Chapter 0 — Formal State Space

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1. #1

   The collection of all admissible topologies is denoted:

   $$\mathrm{Top} := \{(n, E) : n \in \mathbb{N}_{>0}, \, E \subseteq \{1,\ldots,n\}^2 \setminus \mathrm{diag}\}.$$
2. #2

   For fixed graph topology with and :

   $$\boxed{P(n,E) := \mathbb{R}_{\geq 0}^{|E|} \times G^{|E|}}$$
3. #3

   The gauge group acts smoothly on by:

   $$\boxed{(h_1, \ldots, h_n) \cdot (w_{ij}, g_{ij}) := \left(w_{ij}, \, h_i \, g_{ij} \, h_j^{-1}\right)}$$
4. #4

   $$\boxed{C(n,E) := P(n,E) / G^n}$$
5. #5

   $$\boxed{\mathcal{S} := \bigsqcup_{(n,E) \in \mathrm{Top}} C(n,E)}$$
6. #6

   Discrete topology index: For any state , there exists a unique topology such that . Define:

   $$\tau(s) := (n_s, E_s).$$
7. #7

   The observable map is:

   $$\boxed{\mathrm{Obs} : \mathcal{S} \to \mathrm{Obs-Space}}$$
8. #8

   The observable map is: where:

   $$\mathrm{Obs}([p]) := \left( W_{ij}, \, \Omega(\triangle) \right)_{(i,j) \in E, \, \triangle \in T(E)}$$
9. #9

   $$\mathrm{DEFORM}(i,j,\delta) : C(n,E) \to C(n,E)$$
10. #10

    $$\mathrm{CONTACT}(i,j) : C(n, E) \to C(n, E \cup \{(i,j), (j,i)\})$$
11. #11

    $$\mathrm{BIRTH}(v) : C(n, E) \to C(n+1, E')$$
12. #12

    $$\mathrm{DEATH}(v) : C(n, E) \to C(n-1, E \setminus E_v)$$
13. #13

    The events BIRTH, DEATH, and CONTACT induce transitions between strata:

    $$\tau(s_t) = (n_t, E_t) \xrightarrow{e_t} (n_{t+1}, E_{t+1}) = \tau(s_{t+1})$$
14. #14

    Recall from Chapter 7:

    $$\mathrm{Existence}(F, t) := ([A_{\infty}], \Sigma_t(F), \mathbf{e}_t(F))$$
15. #15

    Given a state transition from to triggered by event , the evidence extraction functional is:

    $$\boxed{\mathrm{Ev}(s_t, s_{t+1}, e_t) := (e_t, \Delta W, \Delta \Omega, \Delta \mathfrak{F}, \Delta \Sigma)}$$
16. #16

    Chapter 8 defines the instantaneous world:

    $$\mathfrak{W}_t = (\mathcal{W}_t, \mathfrak{F}_t, \Sigma_t)$$