Mousse-notes.

Soft Cognitive Cohesion — Canonical Specification (CV-1.11)

The canonical formal text of SCC. The programme's canonical has since advanced to CV-1.17 (68 Cat A + 19 B + 6 C + 5 Retracted = 98 claims, ~70% proved, sealed 2026-05-15); this page is the CV-1.11 (W6, 2026-05-08) formal snapshot — the CV-1.12–1.17 additions are logged in the status page. OMS-2.0 Accepted — Full.

SCC Canonical Spec — Part 1: Foundations & Formal Universe

Sections 0, 2–5 of the SCC canonical specification (CV-1.5.2): the summation convention, the foundational orientation, the formal universe, the primacy of the soft form, and the derived geometric and morphological notions.

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

Sections 6–7 of the SCC canonical specification (CV-1.5.2): the five axiomatic groups (closure, adjacency, co-belonging, distinction, temporal transport) and the proto-cohesion diagnostic vector with its four component predicates.

SCC Canonical Spec — Part 3: Energy Principle & Provisional Operators

Sections 8–9 of the SCC canonical specification (CV-1.5.2): the volume constraint, the four-term canonical energy functional, and the currently favored provisional realizations of closure, distinction, co-belonging, and temporal transport operators.

SCC Canonical Spec — Part 4: Structural Interpretation, Commitments & Open Problems

Sections 10–12 of the SCC canonical spec (CV-1.5.2): structural interpretation, fixed commitments vs open design choices, and open problems by foundational/bridging/extension layer. W4 close (2026-04-24) resolved Critical-3 (F-1, M-1, MO-1); CV-1.5.2 promoted T-L1-F as the first multi-formation Cat A theorem.

SCC Canonical Spec — Part 5: Proved Results Registry & Closing Notes

Sections 13–15 of the SCC canonical specification. Canonical is now CV-1.17 (68 Cat A / 98 claims, sealed 2026-05-15); this part is the CV-1.11 (W6) snapshot — 54 Category A + 14 B + 5 C + 5 retracted = 78 claims, ~69% proved. W6 additions: T-ST-5a (Stereo-SCC), T-OP6-B (OP-0006 RESOLVED), P-F-A1 Package I fully Cat A, OMS-2.0 Full.

Chapter 0 — Formal State Space

Formalises the state space of RelationWorld — the mathematical object describing all possible world configurations and the structure of transitions between them.

SCC Glossary

Plain-language definitions of all formal SCC concepts — formal ID, intuitive description, why it matters, concrete analogy. Aligned with canonical CV-1.5.2: covers multi-formation vocabulary (Commitment 16 K-status, σ-tuple, OP-0008/0009) and L1-J / T-L1-F Hard-Bar / Active-Count Bridge.

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.

SCC Theorem Registry

Canonical is now CV-1.17 (68 Cat A + 19 B + 6 C + 5 Retracted = 98 claims, ~70% proved, sealed 2026-05-15); this registry is the CV-1.11 (W6) snapshot — 54 Cat A / 78 claims, ~69% proved. The CV-1.12–1.17 additions are logged in the status page. W6 additions: T-ST-5a (CV-1.6), T-OP6-B/OP-0006 RESOLVED (CV-1.7), P-F-A1 Package I fully Cat A (CV-1.8–CV-1.9), OMS-2.0 Full.

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.

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.

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.

Claim C-0001 — The Soft Cohesion Field is Primitive

The foundational claim that the graded cohesion field u_t is ontologically prior to discrete objecthood. Aligned with canonical CV-1.5.2 (W5 Day 6 close, 2026-05-02; 2026-05-04 audit pass).

Claim C-0002 — The K-Field Architecture is Necessary

K-field architecture as structural multi-formation realisation. Reframed post-W4: no longer the sole mechanism (T-PreObj-1 (i) gives single-field F≥2 default). Superseded by Commitment 16 (CV-1.5.1): two-tier K_field (architectural cap) / K_act (dynamic stratum) decomposition; OAT-1 closes OP-0009-K. Confidence 65→80%.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

SCC Hero Theorems — 16 Major Results

Curated index of the 16 hero theorems of Soft Cognitive Cohesion (current canonical CV-1.17). Foundation, phase transition + stability, W4 Pre-Objective Mechanism, W5 multi-formation bridge (T-L1-F), W6 stereo + boundary (T-ST-5a, T-OP6-B), and W6 stochastic foundation (P-F-A1 Package I). Full proofs live in the canonical spec.

Overview — The Sweet-Potato-Vine Worldview

Philosophy and roadmap of RelationWorld Theory. Five foundational principles, the central sweet-potato-vine metaphor, and a correspondence dictionary between the discrete formalism and its classical counterparts.

Relation

The fundamental unit of RelationWorld — a relation tuple (i, j, w, g) carrying scalar intensity and group-valued transit, with symmetrised weight, degree, and their basic properties.

Relational Field

Gauge group and action on relational fields, scalar invariants, holonomy and discrete curvature, and the completeness of invariants in the connected case.

Fruit

Low-conductance clusters as the basic unit of cohesive existence. Cheeger conductance, the fruit definition with threshold θ, and Theorems A (Energy Isolation) and D (Metastability).

Stem

The connective tissue between fruits. Stem region, bridge edges, and the no-boundary principle that prevents the exterior from being defined explicitly.

Door

Internal singularities arising from anomalous external contact, detectable only within the fruit interior. The intrinsic data axiom A5, door definitions, and Theorems B, C, G.

Existence

Existence as a gauge-invariant triple combining the optimal gauge class, door locus, and residual energy. Flattening energy, optimal gauge, and Theorems E and H.

World

The world as a three-layer hierarchy of relational fields, fruits with existence triples, and their inter-fruit connectivity. Theorem F (Spectral Stability) closes Part I.

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 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.

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.

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.

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.

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.

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.

Theorem F — Spectral Stability

Small weight perturbations produce bounded conductance changes; strong fruits persist under perturbation with an explicit constant C1 = 2(1+theta).

Theorem G — Door Stability

Under weight perturbation, doors with sufficient margin above the threshold are stable — only the epsilon-boundary layer may change.

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.

Chapter 10 — Worked Examples

Fully worked examples of RelationWorld on the U(1), SU(2), and Z_2 theories. Explicit calculations of holonomies, cohomological invariants, and the three-axis readout on concrete small graphs.

Chapter 11 — Čech Cohomology Framework

Čech cohomology as the natural language of the theory — what holds exactly in the discrete setting, what requires additional topological hypotheses, and how the cocycle condition encodes the closure of relational transit.

Chapter 12 — Three Computational Axes

The three computational axes of invariants: intrinsic cohomology, relative cohomology of the fruit–door pair, and persistence barcodes of the door set.

Chapter 13 — Yang–Mills Flow

Discrete Yang–Mills gradient flow on the space of connections. How curvature-energy minimisation automatically stabilises the world's essential structure over time.

Chapter 14 — Time Evolution

How the instantaneous world W_t evolves with t. Hybrid dynamics on the state space, event-driven transitions, and the emergence of coarse trajectories.

Chapter 15 — Applications

Applications of RelationWorld theory to physics, topology, and combinatorics.

Chapter 16 — Open Problems

All open problems and conjectures collected from throughout the theory. Organised by part and difficulty.

Chapter 17 — End-to-End Mobile Manipulator

Instantiates RelationWorld theory for a practical mobile manipulator: 4-wheel swerve base, 4-DOF arm, and the full control pipeline in the theory's language.

Appendix A — Unified Notation Table

A single comprehensive symbol reference for the entire theory.

Appendix B — Prerequisites

Mathematical prerequisites assumed by the main text.

Appendix C — Discrete–Continuous Correspondence Dictionary

A thirty-row dictionary mapping discrete RelationWorld constructions to their continuous differential-geometry counterparts, plus theorem and axiom correspondence tables.

Appendix D — Hybrid Dynamical Systems Background

Mathematical background on hybrid dynamical systems and their application to RelationWorld's state space.