Density Thresholds & Erdős Phase Transitions
At what edge probability $p_c = 1/|V|$ does the cycle rank $\beta_1$ transition from zero to non-zero, forcing field extensions beyond $\mathbb{R}$?
ERDŐS–RÉNYI GRAPH EVOLUTION PHASES
Phase 1: Subcritical ($p < 1/n$)
The graph is a forest of small trees ($\beta_1 = 0$). All contextual transition assignments are strictly $\mathbb{R}$-realizable.
Phase 2: Critical ($p = 1/n$)
The minimal forbidden subgraph $K_3$ (triangle) emerges with high probability ($\beta_1 = 1$). Complex field amplitudes $\mathbb{C}$ become necessary.
Phase 3: Supercritical ($p > 1/n$)
A giant component containing dense cycle networks ($\beta_1 \ge 4$) forms, forcing higher-dimensional field extensions ($\mathbb{H}, \mathbb{O}$).