Taylor normalization in elliptic blow-up arguments ($v(z)=\frac{u(x+rz)-T_x^{2}u(x+rz)}{A{r}^{2+\mu }}$)

Subtracting the quadratic Taylor polynomial at the rescaling centre makes the function, gradient and Hessian matrix vanish there. Dividing by $A{r}^{2+\mu }$ preserves the normalized Hölder oscillation of the Hessian matrix while turning a small forcing-to-oscillation ratio into a harmonic limiting equation. The three vanished Taylor coefficients provide local bounds needed by the Arzela-Ascoli theorem.