Uniform Interior $C^{1,\alpha}$ Estimates for Infinity-Harmonic Functions

We prove a uniform interior $C^{1,\alpha_d}$ estimate for bounded infinity-harmonic functions in every dimension $d\ge3$. For some $\alpha_d\in(0,1/3]$ depending only on the dimension, the gradient's supremum norm and $\alpha_d$-Hölder seminorm on $B_{1/2}$ are bounded by a dimension-dependent constant times the oscillation on $B_1$. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally $C^1$ on every open subset of $\mathbb{R}^d$, for $d\ge3$.