The Zeta Axis II: A Geometric Framework for Complex Analysis via Formal Puiseux Series and Exact Certification

This paper presents a comprehensive mathematical framework that reimagines classical complex analysis through the integration of a novel number system and its precise geometric visualization. By replacing the classical zero with an invertible infinitesimal unit $z$, we establish a robust algebraic foundation utilizing the field of formal Puiseux series, $\mathbb{C}\{\{z\}\}$. Within this system, classical indeterminate forms acquire specific values, and poles are systematically transformed into elements of negative order. Geometrically, we introduce a rigorous visualization convention utilizing a third dimension—the "Zeta axis". Rather than arbitrary plotting, this axis represents a differential invariant selected from a structured ladder indexed by the five connected algebraic subgroups of $PGL_2(\mathbb{C})$. The resulting surfaces are harmonic graphs whose geometric properties are deeply governed by these invariants. The framework utilizes Gaussian and mean curvature measures to analyze the surface's topology, revealing exactly three exceptional sets—folds, saddles, and flat points—which directly correlate to the poles, zeros, and critical points of a single rational map. Finally, we detail a computational methodology that prioritizes exact numerical certification over standard sampling or approximation algorithms. By employing a unified local-to-global topological identity across eight distinct mathematical groups, critical data such as divisors, Puiseux exponents, and tangent-cone monodromy are rigorously extracted using exact orders and winding numbers, entirely avoiding the use of resultants.