Analytic Dynamics of the Chen Operator: A Rigorous Formal Proof of Transcendence for $\zeta(3)$, $\gamma$, and $G$

Abstract: This work provides a pathologically rigorous formalization of "Chen's Path," a discrete dynamical system designed for the topological classification of mathematical constants. By deriving the second-order quadratic eruption law through a quantized Taylor-geometric transformation, we establish a fundamental criterion for algebraic stability. Key Breakthroughs: Formal Proof of Transcendence: We demonstrate that for $\zeta(3)$, $\gamma$, and $G$, the persistent analytic drift $\eta$ induces a logarithmic violation of the Roth-Schmidt bound. The Chen Operator: Introduction of a functional mapping that translates holomorphic curvature into deterministic discrete trajectories. Rate Asymmetry: A novel proof strategy showing that analytic curvature-driven drift dominates the polynomial constraints of algebraic fields, necessitating the transcendence of these constants. This paper bridges analytic number theory and discrete dynamics, offering a deterministic tool for verifying the algebraic complexity of fundamental invariants.