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

We present a graded extension of the complex numbers, integrating both infinitary and infinitesimal scales in a unified, hierarchical framework, which we refer to as the \emph{Omega number system}. Our construction is anchored by a fundamental scaling element, \(\Omega\), defined as the hyperreal corresponding to the equivalence class of the standard sequence \((1,2,3,\dots)\) via the ultrapower construction. Central to our approach is the lifting function \(L^\Omega(n)\), which elevates \(\Omega\) into a graded hierarchy so that the index \(n=0\) corresponds to the classical (finite) complex domain, \(n>0\) to increasingly large (infinitary) magnitudes, and \(n<0\) to infinitesimal values. From \(\Omega\) and \(L^\Omega(n)\) we derive key foundational objects, including the \emph{absolute zero} \(\underline{0}\), the graded continuum of \emph{almost zero} \(\overline{0}\), and its distinguished member, the \emph{canonical zero}, \(0^*=\Omega^{-1}\); which together with the identity \(1\), extend classical arithmetic coherently. We illustrate our approach through two foundational models: a linear (polynomial) model that extends familiar arithmetic via a straightforward graded extension and a non-linear (exponential) model incorporating exponential transfinite index growth. Moreover, by analysing selected elementary functions, we present concrete examples that include multivalued and probabilistic interpretations. We also introduce an enhanced non-linear system featuring multiplicative product-lifting, in which the generator function is defined through the additive components of Riemann’s zeta function, \(\zeta(s)\). Using this connection, we obtain the result that the negative and positive real \(\Omega\) limits of \(\zeta(s)\) are precisely characterized by the complete sequence of infinitesimal and infinitary canonical base elements, respectively. We further define a \emph{rank-lifting} of model families by introducing a rank-ordering of the fundamental scaling elements, \(^{r}\Omega\), for each tetration rank of \(\Omega\). For each rank \(r\), we associate a corresponding family of models, thereby extending \(\mathbb{O}\) to \({}^{r}\mathbb{O}\). We then analyze the special graded functions in \(\mathbb{O}\), including the \emph{reflection} operator \(R\!\bigl(\sum a_nL^\Omega(n)\bigr):=\sum a_nL^\Omega(-n)\), with selected examples showing deep and unexpected symmetries between infinitesimals and infinitaries. Building on these dualities, we introduce an $\Omega$–operator toolkit on the graded basis (grade-shifts, reflection, parity, phase twists). In the formal setting this yields discrete ladder

symmetries and a valuation-driven calculus suited for regularisation; in a hyperreal realisation it separates infinitesimal from infinitary contributions. We illustrate the approach on elementary functions and on the Dirichlet series for $\zeta(s)$; in the latter case we give a conditional $\Omega$–reflection principle mirroring the classical functional equation across grades. The present paper establishes the formal graded framework; analytic and operator-algebraic refinements are left for future work.