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

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1. #1

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

   $${\mathrm{Cl}}_t(u)(x)\ge u(x)\text{whenever }u(x)\le c^{\ast }\text{ and }(P_{t}u)(x)\ge u(x)$$
2. #2

   A2. Monotonicity. For all ,

   $$u\le v⟹{\mathrm{Cl}}_t(u)\le {\mathrm{Cl}}_t(v)$$
3. #3

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

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

   B1. Nonnegativity. For all ,

   $${\mathbf{N}}_t(x,y)\ge 0.$$
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 ,

   $$\sum _{y\in X_s}{\mathbf{M}}_{t\to s}(x,y)\le 1.$$
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:

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

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

    $${\mathsf{Bind}}_t(u_t)=1-\frac{\Vert u_t-{\mathrm{Cl}}_t(u_t){\Vert }_2}{\sqrt{n}}$$
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 :

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

    Proto-cohesion is reported as a diagnostic vector :

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

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

    $${\mathsf{ProtoCoh}}_W^{\mathrm{soft}}(\mathbf{u})⟺{\mathsf{Bind}}_t\ge {\epsilon }_{\mathrm{cl}}\wedge {\mathsf{Sep}}_t\ge {\delta }_{\mathrm{sep}}\wedge {\mathsf{Inside}}_t\ge {\mu }_{\mathrm{in}}\wedge {\mathsf{Persist}}_W\ge {\rho }_{\mathrm{persist}}$$