Integrated Architecture — SCC × ONN as one cognitive-reasoning system

The ultimate target of the entire programme — the reason every other page on this site exists.

The claim#

Perception, representation, reasoning, and control are one continuous mathematical object. The claim of this research is that Soft Cognitive Cohesion (SCC) and Ontology Neural Networks (ONN) — together with their mathematical substrate (RelationWorld Theory) and their control-theoretic closure (ORTSF) — can be composed into a single cognitive-reasoning architecture that carries structure without loss from the pre-objective soft field all the way down to an embodied control signal.

This is the design target. The rest is bookkeeping.

The four layers#

Layer 1 — Soft Cognitive Cohesion (pre-objective)#

Primitive. A soft cohesion field $u_t:X_t\to [0,1]$ over a relational support space; the value $u_t(x)$ is the degree to which site $x$ participates in a cohesive formation.

Operators. Closure ${\mathrm{Cl}}_t$ (self-completion), distinction ${\mathbf{D}}_t$ (self-contrast), temporal transport ${\mathbf{M}}_{t\to s}$.

Energy. $E={\lambda }_{\mathrm{cl}}{E}_{\mathrm{cl}}+{\lambda }_{\mathrm{sep}}{E}_{\mathrm{sep}}+{\lambda }_{\mathrm{bd}}{E}_{\mathrm{bd}}+{\lambda }_{\mathrm{tr}}{E}_{\mathrm{tr}}$ on the volume-constrained simplex ${\Sigma }_m$.

Output. A proto-cohesion diagnostic vector $\mathbf{d}=(\mathrm{Bind},\mathrm{Sep},\mathrm{Inside},\mathrm{Persist})\in [0,1{]}^4$.

Ontological commitment. The field is primitive; objects are derivative. Deeper dive →

Layer 2 — RelationWorld (structural discovery)#

Substrate. A weighted relational field ${\mathcal{W}}_t=(V,W_t,g_t)$ where every edge carries both a positive weight and a group-valued transit.

Discovery. Fruits — low-conductance clusters detectable via the Cheeger condition ${\varphi }_t(F)\le \theta$ — are the natural stable structures in the relational field. Doors are the boundary-adjacent singularities that signal contact with an exterior not explicitly modelled.

Output. A fruit set ${\mathfrak{F}}_t$ together with its existence triple $([A_{\infty }],\Sigma ,\mathbf{e})$ — a gauge-invariant portrait.

Role in the architecture. This is the layer that makes SCC concrete: the graded soft field acquires a relational substrate with a theory of what counts as existing. Deeper dive →

Layer 3 — Ontology Neural Networks (relational semantics)#

Substrate. A fruit-node semantic graph — fruits as nodes, their inter-fruit relations as edges, the ontology's type structure as type algebra on both.

Process. Constraint-based semantic inference on this graph, projecting onto a topology-aware manifold. The ONN's internal state is not a flat vector but a typed object carrying the structure of the target ontology.

Output. A semantic structure augmented with action predicates — what follows from what the fruits mean.

Role in the architecture. ONN is what turns discovered structure into actionable meaning, with invariants that can be read as cohomology classes. Deeper dive →

Layer 4 — ORTSF (embodied action)#

Substrate. A delay-robust control fabric consuming ONN semantic output.

Process. Meaning-preserving trajectory synthesis under stochastic communication, sensing, and compute delay.

Output. A real-time embodied control signal whose stability margins are stated in cohomological terms, not raw state norms.

Role in the architecture. This is what closes the loop from cognition back to the world. Deeper dive →

Why each layer requires the next#

Layer 1 — SCC
  "Objects are given, but how does anything first hold together?"
  → Soft field, four energy terms, graded cohesion.
      │ "u_t is abstract. What is the relational substrate?"
      ▼
Layer 2 — RelationWorld
  "If cohesion is relational, what makes relations cohere?"
  → Weighted gauged graph, fruits, doors, existence.
      │ "Fruits are discovered. How do they interact semantically?"
      ▼
Layer 3 — ONN
  "Fruits are topological units — how do we reason between them
   while preserving their internal structure?"
  → Topology-aware constraint inference, type-preserving projection.
      │ "Semantics are inferred. How do we embody them in action?"
      ▼
Layer 4 — ORTSF
  "Semantic structure is defined. How do we preserve that meaning
   in real-time control under stochastic delay?"
  → Delay-robust control, Lyapunov from loss, cohomology-stated bounds.

Forward flow and backward flow#

Forward flow is the bottom-up emergence of meaning:

Backward flow is the top-down validation and refinement:

The architecture is only whole when both flows are closed — the system not only produces behaviour but uses the outcomes of that behaviour to update its beliefs about what exists, what is, and what it should do next.

Why this integration matters#

Most architectures for perception-to-control keep representation learning, semantic reasoning, and control synthesis as separate problems with separate guarantees. The integrated architecture proposed here insists that the three share a single mathematical substrate — topology — and that guarantees in one layer translate directly into guarantees in the others.

Whether this proposition is provable in its strongest form is the open question the programme is organised around. The theorems of Part II (stability, localisation, flow convergence) and the ONN + ORTSF framework are the pieces already in place; what remains is the composition theorem that ties them together into one certificate.

Where this stands today#

• Layer 1 (SCC) — 46 Cat A theorems (61 total claims), 4 active HIGH OPs (OP-0005 K-Selection, OP-0006 Boundary precision, OP-0008 σ^A K-jump non-determinism, OP-0009 Multi-Formation Foundations). The W4-original Critical OPs F-1 / M-1 / MO-1 were resolved/clarified/sidestepped on 2026-04-24. The current canonical/ directory contains canonical.md + theorem_status.md (Open Problems Catalog merged 2026-05-04) + figures/ + README.md. Layer 1 now also includes the σ-framework with Commitment 14 (multi-static σ formulation) and Commitment 16 (K_field architectural cap vs K_act dynamic stratum index two-tier K-status decomposition), the multi-formation σ static promotions D-6a (CV-1.5.1, 2026-04-29), and T-L1-F (CV-1.5.2, 2026-05-02) — the first multi-formation Cat A conditional theorem (Hard-Bar / Active-Count Bridge under L1-J Regime (P0)–(P11)) on shared-pool architecture I9'. See the status page.
• Layer 2 (RelationWorld) — Theorems A–H proved. See Part I and Part II summary.
• Layer 3 (ONN) — Framework paper accepted at Int. J. Topol. (paper); empirical follow-up reaching 99.75 % of theoretical optimality (paper).
• Layer 4 (ORTSF) — Constructive Lyapunov functions proved (paper); ${\tau }_{\max }=177\mu$s for 3M-node systems.
• Integration theorem — in progress. The piece that turns four separate results into one architecture.