Boundary-Driven Fractal Generation Mechanism: From Time Scale Mother Ruler to Feedback Scattering and Critical Interfaces

Within unified boundary--time--scattering framework, analyze widespread ``fractals grow on boundaries'' phenomenon in nature. Provide rigorous characterization from three complementary threads: (i) On total space Y=M\times X^\circ of manifold with boundary and parameter space, take time scale mother ruler \kappa(\omega)=\varphi'(\omega)/\pi=\rho_{rel}(\omega)=(2\pi)^{-1}trQ(\omega), relative topological class [K]\in H^2(Y,\partial Y;Z_2), scattering family K^1 class [u]\in K^1(X^\circ) as fundam