= Solution
Put $h=S_0-\mu/R$. The initial capital of the <one-period quadratic hedge> is
$$
X_0^*=\phi^*+(\pi^*)^TS_0
=\frac{m}{R}+h^TV^{-1}c.
$$
Define the <minimum-norm one-period pricing weight>
$$
\boxed{\rho^*=\frac1R+h^TV^{-1}(S_1-\mu).}
$$
It is square integrable, and $\mathbb E[(S_1-\mu)\xi_1]=c$, so
$$
\mathbb E[\rho^*\xi_1]=\frac{m}{R}+h^TV^{-1}c=X_0^*.
$$
The weight depends only on the asset prices and their first two moments, not on the claim. It can be negative, so this identity alone does not supply an <equivalent martingale measure> or a positive <state-price density>.
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