= Solution
At $\alpha=0$, bounded Hessians have no uniform modulus of continuity, so the <Arzela-Ascoli theorem> does not supply the compact $C^2$ limit used in the contradiction. At $\alpha=1$, the blow-up has cubic growth, and the <Liouville theorem> no longer forces its Hessian to be constant: nonzero harmonic cubic polynomials are possible. These are the two endpoint failures behind the restriction $0<\alpha<1$; ordinary Schauder theory replaces neither endpoint by a general $C^{2,\alpha}$ estimate of the displayed form.
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