Holomorphic Wronskian for abstract rigidity inequalities

We introduce a holomorphic Wronskian framework for the asymptotic rigidity of solutions to non‑homogeneous linear complex differential equations. Starting from the exact identity obtained in \cite{Oukil}, we replace the standard squared‑modulus weight $|t^{-1}\widehat{\delta}_s(t)|^2$ by a general holomorphic observable $H(f(t))$, where $H'(z)=h(z)$ and $f(t)=t^{-1}\widehat{\delta}_s(t)$. The resulting Wronskian structure yields a family of limiting periodic profiles $F_h(s,\theta)$ parametrized by arbitrary holomorphic functions $h$. When $h(z)=\overline{z}$ we recover the original rigidity equality; other choices produce new oscillatory criteria for the non‑vanishing of $(\mu_\eta(s),\mu_\eta(1-\overline{s}))$. The method unifies and extends the power‑weight asymptotics developed previously.