Chapter 14 — Time Evolution

function TrackWorldEvolution(W_t, g_t, t_range, theta, tau):
    t = t_range[0]
    fruits = FindFruits(W_t, theta)
    doors = {F: FindDoors(F, W_t, tau) for F in fruits}
    axes = {F: ComputeWorldAxes(F, doors[F], ...) for F in fruits}
 
    events = []
 
    for t_next in t_range[1:]:
        // Check for phase transitions
        fruits_new = FindFruits(W_{t_next}, theta)
        doors_new = {F: FindDoors(F, W_{t_next}, tau) for F in fruits_new}
 
        // Detect changes
        born = fruits_new - fruits
        died = fruits - fruits_new
        door_changes = DetectDoorChanges(doors, doors_new)
 
        if born or died or door_changes:
            events.append((t_next, born, died, door_changes))
            axes = Recompute(fruits_new, doors_new)
 
        fruits, doors = fruits_new, doors_new
        t = t_next
 
    return events, axes
```plaintext
---
 
## 14.6 Summary
 
The time evolution of the world is governed by:
 
1. **Continuous variation** of the relational field $\mathcal{W}_t$.
2. **Piecewise-constant topology**: discrete invariants (fruits, doors, Betti numbers) change only at isolated phase-transition times.
3. **Yang--Mills flow** (Chapter 13) drives the gauge connection toward a stable equilibrium within each topological phase.
4. **Phase transitions** create, destroy, split, or merge fruits, and create or annihilate doors.
 
The overall picture: **smooth dynamics punctuated by discrete topological events**.
 
---
 
## 14.7 Hybrid Dynamical Systems Formulation
 
### Motivation
 
The description above ("smooth + discrete") is intuitive but informal. To make it rigorous, we employ the **hybrid dynamical systems** framework (see Appendix D for background). A hybrid system naturally captures:
- **Continuous state** within each topological phase.
- **Discrete transitions** between topologies.
- **Guards and resets** specifying when and how transitions occur.
 
### Definition 14.5 (World evolution as hybrid system)
 
The **world evolution** is a hybrid dynamical system $(Q, X, f, G_e, R_e)_{e \in E_{\mathrm{events}}}$ defined as:
 
| Component | Definition |
|-----------|-----------|
| **Discrete state** $Q$ | $Q := \{(n, E) : \text{admissible topologies}\}$ (Definition 0.1 of Ch. 0) |
| **Continuous state fiber** $X(q)$ | For $q = (n, E)$: $X(q) := P(n,E) = \mathbb{R}_{\ge 0}^{|E|} \times G^{|E|}$ (raw parameter space) |
| **Continuous dynamics** $\dot{x} = f_q(x)$ | Yang--Mills gradient flow + exogenous DEFORM forces: $\dot{h} = -\nabla_h \mathcal{E}_{F^\circ}(h)$ within each topological phase |
| **Guards** $G_e$ | Threshold conditions triggering events: $G_{\mathrm{SPLIT}} := \{x : \exists F, \phi(F) = \theta\}$, etc. |
| **Resets** $R_e$ | After event $e$, transition $q \to q'$ and initialize new parameters |
 
### Discrete Topology Transitions
 
The six events from Definition 0.7 (Chapter 0) induce transitions:
 
| Event | Transition | Trigger |
|-------|-----------|---------|
| **DEFORM** | $(n,E) \to (n,E)$ | Continuous flow; operator-commanded |
| **CONTACT** | $(n,E) \to (n,E \cup \{(i,j),(j,i)\})$ | Commanded; new edge initialized |
| **BIRTH** | $(n,E) \to (n+1, E')$ | Commanded; new node + initial edges |
| **DEATH** | $(n,E) \to (n-1, E\setminus E_v)$ | Commanded; incident edges removed |
| **SPLIT/MERGE** | $(n,E) \to (n,E)$ | Guard: $\phi(F)=\theta$ crossing; detected |
 
### Guard Conditions (Formally)
 
For each event type:
 
**$G_{\mathrm{DEFORM}}$** (continuous): Always active within a topology. The continuous flow $\dot{x} = f_q(x)$ evolves parameters.
 
**$G_{\mathrm{CONTACT}}$** (discrete): Triggered by external command. When activated, the system resets to $(n, E \cup \{(i,j),(j,i)\})$ with new edges initialized.
 
**$G_{\mathrm{BIRTH}}$, $G_{\mathrm{DEATH}}$** (discrete): Triggered by external command.
 
**$G_{\mathrm{SPLIT/MERGE}}$** (threshold crossing): The system monitors $\phi_t(F)$ for each fruit $F$. When:
$$
\phi_t(F) \text{ crosses } \theta \text{ from below} \Rightarrow \text{SPLIT event (fruit born)}
$$
$$
\phi_t(F) \text{ crosses } \theta \text{ from above} \Rightarrow \text{MERGE event (fruit dies)}
$$
The topology index $(n, E)$ remains unchanged, but $\mathfrak{F}_t$ is recomputed.
 
### Reset Maps (Formally)
 
After event $e$ transitions the discrete state from $q$ to $q'$:
 
**Reset $R_{\mathrm{DEFORM}}$** (identity): parameters continue; no reinitialization.
 
**Reset $R_{\mathrm{CONTACT}}$** (partial):
- Existing edges $(i,j) \in E$ retain their parameters.
- New edges: $w_{ij} := \varepsilon > 0$ (small), $g_{ij} := e$ (identity).
 
**Reset $R_{\mathrm{BIRTH}}$**, $R_{\mathrm{DEATH}}$** (structural):
- BIRTH: new node $v$ is added; edges to/from $v$ are initialized.
- DEATH: node $v$ is removed; all its incident edges are deleted.
 
**Reset $R_{\mathrm{SPLIT/MERGE}}$** (none): The discrete topology does not change; only the fruit set $\mathfrak{F}_t$ is recomputed.
 
### Theorem 14.5 (Piecewise-Constant Topology Revisited)
 
Within each hybrid stratum (fixed discrete state $(n, E)$), the world evolves continuously under the Yang--Mills flow and any applied DEFORM forces. The discrete state $(n, E)$ remains constant between guard-crossings. When a guard is crossed (e.g., $\phi(F) = \theta$ or a discrete event is commanded), a transition occurs and the world jumps to a new stratum (or remains in the same stratum with recomputed fruit/door structure).
 
---
 
### Example: SPLIT Event in Detail
 
**Setup:** A single fruit $F$ exists at time $t$ with $\phi_t(F) = \theta - \epsilon$ (just above the threshold, about to vanish).
 
**Evolution:**
1. Continuous DEFORM forces reduce inter-community weights, slowly increasing $\phi_t(F)$.
2. When $\phi_t(F)$ crosses $\theta$ from below, guard $G_{\mathrm{SPLIT}}$ triggers.
3. The fruit set $\mathfrak{F}_t$ is recomputed: now $F \notin \mathfrak{F}_t$, and possibly two new fruits $F_1, F_2$ appear.
4. The discrete topology $(n,E)$ is unchanged, but the derived structure (fruits, doors, Existence triples) changes.
5. The evidence log records the SPLIT event with $\Delta \mathfrak{F} = \{F\} \to \{F_1, F_2\}$.
 
---
 
### Continuous--Discrete Coupling
 
The hybrid system captures the essential interplay:
- **Continuous layer** (Yang--Mills flow, weight evolution) is governed by the relational field $\mathcal{W}_t$ within fixed topology.
- **Discrete layer** (topology changes, fruit structure changes) is governed by threshold-crossings and external commands.
- **Coupling** occurs through the observables $\phi(F)$, $b_{F}(i)$, which depend continuously on the parameters but are monitored for discrete transitions.
 
This is formalized rigorously in Appendix D using the theory of hybrid automata.