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" (φᵏ golden-ratio scale ladder) and the "Academic Standard" (e⁻ᵖᵘ logarithmic renormalization scale parameter) for general inverse-power decay fields (g(x) = 1/xᵖ). By mapping microscopic spatial limits (x → 0⁺) to infinite dual scale indices (k → +∞), the formulation provides a smooth geometric resolution to coordinate singularity divergence while preserving macroscopic asymptotic flatness.
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Yu et al. (2026) studied this question.
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