Derivative-Order Geometry of Holomorphic and Meromorphic Functions: Flow, Collapse, Spectral Periodicity, and Asymptotic Fingerprints

Let \(f\) be a nonpolynomial holomorphic or meromorphic function on a plane domain \(U\), and define the derivative-ladder ratios \({{q_{n}:=\frac{f^{({n + 1})}}{f^{(n)}}},{n \geq 0}}.\) We treat \({(q_{n})}_{n \geq 0}\) as a discrete geometry on the derivative-order axis. With the forward difference \({\Delta q_{n}}:={q_{n + 1} - q_{n}}\), the ladder satisfies the exact flow law \({q_{n}^{\prime} = {q_{n}\Delta q_{n}}},\) so order slope is the logarithmic derivative of the rung. The second difference \(K_{n}:={\Delta^{2}q_{n}}\) also satisfies \(K_{n} = {({\log R_{n}})}^{\prime}\), where \(R_{n}:={q_{n + 1}/q_{n}}\), and therefore acts as an order-curvature defect invariant for the departure from the rigid exponential and affine-power regimes. The paper develops three layers of theory. The first is exact structural calculus: inverse realization of compatible ladders, differential-polynomial identities for \(f^{({n + m})}/f^{(n)}\), and local transport laws showing how zeros and poles of successive derivatives propagate through the ladder and how resonance fields record multiplicities. The second is exact rigidity: vanishing curvature at one rung yields local exponential or affine-power structure, and if \({\Delta^{r}q_{n}} \equiv 0\) for some finite \(r\), then automatically \({\Delta^{2}q_{n}} \equiv 0\). In particular, higher polynomial geometries on the derivative-order axis do not exist: every finite-order ladder collapses to the exponential model or the affine-power model. The third is asymptotic rigidity: periodic ladders are classified by constant-coefficient equations \(f^{(p)} = {\kappa f}\), and for exponential-polynomial source models we prove an asymptotic order-dimension law \({{{q_{n}{(z)}} = {\lambda\left( {1 + \frac{d}{n} + {O{(n^{- 2})}}} \right)}},{{{\Delta q_{n}{(z)}} = {{- \frac{d\lambda}{n^{2}}} + {O{(n^{- 3})}}}},{{K_{n}{(z)}} = {\frac{2d\lambda}{n^{3}} + {O{(n^{- 4})}}}}}},\) locally uniformly on compact sets, where \(d\) is the degree of the dominant polynomial prefactor. Thus the ladder asymptotically recovers hidden spectral multiplicity from its own curvature defect. The fields \(q_{0} = {f^{\prime}/f}\) and \(q_{1} = {f^{\operatorname{\prime\prime}}/f^{\prime}}\) are classical, belonging respectively to the logarithmic-derivative and pre-Schwarzian traditions. The novelty here lies in treating the full infinite ladder as a coupled meromorphic object with exact flow, inverse realization, singular fingerprints, finite-order collapse, periodic spectral closure, and asymptotic fingerprint laws.