Chapter 0 — Formal State Space

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1. #1

   The collection of all admissible topologies is denoted:

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

   For fixed graph topology with and :

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

   The gauge group acts smoothly on by:

   $$\boxed{(h_1,\dots ,h_n)\cdot (w_{ij},g_{ij}):=\left(w_{ij},h_i{g}_{ij}{h}_j^{-1}\right)}$$
4. #4

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

   $$\boxed{\mathcal{S}:=\underset{(n,E)\in \mathrm{Top}}{⨆}C(n,E)}$$
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:

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

   The observable map is: where:

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

    $$\mathrm{BIRTH}(v):C(n,E)\to C(n+1,E^{\prime })$$
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:

    $$\tau (s_t)=(n_t,E_t)\stackrel{e_t}{\to }(n_{t+1},E_{t+1})=\tau (s_{t+1})$$
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:

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