#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.

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 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 Superseded; for the active living status page see scc-status-2026-05.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.

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.

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.

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.