SCC Canonical Spec — Part 2: Axiomatic Groups & Proto-Cohesion

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1. #1

   A1'. Conditional Extensivity (Self-Regulation). For all and all ,

   $$\mathrm{Cl}_t(u)(x) \geq u(x) \quad \text{whenever } u(x) \leq c^* \text{ and } (P_t u)(x) \geq u(x)$$
2. #2

   A2. Monotonicity. For all ,

   $$u \leq v \implies \mathrm{Cl}_t(u) \leq \mathrm{Cl}_t(v)$$
3. #3

   A3. Stabilization Tendency (Contraction). Iterated application of satisfies the Cauchy condition:

   $$\|\mathrm{Cl}_t^{(n+1)}(u) - \mathrm{Cl}_t^{(n)}(u)\| \to 0 \quad \text{as } n \to \infty$$
4. #4

   B1. Nonnegativity. For all ,

   $$\mathbf{N}_t(x,y) \geq 0.$$
5. #5

   B2. Symmetry. In the minimal (undirected) case,

   $$\mathbf{N}_t(x,y) = \mathbf{N}_t(y,x).$$
6. #6

   C4. Symmetry. For all ,

   $$\mathbf{C}_t(x,y) = \mathbf{C}_t(y,x).$$
7. #7

   Provisional Realization. The resolvent form

   $$\mathbf{C}_t = (I - \alpha\, W_{\mathrm{sym}})^{-1}$$
8. #8

   E1. Sub-Stochasticity. For each ,

   $$\sum_{y \in X_s} \mathbf{M}_{t \to s}(x,y) \leq 1.$$
9. #9

   E3. Core Inheritance (Solution Constraint). The transport kernel should preferentially preserve the cohesive core. That is, for sites in , the inherited cohesion at the receiving sites should remain high:

   $$\sum_{y \in X_s} \mathbf{M}_{t \to s}(x,y)\, u_s(y) \geq \delta \quad \text{for some } \delta > 0$$
10. #10

    Binding. The predicate assesses how closely the cohesion field at time approximates self-support under closure:

    $$\mathsf{Bind}_t(u_t) = 1 - \frac{\|u_t - \mathrm{Cl}_t(u_t)\|_2}{\sqrt{n}}$$
11. #11

    Separation. The predicate assesses the degree to which the cohesion field is structurally distinguished from its exterior, weighted by cohesion:

    $$\mathsf{Sep}_t(u_t) = \frac{\sum_{x \in X_t} u_t(x)\, \mathbf{D}_t(x;\, 1-u_t)}{\sum_{x \in X_t} u_t(x)}$$
12. #12

    Inside-Structure. The predicate assesses the morphological articulation of the cohesion field:

    $$\mathsf{Inside}_t(u_t) = \mathcal{Q}_{\mathrm{morph}}(u_t)$$
13. #13

    where the morphological quality measure is defined as

    $$\mathcal{Q}_{\mathrm{morph}}(u_t) = \frac{\ell_{\max}(u_t) - c}{1 - c} \cdot \mathrm{Artic}(u_t)$$
14. #14

    Persistence. The predicate assesses the degree to which the cohesive organization is structurally inherited across time throughout the window :

    $$\mathsf{Persist}_W(\mathbf{u}) = \min_{t < s \in W} \frac{\sum_{x \in \mathrm{Core}_t} \sum_{y \in \mathrm{Core}_s} \mathbf{M}_{t \to s}(x,y)\, u_s(y)}{\rho_{\mathrm{persist}}}$$
15. #15

    Proto-cohesion is reported as a diagnostic vector :

    $$\mathbf{d} = \Big(\mathsf{Bind}_t,\ \mathsf{Sep}_t,\ \mathsf{Inside}_t,\ \mathsf{Persist}_W\Big)$$
16. #16

    Boolean recovery. The Boolean proto-cohesion predicate is recoverable by thresholding each component:

    $$\mathsf{ProtoCoh}^{\mathrm{soft}}_W(\mathbf{u}) \iff \mathsf{Bind}_t \geq \varepsilon_{\mathrm{cl}} \;\wedge\; \mathsf{Sep}_t \geq \delta_{\mathrm{sep}} \;\wedge\; \mathsf{Inside}_t \geq \mu_{\mathrm{in}} \;\wedge\; \mathsf{Persist}_W \geq \rho_{\mathrm{persist}}$$