§SCC · Hub
Soft Cognitive Cohesion.
One entry point for the SCC programme — the canonical specification, the current research status, the unification plan with Ontology Neural Networks, and the mathematical results. A living hub rather than a chapter list.
Suggested reading order
- 01Canonical specification
- 02Research status
- 03Integrated architecture
- 04Theorem references as needed
Canonical specifications
The authoritative formal statement of the theory — primitives, axioms, the energy functional, and the proved-results registry.
Research roadmap & status
Dated status reports — theorem ledger, implementation state, iteration history, and the open problems currently blocking consolidation.
- updated 15 May 2026
SCC — Current Research Status (May 2026, CV-1.17)
Active SCC status as of CV-1.17 (sealed 2026-05-15 — W7 close). Ledger: 68 Cat A + 19 B + 6 C + 5 Retracted = 98 claims, ~70% proved; 215 passed + 1 xfailed. H-COMP-KERNEL CLOSED Cat B. H-MORSE partially closed. First _archive/ cohort instantiated.
- weekly · 15 May 2026
Perception · W7 — Six Sealed Versions, Compositional Closure, First Archive
W7 sealed six canonical versions, closed H-COMP-KERNEL with the kernel-composed compositional consistency theorem, partially closed H-MORSE, archived 14 working/parking files into the first _archive cohort, and deliberately deferred the z_t / S_0 / K_read direction via Decision C.
- weekly · 8 May 2026
Perception · W6 — Observer Moduli Space
W6 covers six canonical versions in one day, completes P-F-A1 Package I, launches and fully closes the Observer Moduli Space theory, and resets theory navigation around six epistemological questions.
- weekly · 3 May 2026
Perception · W5 — Multi-Formation Count Bridge
W5 promotes T-L1-F, SCC's first multi-formation Cat A theorem. It adds the hard-count bridge, drafts the L1-M soft-count companion, registers two HIGH open problems, and documents the T-σ-Theorem-4 demotion.
- updated 27 Apr 2026
SCC — Research Status Snapshot (April 2026, W5 Day 1 G0 close, CV-1.5) — HISTORICAL
Historical SCC status snapshot frozen at W5 Day 1 G0 close (2026-04-27, CV-1.5) — 43 Cat A / 57 claims / 75% proved, the W4 resolution of the 3 Critical OPs (F-1, M-1, MO-1), and the W5 Day 1 G0 σ-framework canonical merge. Superseded; for the active living status page see scc-status-2026-05.
- weekly · 26 Apr 2026
Perception · W4 — Critical-3 Resolved
W4 closes the year-old Critical-3: F-1, M-1, and MO-1. The Pre-Objective Mechanism cluster and T-V5b-T enter Cat A, the σ-framework becomes the language for critical-point signatures, and the v2.0 release path unblocks.
Integration & north-star
How SCC is positioned relative to Ontology Neural Networks and the broader research programme.
- updated 4 May 2026
SCC Research Overview
Narrative overview of Soft Cognitive Cohesion — what the theory does, core thesis, conceptual inversion, mathematical problem. Canonical is now CV-1.17 (68 Cat A / 98 claims, sealed 2026-05-15); this overview is the CV-1.11 (W6 EOD, 2026-05-08) snapshot — 54 Cat A / 78 claims, ~69% proved. OMS-2.0 Accepted — Full. OP-0006 RESOLVED. P-F-A1 Package I complete.
- updated 4 May 2026
Integrated Architecture — SCC × ONN as one cognitive-reasoning system
The ultimate design target of the whole research programme: a single cognitive-reasoning architecture that unifies Soft Cognitive Cohesion (SCC) and Ontology Neural Networks (ONN). Four layers, one pipeline, one vision.
Mathematical results
Theorems and proofs. The Part II summary lists the eight main theorems; individual theorem pages will be detached from it as they are cleaned for public reading.
- Part 2 · Chapter 9
Main Theorems A–H
Complete statements and full proofs of the eight core theorems of RelationWorld Theory — energy isolation, door finiteness, self-interpretation, metastability, curvature localisation, spectral stability, door stability, and flow stability.
- Part 2 · Chapter 9
Part II · Main Theorems A–H (summary)
A condensed statement of the eight main theorems of RelationWorld Theory with one-paragraph proof ideas and their logical dependencies. Full proofs live in the research archive.
- Part 2 · Theorem A
Theorem A — Energy Isolation
Internal edge-energy of a fruit is at least (1-theta) of its total volume. A three-step proof from the conductance bound.
- Part 2 · Theorem B
Theorem B — Finiteness of Doors
The number of door nodes is bounded by theta times the volume divided by the threshold. A direct energy-budget argument.
- Part 2 · Theorem C
Theorem C — Self-Interpretation
Under Axiom A5, the door set and door energies are determined entirely by the fruit's intrinsic data — no exterior information is needed.
- Part 2 · Theorem D
Theorem D — Metastability
The expected escape time from a fruit under the lazy walk is at least 1/(2 theta), via the Cheeger inequality and the Sinclair–Jerrum spectral bound.
- Part 2 · Theorem E
Theorem E — Curvature Localisation
Residual curvature under the optimal gauge concentrates exponentially near doors for U(1), and satisfies a contraction bound for general compact G.
- Part 2 · Theorem F
Theorem F — Spectral Stability
Small weight perturbations produce bounded conductance changes; strong fruits persist under perturbation with an explicit constant C1 = 2(1+theta).
- Part 2 · Theorem G
Theorem G — Door Stability
Under weight perturbation, doors with sufficient margin above the threshold are stable — only the epsilon-boundary layer may change.
- Part 2 · Theorem H
Theorem H — Flow Stability
The Yang–Mills gradient flow converges to a stable critical point, with exponential rate when the Lojasiewicz exponent is 1/2, via real-analyticity and compactness.
- Part 0 · SCC Hero · T-L1-F
T-L1-F — Hard-Bar / Active-Count Bridge under L1-J Regime
First multi-formation canonical Cat A conditional theorem. Under hypothesis package (P0)-(P11) on shared-pool Σ̃_M^K_field, the hard-bar count from H_0 superlevel persistence equals the active-slot count, with labelled bijection.
- Part 0 · SCC Hero · T-PreObj-1
T-PreObj-1 — Pre-Objective Multi-Peak Formation Mechanism (W4 Capstone)
Under full SCC on any (G1)-(G4) graph, the F=1 single-disk is not a critical point of full E, and gradient flow attracts to multi-peak F≥2 configurations. The W4 capstone — SCC's pre-objective character is theorem, not modeling choice. Resolves F-1.
- Part 0 · SCC Hero · T-V5b-T
T-V5b-T — Pre-Objective Goldstone on Translation-Invariant Graphs (W4-Extended Capstone)
On translation-invariant graphs (torus T^d, cycle C_n), a sub/super-lattice spectral dichotomy at the F=1/F≥2 transition. 2D: Goldstone doublet with commensurability split; 1D: 1-fold Goldstone; nodal count = 2 universal. W4-extended capstone (8 V5b iterations + NQ-172 reproducibility check).
- Part 0 · SCC Hero · T1
T1 — Existence of Minimizers
The energy E attains its minimum on the constraint manifold Σ_m. The well-posedness foundation of SCC: every other result is a refinement of the bare existence statement.
- Part 0 · SCC Hero · T11
T11 — Sharp-Interface Γ-Convergence
As ε = α/β → 0, the boundary-morphology energy E_bd Γ-converges to a perimeter functional. Minimizers converge to characteristic functions of minimal-perimeter sets. The soft-to-crisp bridge — recovery of object-like sharp interfaces from the soft cohesion field.
- Part 0 · SCC Hero · T14
T14 — Gradient Flow Convergence (Łojasiewicz)
The projected gradient flow on Σ_m converges to a critical point. With analytic energy (b_D = 0), convergence is exponential via the Łojasiewicz inequality. The dynamical existence theorem — variational minimizers are reachable by descent.
- Part 0 · SCC Hero · T20
T20 — Axiom Consistency (A1' resolves A1↔A3 incompatibility)
The axioms A1' / A2 / A3 / A4 of Group A (closure) are mutually consistent. The original A1 (weak extensivity) is incompatible with A3 (contraction) for the sigmoid closure realization; A1' (conditional extensivity) resolves the tension. The theory's foundational legitimacy.
- Part 0 · SCC Hero · T6
T6 — Closure Fixed Point (Banach Contraction)
When the closure steepness parameter a_cl < 4, the sigmoid closure operator has a unique fixed point on [0,1]^n with geometric convergence rate a_cl/4. The convergence guarantee for the central self-referential operator.
- Part 0 · SCC Hero · T7-Enhanced
T7-Enhanced — Non-Idempotent Metastability Advantage
At a non-idempotent closure fixed point with operator norm < 1, the closure Hessian contribution is strictly positive definite (n/n positive eigenvalues). For idempotent closure, only n-k of n eigenvalues are positive. The mathematical payoff of SCC's deliberate non-idempotence commitment.
- Part 0 · SCC Hero · T8-Core
T8-Core — Phase Transition (Spectral Universality)
On any connected graph with Fiedler eigenvalue λ_2 > 0, when β/α exceeds 4λ_2/|W''(c)|, the uniform field becomes unstable and a non-uniform minimizer exists. Formation birth is topologically universal — depending only on the spectral gap.
Related papers
Published and in-progress manuscripts that develop, cite, or depend on the SCC formalism.
- published · 2026
Ontology Neural Network and ORTSF: A Framework for Topological Reasoning and Delay-Robust Control
The advancement of autonomous robotic systems has led to significant capabilities in perception, localization, mapping, and control, yet a critical challenge remains in representing and preserving relational semantics, c
- preprint · 2025
Advanced Topology-Preserving Neural Networks: An Extension of ONN/ORTSF Framework with Dynamic Structural Optimization
This work presents the first comprehensive empirical investigation into the practical realization of performance bounds theoretically established by the Ontology Neural Network (ONN) and Ontological Real-Time Semantic Fa
- preprint · 2025
Constructive Lyapunov Functions via Topology-Preserving Neural Networks
We present a constructive solution to the Lyapunov–Massera–Kurzweil problem via Ontology Neural Networks (ONN), bridging a 60-year gap between existence and construction in stability theory. While Massera (1949) proved t
- preprint · 2025
Ontology Neural Networks for Topologically Conditioned Constraint Satisfaction
Neuro-symbolic reasoning systems face fundamental challenges in maintaining semantic coherence while satisfying physical and logical constraints. Building upon our previous work on Ontology Neural Networks, we present an