Chapter 14 — Time Evolution

Open source document →

1. #1

   A topological phase transition occurs at time if the combinatorial topology of the fruit structure changes discontinuously:

   $$\exists\,F\in\mathfrak{F}_{t^*-\epsilon}\setminus\mathfrak{F}_{t^*+\epsilon}\quad\text{or}\quad\exists\,F\in\mathfrak{F}_{t^*+\epsilon}\setminus\mathfrak{F}_{t^*-\epsilon}.$$
2. #2

   $$\mathrm{Lifespan}(F):=\sup\{T:F\text{ can be continuously tracked over }[t_0,t_0+T]\}.$$
3. #3

   The exact sequence (Theorem 12.8) constrains how Betti numbers can change:

   $$\Delta\beta_k(\text{Axis 2})-\Delta\beta_k(\text{Axis 1})+\Delta\beta_k(\text{Axis 3})=\Delta(\text{connecting map rank}).$$
4. #4

   (threshold crossing): The system monitors for each fruit . When:

   $$\phi_t(F) \text{ crosses } \theta \text{ from below} \Rightarrow \text{SPLIT event (fruit born)}$$