Omega Numbers: Symmetries and Infinitary Regularization in a Graded Transarchimedean Extension of Complex Analysis

Starting from the standard Hahn/Laurent-series construction over \ (C\) with value group \ (Z\), we introduce the Omega number system, a graded transarchimedean interpretation and extension of complex arithmetic designed to represent finite, infinitesimal, and infinitary quantities within a common algebraic setting. The construction is organized around a distinguished infinitary scale \ (\), which may be realized concretely as the hyperreal equivalence class of the sequence \ ( (1, 2, 3, ) \) under an ultrapower construction. A lifting function \ (L^ (n) \), indexed by \ (n\), separates the classical complex component at grade \ (0\), infinitary magnitudes at positive grades, and infinitesimal magnitudes at negative grades. Within this graded setting we distinguish the absolute zero \ (0\), the infinitesimal almost-zero class \ (0\), and its canonical member \ (0^*=^-1\). Together, \ (00\) refines the classical zero of real and complex analysis into a continuum-cardinality graded near-zero class. We develop additive linear and exponential model families, introduce a multiplicative product-lifting model motivated by Dirichlet-type expansions, establish uniqueness of graded representation and basic valuation laws, and describe rank-lifted model families generated by higher-order infinitary base elements \ (^r\). These constructions provide a formal setting in which infinitesimal and infinitary contributions can be compared, reflected, shifted, and separated by grade. We then study selected elementary functions and zeta-type expressions in the Omega domain. Periodic functions at infinitary arguments admit natural ultralimit or distributional interpretations, while divergent or asymptotic expressions may be decomposed into finite, infinitesimal, and infinitary graded components. We also introduce an \ (\) -operator toolkit on the graded basis, including grade shifts, reflection, parity, and phase twists. In particular, the reflection operator exchanges infinitesimal and infinitary grades and leads to conditional \ (\) -reflection principles for zeta-type expressions paralleling the classical functional equation across grades. The present paper develops the formal graded construction and illustrative examples; fuller analytic, operator-algebraic, and application-oriented extensions are left for future work.