Part 0 · Soft Cognitive Cohesion — Historical Archive

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 Research Overview

Historical SCC overview at the CV-1.11 snapshot, with the final CV-1.17 ledger linked. SCC is sealed and archived; current Main research moved to ULR on 2026-07-24.

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

Historical north-star for the proposed SCC × RelationWorld × ONN × ORTSF architecture. The strong integration claim was not established; active main research has moved to ULR.

SCC — Research Status Snapshot (April 2026, W5 Day 1 G0 close, CV-1.5) — HISTORICAL

Historical SCC snapshot at CV-1.5: 43 Cat A / 57 claims / 75% proved, W4 Critical-3 resolution and the σ-framework merge. Superseded by the final sealed CV-1.17 archive; current Main research is ULR.

SCC — Final Sealed Status (May 2026, CV-1.17) — HISTORICAL

Final sealed SCC status at CV-1.17 (2026-05-15): 68 Cat A + 19 B + 6 C + 5 Retracted = 98 claims. Preserved as a historical archive after the 2026-07-24 ULR-only transition; it is not the current research ledger.

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.

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.

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%.

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 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.