Scale-Invariance & Dual-Mapping Cheat Sheet: What is "1"?

This cheat sheet presents a concise, unified mathematical framework bridging scale-invariance, dual-scale mapping, and asymptotic field behavior. It formally establishes the algebraic duality between the "Taiji Transformation" ($\phi^k$ golden-ratio scale ladder) and the "Academic Standard" ($e^{-pu}$ logarithmic renormalization scale parameter) for general inverse-power decay fields ($g(x) = 1/x^p$). By mapping microscopic spatial limits ($x \to 0^+$) to infinite dual scale indices ($k \to +\infty$), the formulation provides a smooth geometric resolution to coordinate singularity divergence while preserving macroscopic asymptotic flatness.