= Solution
Use the <Fourier transform> convention $\widehat\varphi(\xi)=\int\varphi(x)e^{-ix\xi}\,dx$. Nesting and the <orthonormal basis> of $V_1$ give the <scaling refinement equation>
$$
\varphi(x)=\sqrt2\sum_kh_k\varphi(2x-k),\qquad
m(\xi)=\frac1{\sqrt2}\sum_kh_ke^{-ik\xi},\qquad
\widehat\varphi(2\xi)=m(\xi)\widehat\varphi(\xi).
$$
Here $h_k=\langle\varphi,\sqrt2\varphi(2\cdot-k)\rangle$. <Compact support> makes these <inner products> zero for all but finitely many $k$, so $m$ is a <trigonometric polynomial>. Expanding the orthogonality of the integer translates of $\varphi$ in the <orthonormal basis> of $V_1$ yields
$$
\sum_kh_k\overline{h_{k-2n}}=\delta_{n0}.
$$
Consequently
$$
|m(\xi)|^2+|m(\xi+\pi)|^2
=\sum_n\left(\sum_kh_k\overline{h_{k-2n}}\right)e^{-2in\xi}
=\boxed{1}.
$$
This holds everywhere because $m$ is continuous. The invoked properties are nesting, dyadic dilation, orthonormal integer translates, and <compact support>; mere finite-energy refinement would not imply the <quadrature mirror filter> identity.
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