2 The \(z^*\) formula
The formula \(z_k^* = p_{k+1}^{-1}\bigl((s_k p_k + \sum (-1)^{k-j} s_j\, r_{j,k+1})\, i - \sum q_{k+1,l}\, x_l\bigr)\).
The imaginary part of \(z_k^*\) is \(\mathrm{Im}(z_k^*) = \bigl(\mathrm{Re}(s_k)\, p_k + \sum (-1)^{k-j}\mathrm{Re}(s_j)\, r_{j,k+1} - \sum \mathrm{Im}(x_l)\, q_{k+1,l}\bigr) / p_{k+1}\).
Direct computation from the definition.
For all \(s\) with \(\mathrm{Re}(s) {\gt} 0\) and all real \(x\), \(\mathrm{Im}(z_k^*) {\gt} 0\) if and only if \(J\) is stable at step \(k\).
Forward: stability sign conditions make the numerator positive. Backward: taking \(s_j \to +\infty \) extracts the sign of each \(r\)-minor.
\(J\) is stable if and only if \(\mathrm{Im}(z_k^*) {\gt} 0\) at every step \(k\), for all \(s\) with \(\mathrm{Re}(s) {\gt} 0\) and all real \(x\).
Combines step-wise decomposition with the step-wise characterization, plus an induction showing \(p_{k+1} {\gt} 0\) from \(\mathrm{Im}(z_k^*) {\gt} 0\).