Projection-Dependent Geometry of Hierarchical Emergence in Fractal Resonance Systems

We demonstrate that hierarchical emergence can be treated as a geometric object whose statistical expression deforms smoothly under phase-space projection. Using phase rotation of the analytic signal as a controlled projection operator, we show that tail exponents and scale-conditional entropy vary continuously with angle while curvature persistence remains invariant. A sequence of surrogate controls—permutation, block shuffling, time reversal, and Fourier phase randomization—reveals that smooth deformation under projection is lost when nonlinear temporal structure is destroyed. These results establish that hierarchical emergence is not a spectral or distributional artifact, but a nonlinear, multi-scale geometric structure.