SCC Canonical Spec — Part 1: Foundations & Formal Universe

Open source document →

1. #1

   The formal universe of the soft theory is a structured tuple

   $${\mathfrak{C}}^{\mathrm{soft}}=(T,\{X_t{\}}_{t\in T},\{u_t{\}}_{t\in T},\{{\mathrm{Cl}}_t{\}}_{t\in T},\{{\mathbf{N}}_t,{\mathbf{D}}_t{\}}_{t\in T},\{{\mathbf{M}}_{t\to s}{\}}_{t,s\in T})$$
2. #2

   For each , the function

   $$u_t:X_t\to [0,1]$$
3. #3

   For each , the operator

   $${\mathrm{Cl}}_t:[0,1{]}^{X_t}\to [0,1{]}^{X_t}$$
4. #4

   For each , the function

   $${\mathbf{N}}_t:X_t\times X_t\to [0,\infty )$$
5. #5

   For each , the operator

   $${\mathbf{D}}_t:X_t\times [0,1{]}^{X_t}\to [0,1]$$
6. #6

   For each pair , the function

   $${\mathbf{M}}_{t\to s}:X_t\times X_s\to [0,1]$$
7. #7

   The core of a cohesion field at time is the set of sites whose cohesive participation exceeds a high threshold:

   $${\mathrm{Core}}_t(u_t)=\{x\in X_t\mid u_t(x)\ge {\theta }_{\mathrm{core}}\}$$
8. #8

   The interior of a cohesion field is defined by a lower threshold:

   $${\mathrm{Int}}_t(u_t)=\{x\in X_t\mid u_t(x)\ge {\theta }_{\mathrm{in}}\}$$
9. #9

   The boundary of a cohesion field is not a sharp line but a transition region — a band of sites where cohesion is intermediate between full interior participation and effective exteriority:

   $${\mathrm{Bd}}_t(u_t)=\{x\in X_t\mid {\theta }_1<u_t(x)<{\theta }_2\}$$
10. #10

    The boundary band may also be characterized in gradient terms. A local gradient indicator may be defined as

    $$g_t(x;u)=\sum _y{\mathbf{N}}_t(x,y)|u_t(x)-u_t(y)|$$
11. #11

    The exterior is defined by complementarity:

    $${\mathrm{Ext}}_t(u_t)=\{x\in X_t\mid u_t(x)\le {\theta }_{\mathrm{ext}}\}$$