Comparative Analysis of Emergence Geometries: From Heuristic Step Functions to Analytical Sigmoid Manifolds

Abstract: This supplementary paper presents a comparative analysis of emergence dynamics, contrasting standard heuristic-driven cognitive models with the analytically calibrated ESCT v9.4 framework. We investigate the topological shift in resonance shapes from discrete, jagged discontinuities to harmonic sigmoid manifolds. By mapping the transition operator to a frequency-domain selection filter G(X) = \mathcal{F}^{-1}[S(\Theta) \cdot \hat{F}] and applying first-principles derivation to secure the baseline anchor (s_0 \approx 0.099) and transition gain (\beta \approx 3.42), the v9.4 framework successfully eliminates arbitrary heuristics. Our comparative results demonstrate that the analytical approach effectively suppresses high-frequency noise and stabilizes physical boundaries within S \in [0.38, 0.94]. The framework achieves a strictly validated 0.3% error margin, confirming the structural superiority of continuous phase transitions in modeling artificial intelligence emergence.