Scale Quantization Proof Cheat Sheet: Why k Must Be Integer

This cheat sheet provides a rigorous algebraic proof for scale quantization and discrete scale invariance (DSI) within the Taiji Transformation framework ($g(x) = \phi^k$). By evaluating state transition ratio operators, it establishes that the scale index must be strictly constrained to the integer set ($k \in \mathbb{Z}$). It proves by contradiction that any continuous interpolation ($k \in \mathbb{R} \setminus \mathbb{Z}$) breaks the defining recursive identity of the Golden Ratio ($\phi^2 = \phi + 1$), exposing smooth continuum scale models (such as Wilsonian $e^{-pu}$) as analytical smoothing approximations. Consequently, physical scale space is formalized as a discrete point set ($\mathcal{S}_\phi = \{\phi^k \mid k \in \mathbb{Z}\}$), resolving pointwise continuum singularities into discrete, self-similar golden ladders.