A Hierarchical–Fractal Operational Framework for Surreal Numbers

We develop an operational framework for the surreal numbers based on three interacting structures: a hierarchical organization of dominant scales, an algebraic calculus defined via translations, and a generalized positional representation for surreal numbers.

The hierarchical construction provides an explicit multiscale decomposition compatible with the Conway normal form, while the algebraic calculus defines differentiation and integration without limits, recovering classical infinitesimal behavior through truncation across levels. In parallel, the positional representation introduces a resolution principle that ensures existence, completeness, and uniqueness for transfinite bases.

Together, these components establish a coherent structure in which discrete, continuous, infinitesimal, and transfinite phenomena can be interpreted within a single system, extending key features of real analysis to the surreal setting.