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) = φᵏ). By evaluating state transition ratio operators, it establishes that the scale index must be strictly constrained to the integer set (k ∈ Z). It proves by contradiction that any continuous interpolation (k ∈ R Z) breaks the defining recursive identity of the Golden Ratio (φ² = φ + 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 (S_φ = \φᵏ k ∈ Z\), resolving pointwise continuum singularities into discrete, self-similar golden ladders.