SCC Hero Theorems — 16 Major Results

What this is. SCC has 54 Category A theorems (CV-1.11, 78 total claims). Most are technical, conditional, or auxiliary. This page curates the 16 hero theorems — the results that define the theory's identity and would be cited first in any introduction.

Where the full proofs live. The full proofs are in Canonical Spec — Part 5 · §13. W6 additions are narrated in the Week 6 weekly post.

The 10 heroes#

Logical dependency#

Reading guide. Foundation row (T1, T6, T20) is the well-posedness layer. Phase + Stability row (T8-Core, T7-Enhanced, T11) establishes that non-trivial minimizers exist universally and are stable. The Capstone row (T14, T-PreObj-1, T-V5b-T) connects variational + dynamic + symmetric layers: gradient flow converges (T14), pre-objective F≥2 attractor is graph-class-independent theorem (T-PreObj-1, the W4 capstone), and on translation-invariant graphs the Goldstone signature is universal (T-V5b-T). The Multi-Formation Flagship (T-L1-F, W5 Day 6) is the first multi-formation canonical Cat A theorem — under the L1-J regime hypothesis package $(P0)$–$(P11)$, it bridges chart-level active count to aggregate-field topological bar count.

Group 1 — Foundation (well-posedness)#

T1 — Existence of Minimizers The energy $\mathcal{E}$ attains its minimum on the constraint manifold $\Sigma_m$. Why hero: every other result is a refinement of "the minimizer exists."

T6 — Closure Fixed Point When $a_{\mathrm{cl}} < 4$, the sigmoid closure operator has a unique fixed point with geometric convergence rate $a_{\mathrm{cl}}/4$. Why hero: closure is the central self-referential operator and this is its convergence guarantee.

T20 — Axiom Consistency The axioms A1' / A2 / A3 / A4 are mutually consistent. The original A1 (weak extensivity) and A3 (contraction) are incompatible; A1' (conditional extensivity below the scalar fixed point $c^*$) resolves the tension. Why hero: this is the theory's foundational legitimacy — the axioms are not just stated, they're proved consistent.

Group 2 — Phase Transition + Stability#

T8-Core — Phase Transition On any connected graph with Fiedler eigenvalue $\lambda_2 > 0$, when $\beta/\alpha > 4\lambda_2/|W''(c)|$, the uniform field becomes unstable and a non-uniform minimizer exists. Why hero: spectral universality — the phase transition condition depends only on the spectral gap, valid on any connected graph.

T7-Enhanced — Non-Idempotent Metastability Advantage At a non-idempotent closure fixed point with $\|J_{\mathrm{Cl}}\|_{\mathrm{op}} < 1$, the closure Hessian is strictly positive definite ($n/n$ positive eigenvalues). For idempotent closure, only $\leq (n-k)/n$ are positive. Why hero: this is the mathematical payoff of the deliberate non-idempotence commitment — strictly stronger stability.

T11 — Sharp-Interface Γ-Convergence As $\varepsilon = \alpha/\beta \to 0$, the boundary-morphology energy $\mathcal{E}_{\mathrm{bd}}$ Γ-converges to a perimeter functional. Minimizers converge to characteristic functions of minimal-perimeter sets. Why hero: this is the soft-to-crisp bridge — recovery of object-like sharp interfaces from the soft cohesion field.

Group 3 — W4 Capstone#

T14 — Gradient Flow Convergence The projected gradient flow $\dot u = -\Pi_{\Sigma_m}\nabla\mathcal{E}(u)$ converges to a critical point. With analytic energy ($b_D = 0$), convergence is exponential via the Łojasiewicz inequality. Why hero: this is the dynamical existence theorem — variational minimizers are reachable by descent.

T-PreObj-1 — Pre-Objective Multi-Peak Formation Mechanism (W4) Under full SCC parameters on any finite connected graph satisfying (G1)–(G4) hypotheses: (i) the F=1 single-disk minimizer of pure $\mathcal{E}_{\mathrm{bd}}$ is not a critical point of full $\mathcal{E}$; (ii) gradient flow attracts to multi-peak F≥2 configurations; (iii) IC-protocol dichotomy. Why hero: this is the W4 capstone — SCC's pre-objective character is no longer a modeling choice but a graph-class-independent mathematical theorem. Resolves F-1 / M-1.

T-V5b-T — Pre-Objective Goldstone on Translation-Invariant Graphs (W4-extended) On translation-invariant graphs (torus $T^d$, cycle $C_n$): a sub/super-lattice spectral dichotomy holds; 2D Goldstone modes form a 2-fold doublet with commensurability splitting; 1D 1-fold Goldstone; nodal count = 2 universal. Why hero: the W4-extended capstone — symmetry-breaking signature of the pre-objective transition is universal across graph classes (8 V5b iterations to robust statement; mode-indexing artifact NQ-172 identified and resolved en route).

Group 4 — Multi-Formation Flagship#

T-L1-F — Hard-Bar / Active-Count Bridge under L1-J Regime (W5 Day 6, 2026-05-02) Under the L1-J regime hypothesis package $(P0)$–$(P11)$ on the shared-pool multi-formation manifold $\widetilde\Sigma_M^{K_{\mathrm{field}}}$, the hard-bar count $K_{\mathrm{bar}}^{\ell_{\min}}$ from terminal-death $H_0$ superlevel persistence on the aggregate field $U(\mathbf u) = \sum_j u^{(j)}$ equals the active-slot count $K_{\mathrm{act}}^\varepsilon$, with a labelled bijection between active slots and dominant terminal $H_0$ bars. Why hero: the first multi-formation canonical Cat A theorem in SCC theory. Closes the L1-A through L1-L 13-step working chain that was the substantive content of W5. Cat A conditional — does NOT establish $K_{\mathrm{soft}}^\phi = K_{\mathrm{act}}$ globally and does NOT solve OP-0005 (K-Selection) or OP-0008 ($\sigma^A$ K-jump non-determinism).

Group 5 — W6 Stereo + Boundary#

T-ST-5a — Hard-Depth Topological Locking (Stereo-SCC, W6 D4, 2026-05-06) On a depth-disconnected stereo graph (objects A and B at depths $z_1 \ll z_2$), the energy barrier between the two formation basins is $\Delta E = +\infty$ (state-space disconnection via the Goldstone mechanism). The system is forced to $K = 2$ stable formations — one per depth-connected component. Why hero: the first theorem connecting SCC to stereo geometry, and the mathematical proof that perceptual individuation can be forced by depth topology alone.

T-OP6-B — PersRidge Boundary Equivalence — OP-0006 RESOLVED (W6 D4, 2026-05-06) In the phase-separation regime (H1–H5), the PersRidge locus and the formation boundary $\partial\Omega^*$ satisfy $d_H(\mathrm{PersRidge}(u^*), \partial\Omega^*) \leq 2\sqrt{\alpha/\beta}$. Why hero: resolves OP-0006 (boundary definition precision) — the computationally accessible PersRidge is a provably faithful proxy for the theoretical boundary.

Group 6 — W6 Stochastic Foundation (P-F-A1 Package I)#

T-PF-A1-AR — Affine Reduction (CV-1.8, 2026-05-06) The feasibility polytope $\mathcal{F}_M(\mathcal{P})$ is a bounded compact convex polytope (Lions-Sznitman domain condition satisfied). Why hero: the geometric foundation that makes the reflected SDE well-posed.

T-PF-A1-SDE — Lions-Sznitman Reflected SDE (CV-1.8, 2026-05-06) The reflected Langevin SDE on $\widetilde{\Sigma}_M$ is well-posed: existence via Lions-Sznitman Theorem 1 (convex domain), uniqueness via Tanaka pathwise argument. Why hero: establishes that SCC energy has a well-defined stochastic dynamics — the formation can evolve under noise and stay in the constraint manifold.

T-PF-A1-GI — Unique Gibbs Invariant Measure (CV-1.9, 2026-05-06) The Gibbs measure $\pi_{T_*} \propto e^{-\tilde{\mathcal{E}}/T_*}$ is the unique invariant measure of the reflected SDE via zero-current + $L^2(\pi)$ kernel argument (Aronson 1968 heat kernel). Why hero: proves that temperature $T_*$ canonically selects a distribution over formation states.

T-PF-A1-PE — Poincaré Inequality + Exponential Ergodicity (CV-1.9, 2026-05-06) $\lambda_1 \geq (\pi^2/n)\exp(-\mathrm{osc}(\tilde{\mathcal{E}})/T_*)$ (Payne-Weinberger on bounded convex polytope + Holley-Stroock perturbation). Exponential TV convergence with explicit rate. Why hero: the dynamical completeness result — any initial condition converges to $\pi_{T_*}$ exponentially fast.

What's deliberately not on this list#

The full registry has 54 Cat A + 14 Cat B + 5 Cat C. Excluded from heroes:

• T-Bind-Proj/Full, Predicate-Energy Bridge, Deep Core Dominance — diagnostic-bridging results, important but technical.
• T-Persist-1(a/b/c/e), T-Persist-Full, T-Persist-K-Sep/Weak/Unified — persistence machinery, mostly conditional.
• QM-1..4, T-Merge (a/b), Topological Lock, T-Birth-Parametric, Lemma 4, F-1 Resolution Corollary — supporting / corollary results.
• T-P-F-ε0, T-σ-Lemma-1/2/3, T-σ-Theorem-3 — foundations for the heroes above, but machinery rather than headline results.
• T-K-Select-PF/OBS — Cat B (partial), not yet fully proved.

If a result is missing here, it is not unimportant — it is just below the "굵직한 hero" cut. The full inventory is at SCC Theorem Registry.

See also#

• Canonical Spec — Part 5 · §13 Results Registry — full statements + proofs of all 54 Cat A theorems
• SCC Theorem Registry — flat index by canonical version
• SCC Status (May 2026, CV-1.11) — current research state
• Week 6 weekly post — narrative of W6 (T-ST-5a, T-OP6-B, P-F-A1 Package I, OMS-2.0)
• Week 4 Extended weekly post — narrative of T-PreObj-1 and T-V5b-T

Aligned with Perception_theory canonical CV-1.17 (sealed 2026-05-15). 68 Cat A / 19 Cat B / 6 Cat C / 5 Retracted = 98 claims, ~70% proved. W7 additions: T-Temporal-Identity (a) Cat A (CV-1.12, via H-SINK), T-CC-StableK-Kernel Cat B (CV-1.17, H-COMP-KERNEL closed). Next: CV-1.18.