirs and proceeds by different means. It compares the varifold, at every scale and at ${\mathcal{H}}^d$ almost every point, with the graph of a solution of the Euler–Lagrange system of a smoothed area integrand, and thereby derives an excess-decay estimate that is faster than any power of the scale; the conclusion then follows from Santilli’s characterisation of higher order rectifiability together with Whitney’s extension theorem.