§SCC · Historical archive
Soft Cognitive Cohesion — sealed.
Soft Cognitive Cohesion is preserved as a sealed historical programme at CV-1.17. Its mathematics and 98-claim ledger remain available, while active main research has moved to ULR.
Suggested reading order
Canonical specifications
The authoritative formal statement of the theory — primitives, axioms, the energy functional, and the proved-results registry.
Historical roadmap & final status
Frozen status reports, theorem ledger, implementation history, and the open problems that remained when SCC left the active programme.
- updated 9 Aug 2026
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.
- 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 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 Superseded by the final sealed CV-1.17 archive; current Main research is ULR.
- 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.
Historical integration target
The former SCC–ONN north-star, retained with explicit audit markings. It is not the current architecture or Main programme.
- updated 9 Aug 2026
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.
- updated 9 Aug 2026
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.
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 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 · partially-superseded
Ontology Neural Network and ORTSF: A Framework for Topological Reasoning and Delay-Robust Control
Paper of record for the original ONN + ORTSF framing. A later audit found that the higher-order/cohomological thesis adds no information beyond pairwise data under the stated observation contract. The implemented control
- preprint · 2025 · withdrawn
Advanced Topology-Preserving Neural Networks: An Extension of ONN/ORTSF Framework with Dynamic Structural Optimization
The headline empirical figures and transfer results are not reproducible from the current authoritative research source. This preprint is retained as a historical manuscript, but its validation claims are withdrawn.
- preprint · 2025 · withdrawn
Constructive Lyapunov Functions via Topology-Preserving Neural Networks
The claimed constructive cohomological Lyapunov certificate, together with the 177 microsecond / 3M-node and 99.75% figures, is not supported by the current authoritative research source. Those claims are withdrawn.
- preprint · 2025 · withdrawn
Ontology Neural Networks for Topologically Conditioned Constraint Satisfaction
The reported topology-loss and 95% constraint-satisfaction figures are not reproducible from the current authoritative research source. This manuscript is retained as a historical, unverified draft; its empirical claims