= Solution
Let $\sigma=\operatorname{sgn}(x_S)$, using the real <sign function> on the <support of a vector> $x$. The <fixed-sign null space condition> in the PDF uses $S^c$ on its right-hand side. This complement is essential. For any real coordinate $x_j\ne0$, the supporting-line inequality for the <absolute value> is
$$
|x_j+v_j|\ge|x_j|+\operatorname{sgn}(x_j)v_j.
$$
Every distinct feasible <vector> is $x+v$ with $0\ne v\in\ker A$. Summing the coordinate inequalities on $S$ and adding the <L1 norm> on $S^c$ gives
$$
\|x+v\|_1-\|x\|_1\ge\langle\sigma,v_S\rangle+\|v_{S^c}\|_1\ge\|v_{S^c}\|_1-|\langle\sigma,v_S\rangle|>0.
$$
The strict final inequality is precisely the <fixed-sign null space condition>. \b[Hence $x$ is the unique <basis pursuit> <minimizer>.] Unlike the <null space property> of Question 1, this condition concerns the particular <sign function> values of $x$, rather than every <vector> supported in $S$.
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