Pseudo-Exponential Structures and Exponential Algebraic Closedness in the Surreal Numbers

This work proposes a pseudo-exponential model within the domain of surreal numbers (No), asystematic methodology for solving equations of the form P(x,e^x) = 0 in No, and a formal proof that theexponential-algebraic closedness (EC) axiom is structurally stronger and more comprehensive in the surrealsthan in the real and complex domains. Furthermore, an ECNo axiomatic scheme is proposed, tailored to Hahngeometry, incorporating axioms of surreal rotundity, support freeness, and the transfinite existence of solutions.