Shade Projection Metric: A Geometric-Computational Metric for Comparing Ultra-Large Mathematical Objects

Transcomputational numbers are recognized in the literature as the theoretical boundaries of themathematical numerical universe. Notable examples include Graham’s number, TREE[n], and the Busy Beaversequence. Current analysis of these entities remains largely peripheral and superficial, as robust frameworks forunderstanding their limits and operational dynamics are still lacking. This study proposes a graphical approach toinvestigate the underlying complexity of these numbers, enabling their categorization based on internal behaviorand facilitating a structured hierarchical ordering. Drawing an analogy with optical shadows and wavesuperposition, this work introduces two and three-dimensional representations. This methodology represents anoriginal contribution to the field; the findings are presented for peer review and to serve as a foundation forfuture research.