Holomorphic-conjugate Wronskian for abstract rigidity inequalities

We introduce a holomorphic-conjugate Wronskian framework for the asymptotic rigidity of solutions to non‑homogeneous linear complex differential equations. Starting from the identity obtained in \cite{Oukil}, which relies on the non‑holomorphic weight $\overline{z}$ (complex conjugation), we develop a parallel theory using a general holomorphic observable $H(f(t))$, where $H'(z)=h(z)$ and $f(t)=t^{-1}\widehat{\delta}_s(t)$. The original conjugate‑based identity and the new holomorphic one are structurally distinct; the latter yields a family of limiting periodic profiles $F_h(s,\theta)$ parametrized by arbitrary holomorphic functions $h$. This ``holomorphic-conjugate'' viewpoint unifies the classical and the new oscillatory criteria for the non‑vanishing of $(\mu_\eta(s),\mu_\eta(1-\overline{s}))$, and extends the power‑weight asymptotics developed previously.