SCC Canonical Spec — Part 1: Foundations & Formal Universe

Open source document →

1. #1

   The formal universe of the soft theory is a structured tuple

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

   $$\mathrm{Core}_t(u_t) = \{x \in X_t \mid u_t(x) \geq \theta_{\mathrm{core}}\}$$
8. #8

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

   $$\mathrm{Int}_t(u_t) = \{x \in X_t \mid u_t(x) \geq \theta_{\mathrm{in}}\}$$
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:

    $$\mathrm{Ext}_t(u_t) = \{x \in X_t \mid u_t(x) \leq \theta_{\mathrm{ext}}\}$$