Chapter 14 — Time Evolution

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1. #1

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

   $$\exists F\in {\mathfrak{F}}_{t^{\ast }-ϵ}\setminus {\mathfrak{F}}_{t^{\ast }+ϵ}\text{or}\exists F\in {\mathfrak{F}}_{t^{\ast }+ϵ}\setminus {\mathfrak{F}}_{t^{\ast }-ϵ}.$$
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:

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