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

Part 5 of 5 — Soft Cognitive Cohesion canonical specification. This text is the CV-1.11 (W6, 2026-05-06) snapshot; the current canonical is CV-1.17 (68 Cat A / 98 claims, sealed 2026-05-15) — the CV-1.12–1.17 delta is in the status page. See the index for the status note, change log, and links to the other parts.

13. Proved Results Registry#

(Audit history: §13 counts were corrected from 43A/2B/3C/0R to 35A/4B/5C/5R on 2026-04-07 after the 04-06 deep audit, then upgraded to 38A/4B/5C/5R on the W4 + W4-extended close (2026-04-25 to 2026-04-26) with the addition of T-PreObj-1, T-PreObj-1G, Lemma 4, and T-V5b-T. Further upgraded to 43A/4B/5C/5R on W5 Day 1 G0 (2026-04-27) with the σ-framework supporting structures T-σ-Lemma-1/2/3 + T-σ-Theorem-3/4 — 57 claims total, 75% fully proved. Further upgraded to 45A/5B/5C/5R = 60 claims at CV-1.5.1 (2026-04-29) via D-6a Multi-Static (3 Cat A entries) + V5b-T-zero (Cat A def) + T-σ-Theorem-4 retroactive Cat A → Cat B + T-σ-Multi-1 / V5b-F-empirical (Cat B targets). Further upgraded to 46A/5B/5C/5R = 61 claims at CV-1.5.2 (2026-05-02). W6 upgrades (CV-1.6 → CV-1.11, 2026-05-04 to 2026-05-08): +8 Cat A (T-ST-5a, T-OP6-B, T-P-F-ε0, T-PF-A1-AR/SDE/GI/PE) + 9 Cat B (T-P-F-ε0-K, T-ST-5b, D-ST-1..5, T-K-Select-PF, T-K-Select-OBS) = 17 new claims. As sealed in this snapshot: 54A/14B/5C/5R = 78 claims, ~69% fully proved (CV-1.11, 2026-05-06); the canonical has since advanced to CV-1.17 (98 claims).)

The following theorems have been rigorously proved and audited through the W6 close (2026-05-08 CV-1.11). They are listed with precise statements, proof methods, and known caveats. Totals: 54 Category A, 14 Category B, 5 Category C, 5 Retracted (78 claims, ~69% fully proved).

Category A: Fully Proved (46 theorems)#

T1. Energy Minimizer Existence. On the constraint manifold $\Sigma_m = \{u \in [0,1]^n : \sum u_i = m\}$, the energy $\mathcal{E}_t$ attains its minimum. Proof: Extreme value theorem on compact set. (R5.)

T6a. Closure Fixed Point Existence. The sigmoid closure $\mathrm{Cl}_t$ on $[0,1]^n$ has at least one fixed point. Proof: Brouwer fixed-point theorem ($\mathrm{Cl}_t$ maps $[0,1]^n$ to itself continuously). (R4.)

T6b. Closure Fixed Point Uniqueness (Contraction Regime). When $a_{\mathrm{cl}} < 4$, the sigmoid closure has a unique fixed point, and iterates converge geometrically at rate $a_{\mathrm{cl}}/4$. Proof: Banach contraction mapping theorem; Lipschitz constant $\leq a_{\mathrm{cl}}/4 < 1$. (R4.)

T20. Axiom Consistency (Parameter Admissibility). The sigmoid closure satisfies A1' (conditional extensivity with threshold $c^* = \sigma(a_{\mathrm{cl}}(c^* - \tau_{\mathrm{cl}}))$), A2 (monotonicity, unconditional), A3 (contraction when $a_{\mathrm{cl}} < 4$), A4 (continuity, unconditional). The original A1 and A3 are incompatible; A1' resolves the tension. Proof: Direct computation. A1' from the sign of $g(u) = \sigma(a_{\mathrm{cl}}(u - \tau)) - u$ on $[0, c^*]$. A2 from monotonicity of $\sigma$ and $P_t$. A3 from $\max \sigma' = 1/4$. A1 failure from $\sigma^{-1}(0.9)/z > 4$. (R4.)

T-A2. Monotonicity of Sigmoid Closure. $u \leq v$ pointwise $\implies \mathrm{Cl}_t(u) \leq \mathrm{Cl}_t(v)$ pointwise, for any parameters. Proof: $P_t$ preserves ordering; $\sigma$ is monotone. (R4.)

T8-Core. Non-Trivial Minimizer Existence (Phase Transition). Let $X_t$ be a finite connected graph with Fiedler eigenvalue $\lambda_2 > 0$, $m = cn$ with $c \in \left(\frac{3-\sqrt{3}}{6}, \frac{3+\sqrt{3}}{6}\right)$, and let $W(u) = u^2(1-u)^2$ be the double-well potential. If $\beta/\alpha > 4\lambda_2/|W''(c)|$, the global minimizer of $\mathcal{E}_{\mathrm{bd}}|_{\Sigma_m}$ is non-uniform. Proof: The ordered-pair summation gives smoothness functional $2\alpha v^T L v$ with Hessian $4\alpha L$. Second variation at $u \equiv c$ has eigenvalue $4\alpha\lambda_2 + \beta W''(c) < 0$ when $\beta/\alpha > 4\lambda_2/|W''(c)|$. The uniform state is a saddle point; the global minimizer (which exists by T1) must be non-uniform. (R5, ratio corrected R13.)

Scaling Caveat. The phase transition threshold $\beta_{\mathrm{crit}} = 4\alpha\lambda_2/|W''(c)|$ depends on the spectral gap $\lambda_2$ of the graph Laplacian. For graph families where $\lambda_2 \to 0$ as $n \to \infty$ (e.g., $k \times k$ grids with $\lambda_2 \sim \pi^2/k^2$), the threshold vanishes and the criterion is trivially satisfied for any fixed $\beta > 0$. In this regime, T8-Core guarantees non-trivial minimizer existence but does not provide a meaningful selection criterion for parameters.

On graphs with $\lambda_2 = \Omega(1)$ (expander graphs, bounded-diameter graphs, fixed-size applications), the phase transition is a genuine constraint that separates formation-supporting from formation-suppressing parameter regimes.

On large grid-like graphs, the proto-cohesion diagnostic vector $\mathbf{d} = (\mathsf{Bind}, \mathsf{Sep}, \mathsf{Inside}, \mathsf{Persist})$ replaces the phase transition as the primary assessment tool. Formation quality, not formation existence, is the meaningful question at scale.

Remark (Finite-element rescaling). In the continuum limit interpretation, $\alpha$ plays the role of a diffusion coefficient and should scale with the mesh resolution: $\alpha = \alpha_0 / h^2$ where $h$ is the mesh spacing. For a $k \times k$ grid with $h = 1/k$, this gives $\alpha = \alpha_0 k^2$, and $\beta_{\mathrm{crit}} = 4\alpha_0 k^2 \lambda_2 / |W''(c)| \approx 4\alpha_0 \pi^2 / |W''(c)| = O(1)$, recovering a mesh-independent threshold. This rescaling is standard in numerical PDE but constitutes a parameter prescription rather than a theorem. It is recommended for computational implementations on large grids.

C-Axioms. Co-belonging Axiom Satisfaction. The resolvent realization $\mathbf{C}_t = (I - \alpha W_{\mathrm{sym}})^{-1}$ satisfies C1, C2, C3'', C4. Proof: C1 by construction; C2 by explicit witnesses (3 orders of magnitude discrimination); C3'' via conjugation identity and Schur complement analysis (Phase 9, 2026-04-03); C4 automatic from symmetrized kernel. (R6, C3-SYMMETRIZATION-COMPLETE.md.) C3'' closure: The u-Jacobian of the symmetrized operator $D^{-1/2}(u) W_{\mathrm{sym}} D^{-1/2}(u)$ was rigorously proved via Schur complement decomposition: the Neumann series inherits strict monotonicity from exact algebraic cancellation in the operator conjugation, eliminating the need for term-by-term analysis. Verified on all grid graphs with min degree ≥ 2. Spot-checks: 1e-8 FD agreement on 5×5 and 10×10 grids.

QM1–4. Q_morph Axiom Satisfaction. The normalized morphological quality $\mathcal{Q}_{\mathrm{morph}}(u) = \frac{\ell_{\max} - c}{1 - c} \cdot \mathrm{Artic}$, where $c = \sum_x u(x) / n$ is the mean field value, satisfies: QM1 (vanishing on uniform fields), QM2 (monotonicity in formation quality), QM3 (continuity in $u$), QM4 (discrimination of stratified vs. non-stratified fields). Proof: QM1 — for $u \equiv c$, the surviving component has $\ell_{\max} = c$, so the numerator $\ell_{\max} - c = 0$ and $\mathcal{Q}_{\mathrm{morph}} = 0$. (Note: the unnormalized form $\ell_{\max} \cdot \mathrm{Artic}$ yields $c \neq 0$ on uniform fields because $\mathrm{Artic} = 1$ when all merge bars have zero length; the normalization corrects this.) QM2 — both factors increase with structure. QM3 — $\ell_{\max}$ is Lipschitz via persistence stability theorem. QM4 — product structure. (R7.)

T14. Gradient Flow Convergence. The projected gradient flow $\dot{u} = -\Pi_{\Sigma_m} \nabla \mathcal{E}(u)$ on $\Sigma_m$ converges to a critical point. For analytic energy (sigmoid + polynomial terms, with $b_D = 0$), convergence is exponential via Łojasiewicz inequality. Proof: $\mathcal{E}$ is bounded below on compact $\Sigma_m$; gradient flow decreases energy monotonically; Łojasiewicz–Simon inequality for analytic functions on compact semi-algebraic sets. (R9.)

T3/T6-Stability. Non-Idempotent Stability Advantage. At a non-idempotent closure fixed point with $\|J_{\mathrm{Cl}}\|_{\mathrm{op}} < 1$, the closure Hessian $2(I - J_{\mathrm{Cl}})^T(I - J_{\mathrm{Cl}})$ is strictly positive definite ($n/n$ positive eigenvalues). For idempotent closure, the Hessian has zero eigenvalues in the range direction ($\leq (n-k)/n$ positive eigenvalues for $k$-dimensional range). Proof: Gram matrix analysis. (R4–R5.)

T7-Enhanced. Enhanced Metastability. Non-trivial constrained minimizers of SCC energy have strictly larger minimum Hessian eigenvalue than corresponding Allen-Cahn minimizers, due to the self-referential closure correction. This is a local curvature result; the gap between Hessian eigenvalue and actual energy barrier height (saddle energy minus minimizer energy) requires Morse-theoretic analysis that has not been carried out. Proof: The closure term $\lambda_{\mathrm{cl}}\mathcal{E}_{\mathrm{cl}}$ adds a positive-definite Hessian contribution at closure fixed points (Gram matrix $2(I - J_{\mathrm{Cl}})^T(I - J_{\mathrm{Cl}})$), increasing the minimum Hessian eigenvalue. (R9.)

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 sets with minimal perimeter subject to volume constraint. The self-referential correction terms modify the effective surface tension in the sharp-interface limit. Proof: Standard Modica-Mortola for the leading term; perturbation analysis for corrections. (R11.)

T8-Full. Non-Trivial Minimizer under Full Energy. (Upgraded from Category B, 2026-04-02.) Adding $\lambda_{\mathrm{cl}}\,\mathcal{E}_{\mathrm{cl}} + \lambda_{\mathrm{sep}}\,\mathcal{E}_{\mathrm{sep}}$ to $\mathcal{E}_{\mathrm{bd}}$ does not destroy the non-uniform minimizer for $\lambda_{\mathrm{cl}}, \lambda_{\mathrm{sep}}$ small relative to $\beta$. The IFT gives a smooth family of perturbed minimizers $\hat{u}(\lambda_{\mathrm{cl}}, \lambda_{\mathrm{sep}})$ with $\|\hat{u} - \hat{u}_0\| = O(\lambda_{\mathrm{cl}} + \lambda_{\mathrm{sep}})$. Proof: IFT on bordered KKT. Non-degeneracy ($\mu_0 > 0$) confirmed: at the $\mathcal{E}_{\mathrm{bd}}$ minimizer, $\mu_0 \in [0.96, 60.2]$ across all tested configurations. The earlier reported negative eigenvalue was at the $\mathcal{E}_{\mathrm{full}}$ minimizer (different point). Anti-concentration on the transition layer validates genericity. (GAP-CLOSURES.md §G9, CATEGORY-B-UPGRADES.md.)

Predicate-Energy Bridge. (Upgraded from Category B, 2026-04-02.) $\mathsf{Sep} = 1 - \mathcal{E}_{\mathrm{sep}}/m$ (exact bidirectional equality for $u$-weighted Sep). $\mathsf{Bind} \geq 1 - \sqrt{\mathcal{E}_{\mathrm{cl}}/n}$ (Cauchy-Schwarz; reverse at minimizers via KKT). Proof: Sep identity: direct algebraic verification. Bind: forward by Cauchy-Schwarz; reverse at constrained minimizers from the KKT equilibrium. (CATEGORY-B-UPGRADES.md.)

Deep Core Dominance 2b. (Upgraded from Category B, 2026-04-02.) $|\mathrm{Core}^2|/|\mathrm{Core}| \geq 1 - 4C/\sqrt{m}$ where $C$ is the isoperimetric ratio. On $\mathbb{Z}^d$ grids, $C$ is bounded by the discrete isoperimetric constant unconditionally. Proof: The isoperimetric inequality on $\mathbb{Z}^d$ provides $|\partial_V\mathrm{Core}| / |\mathrm{Core}| \leq C/\sqrt{|\mathrm{Core}|}$ with $C$ depending only on dimension. Combined with Deep Core Dominance 2a ($|\mathrm{Core}^2| = |\mathrm{Core}| - |\partial_V\mathrm{Core}|$), this gives the bound unconditionally for grid graphs. (CATEGORY-B-UPGRADES.md.)

T-Merge (a). K-Formation Local Minimality. (Phase 9, 2026-04-03.) Well-separated K-formations are local minima of the K-field energy on $\Sigma^K_M$. Hessian analysis at K-formation critical point; positive definiteness under $\mu_1\mu_2 > \lambda_{\mathrm{rep}}^2$. Proof: Minimizer geometry + overlap analysis + Hessian non-degeneracy. (MERGE-THEOREM.md §1.) Status: Proved, Cat A.

T-Merge (b). Energy Ordering (Isoperimetric). K=1 has lower energy than K=2 on connected graphs (isoperimetric consequence). K=1 global preference is an isoperimetric consequence (perimeter minimization via Γ-convergence); K>1 coexistence is kinetic, not thermodynamic. Proof: Perimeter minimization via Γ-convergence. (MERGE-THEOREM.md §2.) Status: Proved, Cat A (on grid graphs; general connected graphs by standard isoperimetric results).

Topological Lock. Merge Impossible on $\Sigma^K_M$. On the per-formation mass-constrained manifold $\Sigma^K_M$, merge endpoint $(u_{\mathrm{merged}}, 0) \notin \Sigma^K_M$ because $0 \notin \Sigma_{m_2}$ for $m_2 > 0$. Proof: Topological — the merge endpoint doesn't lie on the manifold. Cat A but vacuous as a merge barrier result (consequence of per-field mass constraint, not energy barrier). Status: Proved, Cat A (trivially true by construction of K-field architecture).

(Erratum 2026-04-07: T-Merge parts (c)(d)(e) RETRACTED — merge path doesn't exist on $\Sigma^K_M$, Mountain Pass theorem inapplicable. See Retracted section. Barrier exponent $\gamma_{\mathrm{eff}} \approx 0.89$ moved to Category B — empirical fit, no analytical derivation.)

T-Birth-Parametric. Supercritical Pitchfork Bifurcation in Formation Stability. At the critical parameter $\beta_{\mathrm{crit}} = 4\alpha\lambda_2/|W''(c)|$, a supercritical pitchfork bifurcation occurs: for $\beta < \beta_{\mathrm{crit}}$, only K=1 (uniform) minimizer; for $\beta > \beta_{\mathrm{crit}}$, two additional non-uniform minimizers appear with amplitude $|u_{\mathrm{nonunif}} - u_{\mathrm{unif}}| \propto (\beta - \beta_{\mathrm{crit}})^{1/2}$. The bifurcation is supercritical (stable branch) by Crandall-Rabinowitz theorem with cubic coefficient $A > 0$ from equivariant analysis (D₄ symmetry on square grids, cubic power symmetry via $W(u) = u(1-u)(1/2 - u)$). Proof: Crandall-Rabinowitz parametric bifurcation theorem. Cubic coefficient $A = 4\beta_{\mathrm{crit}} \cdot \Phi_4 > 0$ where $\Phi_4 = \int_\Omega \psi^4(x) dx$ with $\psi$ the eigenmode. Verified exp37 (zero hysteresis) and exp39 (topological birth). (FORMATION-BIRTH-THEOREM.md, exp37, exp39.) Status: Proved (D₄-symmetric, supercritical), Cat A. (General non-symmetric graphs: Cat B — supercriticality proved for D₄, general graph case incomplete; requires Cheeger/spectral clustering analysis.)

T-Beyond-Weyl. Structured Spectral Perturbation Bound for Multi-Formations. The joint K-formation Hessian spectral gap tightens from the standard Weyl bound $\mu_{\mathrm{joint}} \geq \min_k \mu_k - (K-1)\lambda_{\mathrm{rep}}$ to a formation-aware bound exploiting soft-mode localization:

The mathematical structured perturbation bound (Davis-Kahan + variational, under the relevant second-gap/localization hypotheses) is rigorous. The previously quoted 33× improvement is the special case $1/\omega_{\mathrm{soft}} \approx 33$ measured on a limited 12×12 well-separated configuration, not a universal constant. (Erratum 2026-04-10: The improvement factor should be reported as $1/\omega_{\mathrm{soft}}(B,O)$ or as a branch/geometric interval.) Status: (Erratum 2026-04-07: Downgraded to Category B. The mathematical bound formula is rigorous, but the "33× improvement" quantification is configuration-specific. See Category B section.) Moved to Category B.

T-d_min-Formula. Analytical Formula for Critical Inter-Formation Distance. The branch-conditioned minimum inter-formation distance enabling metastable coexistence should be written $d_{\min}^*(B,\mathrm{rule},\mathrm{params},G)$. The earlier scalar fit

has empirical support on the tested branch-selection protocol, but its coefficients are not analytically derived and are branch/graph/protocol dependent. With closure (SCC), tested protocols give $d_{\min}^* \approx 5$ nodes at $\beta=30,\alpha=1$; without closure (Allen-Cahn), $d_{\min}^* \approx 7$ nodes. (Erratum 2026-04-10: $d_{\min}^*$ is not a universal branch-free scalar. The qualitative closure-driven reduction is retained; coefficient-level formulas remain Category B.) Status: (Erratum 2026-04-07: Downgraded to Category B. The formula is a least-squares regression fit, not an analytical derivation.) Moved to Category B.

T-Bind-Proj. Tangential Residual Bound at Constrained Minimizers. (Moved here from former Category B — Phase 13 upgrade to Cat A for all τ.) Let $G = (X, \mathbf{N})$ be a finite connected graph with $n = |X|$ vertices. Let $\hat{u}$ be a global minimizer of $\mathcal{E} = \lambda_{\mathrm{cl}} \mathcal{E}_{\mathrm{cl}} + \lambda_{\mathrm{sep}} \mathcal{E}_{\mathrm{sep}} + \lambda_{\mathrm{bd}} \mathcal{E}_{\mathrm{bd}}$ on $\Sigma_m$ with strict interiority ($0 < \hat{u}_i < 1$ for all $i$), and suppose $a_{\mathrm{cl}} < 4$ (contraction regime). Let $r = \mathrm{Cl}(\hat{u}) - \hat{u}$ be the closure residual, $r_T = \Pi_T(r)$ its tangential component, and $\bar{r}_0 = |\mathbf{1}^T r|/n$ the per-site mean residual. Then: $\|r_T\|_2 \leq \frac{\lambda_{\mathrm{sep}} G_{\mathrm{sep}} + \lambda_{\mathrm{bd}} G_{\mathrm{bd}}}{2\lambda_{\mathrm{cl}}(1 - a_{\mathrm{cl}}/4)} + \frac{(1 + a_{\mathrm{cl}}/4)\sqrt{n}\,\bar{r}_0}{1 - a_{\mathrm{cl}}/4}$. Proof: KKT projection + Banach inversion of restricted operator with $\sigma_{\min} \geq 1 - a_{\mathrm{cl}}/4$. General $\tau$ via binary mass-balance formula $\Phi(\tau; a_{\mathrm{cl}}, c)$. Status: Proved, Cat A for all $\tau_{\mathrm{cl}} \in (0,1)$.

T-Bind-Full. Bind Lower Bound at Constrained Minimizers. $\mathsf{Bind}(\hat{u}) \geq 1 - f(\text{params})$, $n$-independent when parameters are $O(1)$. Proof: Follows from T-Bind-Proj + universal gradient bounds. Status: Proved, Cat A.

Proposition 1.1. Constraint Manifold Structure. (New, 2026-04-06 Stratified Morse Analysis.) $\Sigma_m$ is convex polytope, manifold with corners, contractible. Proof: Standard convex geometry. Status: Proved, Cat A.

Proposition 1.2. Fiber Dimension. (New, 2026-04-06 Stratified Morse Analysis.) Fiber dimension = $2n-2$ for interior mass splits; cone singularity at $m_2 \in \{0, M\}$. Proof: Direct dimension counting + singularity analysis. Status: Proved, Cat A.

Theorem 3.1(a,b,d). Landscape at Symmetric Point. (New, 2026-04-06 Stratified Morse Analysis.) (a) Tangent space decomposes $T = T_{\mathrm{intra}} \oplus T_{\mathrm{transfer}}$. (b) Intra-formation Hessian PD. (d) Symmetric point has $\mu_1 = \mu_2$, is critical on $\Sigma_M^{\mathrm{relax}}$. Proof: (a) Linear algebra. (b) Per-formation stability. (d) By symmetry. Status: Proved, Cat A.

Persistence Threshold Equation. (New, 2026-04-06 Stratified Morse Analysis.) Exact formula: $\beta > \Gamma(a_{\mathrm{cl}}, \tau, \eta, \lambda_{\mathrm{cl}}, \theta) \cdot \varepsilon_1^2 \cdot \alpha$ with $\Gamma = 4/(C_1^2 \cdot C_2^2)$ derived from KKT + closure recurrence + spectral scaling. Proof: Rigorous derivation verified on 87 test cases with zero violations for $\varepsilon_1 \leq 0.1$. Analytically derived (not fitted). Status: Proved, Cat A.

W4 close additions (2026-04-24)#

T-PreObj-1. Pre-Objective Multi-Peak Formation Mechanism. (New, W4 close, 2026-04-24.) Under full SCC parameters on a 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 the full energy $\mathcal{E}$; (ii) gradient flow attracts to multi-peak F≥2 configurations; (iii) IC-protocol dichotomy — adaptive bounded protocols converge to bounded endpoints, random initialization scales as ~ $L^{2.8}$. Proof: Lemma 4 (M positive definite) + destabilization analysis $\Lambda^T M \Lambda > 0$ + gradient flow concavity argument. Numerical verification: E-0090 (L=12, 3-digit theory–experiment agreement), E-0091 (L=32 dichotomy). Status: Proved, Cat A. (C-0700, P-0700.)

T-PreObj-1G. Pre-Objective Mechanism — Graph-Class Independent. (New, W4 close, 2026-04-24.) Conclusions (i) and (ii) of T-PreObj-1 hold on any finite connected graph under (G1)–(G4) hypotheses, not only on the test grid families. Proof: T-PreObj-1 argument upgraded to graph-class independence via abstract spectral / isoperimetric reasoning; theoretical and qualitative empirical confirmation. Status: Proved, Cat A. (C-0701, P-0701.)

Lemma 4. Quadratic Form Positive Definite (M matrix). (New, W4 close, 2026-04-24.) Let $M \in \mathbb{R}^{2 \times 2}$ be the inner-product matrix of the gradients $g_{\mathrm{cl}}, g_{\mathrm{sep}}$ at the F=1 candidate. Under linear independence of $g_{\mathrm{cl}}$ and $g_{\mathrm{sep}}$, $M$ is positive definite, and the destabilization magnitude $\Lambda^T M \Lambda > 0$ for any nonzero direction $\Lambda$. Proof: Standard PD criterion ($\det M > 0$, $\mathrm{tr}\,M > 0$) under the linear independence hypothesis. Status: Proved, Cat A. (C-0702, P-0702.)

F-1 Resolution Corollary. F-1 SPLIT-RESOLVED via T-Merge (b) + T-PreObj-1. (New, W4 close, 2026-04-24.) The F-1 problem (K=2 vacuity, OP-0001 in Part 4 · §12) decomposes into two layers, each Cat A: (i) the pure $\mathcal{E}_{\mathrm{bd}}$ portion is the correct theorem T-Merge (b) (isoperimetric ordering), not an open problem — original framing as "open problem" was a misclassification; (ii) the full SCC portion is resolved by T-PreObj-1 (i): under full SCC parameters, the F=1 single-disk minimizer of pure $\mathcal{E}_{\mathrm{bd}}$ is non-critical, so the dichotomy "K=1 cheaper vs observed K>1" does not arise. The premise of F-1 collapses. Status: Cat A corollary of T-Merge (b) + T-PreObj-1. M-1 (OP-0002) is similarly LAYER-CLARIFIED (proved theorem misframed as problem); MO-1 (OP-0003) is SIDESTEPPED by the σ-framework operating on $\Sigma_m$. Critical blocker count: 3 → 0.

W4-extended addition (2026-04-26)#

T-V5b-T. Pre-Objective Goldstone on Translation-Invariant Graphs. (New, W4-extended close, 2026-04-26.) On graph classes with full translation symmetry (torus $T^d$, cycle $C_n$): a sub/super-lattice spectral dichotomy holds at the F=1 / F≥2 transition; in 2D the Goldstone modes form a 2-fold doublet with commensurability splitting; in 1D a 1-fold Goldstone branch; the Goldstone nodal count = 2 universal. The crossover scale $\zeta_*(G)$ is graph-class dependent. Proof: Bloch-mode decomposition + symmetry argument + ζ-scan numerical verification. After 8 V5b iterations (V1 → V5b''). Numerical verification: E-0095 (NQ-170b ζ-scan), E-0096 (NQ-170c graph-class extension + nodal count), E-0097 (NQ-172 reproducibility check after mode-indexing artifact identified+resolved). Status: Proved, Cat A. (C-0710, P-0710.)

W5 Day 1 G0 — σ-Framework Supporting Structures (CV-1.5, 2026-04-27)#

The σ-framework signature $\sigma(u^*) = (\mathcal{F}; \{(n_k, [\rho_k], \lambda_k)\})$ — declared in Part 4 · §11.1 Commitment 14 as a Cat A definitional commitment in the W4 close — is grounded by five Cat A supporting structures canonical-merged on 2026-04-27 (W5 Day 1 G0). Each is individually canonical-visible (Option α decision: mathematically independent statements deserve separate §13 entries).

T-σ-Lemma-1. σ-Framework Irrep Decomposition Well-Defined. (New, W5 Day 1 G0, 2026-04-27.) The Hessian $H(u^*)$ commutes with the stabilizer $G_u = \mathrm{Stab}_{\mathrm{Aut}(G)}(u^*)$ action on $T_{u^*} \Sigma_m$, so its spectral decomposition admits a canonical isotypic projector along irreducible representations of $G_u$. Each eigenspace splits into $G_u$-isotypic components. Finite-graph hypothesis essential: Maschke's theorem fails on infinite groups absent compact-Lie or amenable extension. Trivial-stabilizer case is vacuous (single irrep, the trivial one). Proof: Maschke's theorem + Schur orthogonality on the finite group $G_u$ acting unitarily on $T_{u^*} \Sigma_m$. Status: Proved, Cat A. (C-0712, P-0712.)

T-σ-Lemma-2. σ-Framework Nodal Count Properties. (New, W5 Day 1 G0, 2026-04-27.) For $\phi_k \in \mathbf{1}^\perp$ a Hessian eigenvector, the nodal count $\mathcal{N}(\phi_k)$ satisfies: (i) graph-intrinsic (independent of basis choice); (ii) $\mathrm{Aut}(G)$-equivariant under $G_u$-orbit action; (iii) lower bound $\mathcal{N}(\phi_k) \geq 2$ on $\mathbf{1}^\perp$ (from the constraint $\sum \phi_k = 0$ — corrected from the plan-template wording "$n_k = 1$ iff constant", which was incorrect: a constant in $\mathbf{1}^\perp$ requires $\phi_k = 0$); (iv) sign-flip invariance. Sub-statements (i)-(iv) Cat A. Sub-statement riders Cat C: (v) Courant nodal upper bound; (vi) $G_u$-orbit divisibility (vacuous for $G_u$-invariant $\phi_k$). Proof: (i)-(iv) graph-theoretic + variational; numerical confirmation E-NQ-141 (W4-04-25, R23 32×32). (v)-(vi) deferred Cat C riders within parent Cat A entry. Status: Proved (i)-(iv) Cat A; (v)-(vi) Cat C riders bundled. (C-0713, P-0713.)

T-σ-Lemma-3. Goldstone–ℓ=1 Angular Saturation. (New, W5 Day 1 G0, 2026-04-27.) In the continuum limit, the IBP saturation identity $\mathcal{P}_{\ell=1}[\delta u_x] = (-c_d \int_0^\infty u^*(r)\, dr,\ 0)$ (with dimension constants $c_1 = 2$, $c_2 = \pi$, $c_3 = 4\pi$) shows the Goldstone basis automatically lies in the $\ell=1$ angular subspace and saturates Cauchy–Schwarz. Consequence: Goldstone nodal count = 2 universal on translation-invariant graph classes (anchors T-V5b-T-(e), removing a forward reference). Holds for general dimension $d$ (1D cycle, 2D/3D bulk, torus). Proof: IBP polar-Cartesian conversion on tanh-profile localized minimizer + rank/injectivity argument. (Erratum 2026-04-27 evening: initial canonical merge stated $-m$ with mass; the correct identity has factor $\int u^*(r)\, dr \approx r_0$, differing from $m = c_d^{(\mathrm{vol})} r_0^d$ by a factor of $r_0$. W4-04-24 source had a Jacobian error in polar-Cartesian conversion.) Status: Proved Cat A in continuum, discrete correction NQ-180 (W6+). (C-0714, P-0714.)

T-σ-Theorem-3. σ at Uniform on $D_4$ Free-BC Grid (Closed Form). (New, W5 Day 1 G0, 2026-04-27.) At the uniform configuration $u^* \equiv c$ on a $D_4$-symmetric free-BC grid, the Hessian eigenvalues are $\mu_k = 4\alpha\lambda_k^{\mathrm{Lap}} + \beta W''(c)$ where $\lambda_k^{\mathrm{Lap}}$ are the Laplacian eigenvalues. Full $D_4$ irrep table on the cosine basis $\phi_{(p,q)} = \cos(p\pi i/L)\cos(q\pi j/L)$ with nodal count $\mathcal{N}(\phi_{(p,q)}) = (p+1)(q+1)$. Numerical verification: NQ-141 ($L = 4$ to $32$, $< 10^{-9}$ precision). Proof: Diagonalization in cosine basis + character-orthogonality calculation for $D_4$ irrep assignment. (Erratum 2026-04-27 evening: initial irrep-table off-diagonal-pair entries replaced with rigorous Schur-orthogonality character calculation: both-odd → $A_2 \oplus B_2$; mixed parity → single $E$; even pair → $A_1 \oplus B_1$.) Status: Proved, Cat A. (C-0715, P-0715.)

T-σ-Theorem-4. σ at First Pitchfork on $D_4$ Free-BC Grid (Leading Order). (New, W5 Day 1 G0, 2026-04-27. Retroactive Cat A → Cat B at CV-1.5.1, 2026-04-29.) At the first pitchfork on a $D_4$-symmetric free-BC grid, the symmetry breaks $D_4 \to \mathbb{Z}_2$. Mode 0 lies in the trivial irrep $[+1]$ of the residual $\mathbb{Z}_2$; Mode 1 lies in the sign irrep $[-1]$. (Erratum 2026-04-27 evening: initial entry stated "$0 < K_1 < K_0$ would-be transverse Goldstone"; corrected to $K_1 = K_0$ at leading order. Refinement 2026-04-27 night: $K_0 = K_1$ degeneracy requires Commitment 14 (O7) tie-breaking convention.) Status: Cat B target. Retroactive demotion at CV-1.5.1 per Critic 7-agent verdict + NQ-187 numerical refutation (continuum-vs-discrete caveat: NQ-187 measured $\mu_1/\mu_0 \approx 2$ not 1; effective coefficient $\approx 1\cdot|W''(c)|$ not $4\cdot|W''(c)|$). Three reconciliation hypotheses (α/β/γ) under audit; Cat A re-promotion deferred to CV-1.7+ via γ-path Σ_m-Hessian convention audit. See Category B section for the row-level Cat B flag. (C-0716, P-0716.)

W5 Day 3 EOD — D-6a Multi-Static Cluster (CV-1.5.1, 2026-04-29)#

The D-6a cluster extends Commitment 14 (σ-signature) to the multi-formation interior $\widetilde{\Sigma}^{K_{\mathrm{field}},\,\circ}_M = \widetilde{\Sigma}^{K_{\mathrm{field}}}_M \setminus \partial\widetilde{\Sigma}$ via three Cat A definitional entries. Corner-saturated boundaries are excluded per Option A pragmatic — full corner handling via Option B stratified Morse deferred to NQ-248 (W7+); see §11.1 Commitment 16 K-status decomposition. Dynamic extension $\sigma_{\mathrm{multi}}^A(t)$ deferred to CV-1.6 via NQ-242 (D-6b path: full Hessian σ-tuple time-series + rigorous K-jump theory; OP-0008 σ^A K-jump non-determinism).

T-Commitment-14-Multi-Static. Multi-Formation σ-Signature on Interior. (New, CV-1.5.1, 2026-04-29.) Extends Commitment 14 to multi-formation regime on $\widetilde{\Sigma}^{K_{\mathrm{field}},\,\circ}_M$. For a K-field minimizer $\mathbf{u}^* = (u^{(1)*}, \ldots, u^{(K_{\mathrm{field}})*}) \in \widetilde{\Sigma}^{K_{\mathrm{field}},\,\circ}_M$ with $K_{\mathrm{act}}(\mathbf{u}^*) \leq K_{\mathrm{field}}$, the σ-signature is well-defined on the smooth interior (no corners). Status: Cat A definitional (extension of Commitment 14 to multi-formation; well-defined on interior $\widetilde{\Sigma}^{K_{\mathrm{field}},\,\circ}_M$ per Option A; corner-saturated regime open per OP-0008 + OP-0009).

T-σ-multi-A-Static. Multi-Formation σ-A Component (Static). (New, CV-1.5.1, 2026-04-29.) σ-A component of the multi-formation Hessian signature, well-defined in the well-separated regime. Status: Cat A in well-separated regime; Cat B target in weakly-interacting regime (T-Persist-K-Weak overlap; per-block Hessian no longer block-diagonal).

T-σ-multi-D-Static. Multi-Formation σ-D Component (Static). (New, CV-1.5.1, 2026-04-29.) σ-D component of the multi-formation Hessian signature, well-defined in the well-separated regime. Status: Cat A in well-separated regime; Cat B target in weakly-interacting regime.

W5 Day 3 EOD — V5b-T sub-statement (CV-1.5.1, 2026-04-29)#

V5b-T-zero. Sub-Spinodal Goldstone Zero (Exact). (New, CV-1.5.1, 2026-04-29.) Sub-spinodal exact statement: $\mu_{\mathrm{Gold}} = 0$. New Cat A definitional sub-statement under V5b-T parent Cat A entry. Replaces V5b-T' (WITHDRAWN at CV-1.5.1 due to NQ-198f phantom on torus). Status: Cat A definitional, sub-spinodal exact.

W5 Day 6 — T-L1-F Hard-Bar / Active-Count Bridge (CV-1.5.2, 2026-05-02)#

T-L1-F. Hard-Bar / Active-Count Bridge under the L1-J Regime. (New, CV-1.5.2, 2026-05-02; first multi-formation canonical Cat A theorem.) Let $G=(X,E)$ be a finite graph and $\mathbf u\in\widetilde{\Sigma}^{K_{\mathrm{field}}}_M(G)$ a shared-pool multi-formation state. Let $U(\mathbf u)=\sum_{j=1}^{K_{\mathrm{field}}}u^{(j)}$, $A^\varepsilon(\mathbf u)=\{j:m_j(\mathbf u)>\varepsilon\}$, $K_{\mathrm{act}}^\varepsilon=|A^\varepsilon|$, and $K_{\mathrm{bar}}^{\ell_{\min}}(U;G)=\#\{[d,b]\in\mathrm{Bars}_0^{\mathrm{term}}(U;G):b-d\ge\ell_{\min}\}$ under terminal-death $H_0$ superlevel persistence. Under the L1-J regime hypothesis package $(P0)$–$(P11)$ — namely P0 terminal-death convention, P1 deterministic tie convention $\prec$, P2 active mass + connected $\delta$-support, P3 LG-1 disjoint active neighborhoods $N_j^r\cap N_k^r=\emptyset$, P4 LG-2 low boundary collar $\max_{\partial N_j^r}U\le b_j-\ell_{\min}-r_{\mathrm{assoc}}$, P5 LG-4 background suppression on $U$ (not just $R_{\mathrm{inact}}$), P6 birth height $b_j\ge h_{\min}\ge\ell_{\min}$, P7 decay-to-cut (heterogeneous), P8 tightened H6 on $G_j^r$, P9 NE-2 perturbation, P10 inactive residual, P11 margin ledger — we have

$K_{\mathrm{bar}}^{\ell_{\min}}(U(\mathbf u);G)=K_{\mathrm{act}}^\varepsilon(\mathbf u),$

AND a labelled bijection $\mathcal{A}_{\mathrm{bar}}: A^\varepsilon \to \mathrm{Bars}_0^{\mathrm{term}}$ defined by $\mathcal{A}_{\mathrm{bar}}(j)$ = unique dominant bar with birth at $q_j^U$.

Proof: Lower bound $K_{\mathrm{bar}}\ge|A|$ via LG-2 boundary collar, LG-3 inter-neighborhood bridge, $h_{\min}\ge\ell_{\min}$ (L1-H §8 step 2). Upper bound $K_{\mathrm{bar}}\le|A|$ via (α) LG-7 coverage derived from LG-4 + terminal-death; (β) per-neighborhood at-most-one-dominant-bar via L1-H2 Lemma 1 (graph-inclusion: $\ell_{\mathrm{glob}}\le\ell_{\mathrm{loc}}$ on $G_j^r\subseteq G$) + L1-H2 Lemma 2 (contradiction-based bottleneck-stability). PO-1 decay-to-cut (P7) bounds $\theta_{\mathrm{bridge}}^{jk}(U)$ via L1-J §8.1 + L1-B Cat-A cut lemma. (L1-A through L1-L chain in THEORY/working/MF/; post-L1-K external audit + L1-K-REPAIR R-1..R-4; post-L1-L P7 status decision.)

Status: Proved, Cat A conditional under the L1-J regime package $(P0)$–$(P11)$. NOT a global identity. Does NOT establish $K_{\mathrm{soft}}=K_{\mathrm{act}}$ (additionally requires $\phi\in\Phi_{\mathrm{res}}$ per WQ-LAT-1.B). Does NOT solve OP-0005 (K-Selection) or OP-0008 ($\sigma^A$ K-jump non-determinism). P7 adopted as safe technical regime hypothesis; L1-L Combes-Thomas / Agmon analysis provides backing under strong stationarity but P7 is not asserted for all SCC states. The L1-J regime is empirically non-vacuous (L1-I 439/1920 = 22.9% feasible on $T^2_{20}$ with raw_gaussian initial states); production WQ-1 trajectories with mass-projection initial states typically exit the regime, so T-L1-F's reach in production dynamics is narrow.

W6 Stereo-SCC + Boundary Precision (CV-1.6 / CV-1.7, 2026-05-04 to 2026-05-06)#

T-ST-5a. Hard-Depth Topological Locking. (New, CV-1.6, 2026-05-04; Stereo-SCC branch.) On a depth-disconnected stereo graph where objects A and B are at depths $z_1 \ll z_2$ (graph-metric gap $\Delta_{\mathrm{depth}} \geq \Delta_{\mathrm{min}}^{\mathrm{ST}}$), the energy barrier between the two formation basins is $\Delta E = +\infty$ — the two depth-connected components are state-space disconnected via the Goldstone mechanism. The system is forced to $K=2$ stable formations, one per depth-connected component. Proof: Depth-disconnected graph has a block-diagonal Laplacian; the Goldstone mode connects only within each block. The merge path between depth-components requires traversing a forbidden region; the potential barrier diverges as the depth gap grows. (C-ST-5a, P-ST-5a.) Status: Proved, Cat A. First theorem connecting SCC to stereo geometry; proves perceptual individuation is forced by depth topology alone.

T-OP6-B. PersRidge Boundary Equivalence — OP-0006 RESOLVED. (New, CV-1.7, 2026-05-06.) In the phase-separation regime (hypotheses H1–H5), the PersRidge locus and the formation boundary $\partial\Omega^*$ satisfy: $d_H(\mathrm{PersRidge}(u^*), \partial\Omega^*) \leq 2\sqrt{\alpha/\beta}.$ Proof: Persistence stability theorem (Cohen-Steiner et al.) + Γ-convergence profile (T11) give boundary localization; PersRidge birth–death gap controls the Hausdorff distance. (C-OP6-B, P-OP6-B.) Status: Proved, Cat A. OP-0006 (Boundary Definition Precision) RESOLVED: the computationally accessible PersRidge is a provably faithful proxy for the theoretical boundary, with explicit geometric error bound.

W6 Stochastic Foundation — P-F-A1 Package I (CV-1.7 / CV-1.8 / CV-1.9, 2026-05-06)#

T-P-F-ε0. Gibbs Measure Continuity ($\varepsilon \to 0$). (New, CV-1.7, 2026-05-06.) The Gibbs measure $\pi_T^\varepsilon$ on $\widetilde{\Sigma}_M(\varepsilon)$ converges weakly to $\pi_T^0$ (a probability measure concentrated on the set of energy minimizers of $\tilde{\mathcal{E}}$ on $\partial\widetilde{\Sigma}_M$) as $\varepsilon \to 0$. This establishes Gibbs continuity at the constraint manifold boundary. Proof: Laplace method on bounded convex polytope (T-PF-A1-AR) + weak convergence of normalizing constants. (C-PF-A1-ε0, P-PF-A1-ε0.) Status: Proved, Cat A.

T-PF-A1-AR. Affine Reduction — Bounded Convex Polytope. (New, CV-1.8, 2026-05-06.) The feasibility polytope $\mathcal{F}_M(\mathcal{P}) = \{\mathbf{u} \in \widetilde{\Sigma}_M : \mathcal{P}(\mathbf{u}) \leq 0\}$ (where $\mathcal{P}$ encodes the affine constraints of the shared-pool architecture) is a bounded compact convex polytope. The Lions-Sznitman domain condition (uniform exterior sphere condition) is satisfied with explicit radius $r_{\mathrm{ext}}$. Proof: Linear inequalities on a simplex product; boundedness from simplex compactness; convexity from affine constraints; exterior sphere from polyhedral geometry. (C-PF-A1-AR, P-PF-A1-AR.) Status: Proved, Cat A. Geometric foundation for the well-posed reflected SDE.

T-PF-A1-SDE. Lions-Sznitman Reflected SDE. (New, CV-1.8, 2026-05-06.) The reflected Langevin SDE on $\widetilde{\Sigma}_M$, $d\mathbf{u}_t = -\nabla\tilde{\mathcal{E}}(\mathbf{u}_t)\,dt + \sqrt{2T_*}\,dW_t + d\mathbf{k}_t,$ where $\mathbf{k}_t$ is the inward-pointing reflection process, is well-posed: existence via Lions-Sznitman Theorem 1 (convex domain, T-PF-A1-AR), uniqueness via Tanaka pathwise argument. Proof: T-PF-A1-AR supplies the domain condition; Lions-Sznitman (1984) Theorem 1 gives strong solution existence; Tanaka formula provides pathwise uniqueness. (C-PF-A1-SDE, P-PF-A1-SDE.) Status: Proved, Cat A. Establishes that SCC energy has a well-defined stochastic dynamics on the constraint manifold.

T-PF-A1-GI. Unique Gibbs Invariant Measure $\pi_{T_*}$. (New, CV-1.9, 2026-05-06.) The Gibbs measure $\pi_{T_*} \propto \exp(-\tilde{\mathcal{E}}(\mathbf{u})/T_*)$ on $\widetilde{\Sigma}_M$ is the unique invariant measure of the reflected SDE (T-PF-A1-SDE). Uniqueness follows from: (i) zero-current condition (detailed balance for reflected diffusion); (ii) $L^2(\pi_{T_*})$ kernel triviality (Aronson 1968 heat kernel bounds + irreducibility of the reflected process on connected polytope). Proof: Kolmogorov criterion for reversibility; Aronson heat kernel lower bound $p(t,x,y) \geq c t^{-n/2}$ rules out absorption; irreducibility on the connected polytope (T-PF-A1-AR) gives uniqueness. (C-PF-A1-GI, P-PF-A1-GI.) Status: Proved, Cat A. Proves that temperature $T_*$ canonically selects a distribution over formation states.

T-PF-A1-PE. Poincaré Inequality + Exponential Ergodicity. (New, CV-1.9, 2026-05-06.) The reflected SDE (T-PF-A1-SDE) with invariant measure $\pi_{T_*}$ (T-PF-A1-GI) satisfies a Poincaré inequality with explicit constant: $\lambda_1 \geq \frac{\pi^2}{n}\exp\!\left(-\frac{\mathrm{osc}(\tilde{\mathcal{E}})}{T_*}\right)$ (Payne-Weinberger on bounded convex polytope + Holley-Stroock perturbation lemma). Consequently, the law $\mu_t$ of $\mathbf{u}_t$ converges to $\pi_{T_*}$ exponentially in total variation: $\|\mu_t - \pi_{T_*}\|_{\mathrm{TV}} \leq C e^{-\lambda_1 t}$. Proof: Payne-Weinberger spectral gap for convex domains applied to $\mathcal{F}_M(\mathcal{P})$ (T-PF-A1-AR); Holley-Stroock logarithmic Sobolev perturbation by $\mathrm{osc}(\tilde{\mathcal{E}})/T_*$; exponential decay from spectral gap + detailed balance. (C-PF-A1-PE, P-PF-A1-PE.) Status: Proved, Cat A. Dynamical completeness result: any initial condition converges to $\pi_{T_*}$ exponentially fast, with explicit Poincaré constant.

Category B: Proved with Explicit Structural Parameter (14 theorems)#

(Erratum 2026-04-07: T-Bind-Proj/Full moved to Category A above. T-Persist-K-Sep moved to Category C — regime conditions are non-removable structural hypotheses, making it conditional. T-Beyond-Weyl, T-d_min-Formula, barrier exponent γ_eff, and general-graph birth moved here from former Category A. CV-1.5.1 update 2026-04-29: T-σ-Theorem-4 retroactive Cat A → Cat B per Critic verdict; T-σ-Multi-1 registered as Cat B target for Goldstone-pair instability. W6 update (CV-1.7 → CV-1.11, 2026-05-06): T-P-F-ε0-K (Kramers exponent stability), T-ST-5b (smooth-depth barrier raising), D-ST-1..5 (stereo-SCC diagnostic bridge entries), T-K-Select-PF (partial OP-0005-EQ), T-K-Select-OBS (partial OP-0005-OBS).)

W6 Category B additions (CV-1.7 → CV-1.11, 2026-05-06)#

T-P-F-ε0-K. Kramers Exponent Stability. (New, CV-1.7, 2026-05-06.) The Kramers escape rate exponent for tunneling between formation basins is stable under small $\varepsilon$ perturbations of the constraint manifold boundary: the leading-order exponent is preserved and exponential-rate corrections are sub-leading. Cat B — requires explicit spectral-gap lower bound condition $\lambda_1 \geq \lambda_{\min}^{\mathrm{Kr}}$. Status: Cat B.

T-ST-5b. Smooth-Depth Barrier Raising. (New, CV-1.11, 2026-05-07.) On smooth-depth stereo graphs (objects at depths $z_1 < z_2$ not sharply separated), the merge barrier $\Delta E_{\mathrm{merge}}(\Delta z)$ is monotonically increasing in $\Delta z = z_2 - z_1$, with $\Delta E_{\mathrm{merge}} \to +\infty$ as $\Delta z \to \infty$ (recovering T-ST-5a in the limit). Cat B — requires explicit depth-gap lower bound condition $\Delta z \geq \Delta_{\min}$. Status: Cat B.

D-ST-1..5. Stereo-SCC Diagnostic Bridge Entries. (New, CV-1.10, 2026-05-06/07.) Five Cat B diagnostic bridge entries connecting the stereo-graph depth structure (depth-disconnected vs smooth-depth graphs) to the SCC formation diagnostics $\mathbf{d} = (\mathrm{Bind}, \mathrm{Sep}, \mathrm{Inside}, \mathrm{Persist})$ and the σ-signature. Each entry is conditional on its corresponding depth-regime hypothesis. Status: 5 × Cat B.

T-K-Select-PF. Equilibrium K-Selection via Gibbs Measure. (New, CV-1.10, 2026-05-06/07.) Under the Gibbs measure $\pi_{T_*}$ (T-PF-A1-GI), the active formation count $K_{\mathrm{act}}$ concentrates near the Gibbs-weighted modal value at low temperature. This partially resolves OP-0005-EQ (equilibrium branch of the K-Selection problem). Does NOT solve OP-0005-DYN (Eyring-Kramers, W9+). Status: Cat B — conditional on low-temperature regime and spectral gap condition.

T-K-Select-OBS. Observation-Conditioned K-Selection. (New, CV-1.10, 2026-05-06/07.) Under observation conditioning $\pi_{T_*}(\cdot \mid \mathcal{O})$ (where $\mathcal{O}$ is an observation event), the posterior over $K_{\mathrm{act}}$ is computable via Bayes theorem on the Gibbs measure. This partially resolves OP-0005-OBS (observation branch). Does NOT solve OP-0005-DYN. Status: Cat B — conditional on observation model regularity conditions.

T-σ-Theorem-4. σ at First Pitchfork on $D_4$ Free-BC Grid (Leading Order). (Retroactive Cat A → Cat B at CV-1.5.1, 2026-04-29.) The leading-order $K_0 = K_1$ degeneracy with cubic-coefficient ratio $A_2/A_1 = 4$ on $D_4$ at the first pitchfork is the original CV-1.5 (W5 Day 1 G0) entry, retroactively demoted at CV-1.5.1 per Critic 7-agent verdict + NQ-187 numerical refutation. Continuum-vs-discrete caveat: NQ-187 measured $\mu_1/\mu_0 \approx 2$ not 1; effective coefficient $\approx 1\cdot|W''(c)|$ not $4\cdot|W''(c)|$. Three reconciliation hypotheses under audit: (α) discrete eigenmode normalization correction, (β) $D_4$ characters indexing convention drift, (γ) Σ_m-Hessian projection convention (current candidate; full audit deferred to CV-1.7+). Status: Category B target. Cat A re-promotion attempt deferred to CV-1.7+ via γ-path Σ_m-Hessian convention audit (W6 G3 candidate).

T-σ-Multi-1. σ-Framework Multi-Formation Goldstone-Pair Instability. (New Cat B target, CV-1.5.1, 2026-04-29.) Multi-formation σ-framework Goldstone-pair instability statement. Working-grade target for canonical promotion; conditional on Commitment 14-Multi machinery (D-6a Cat A entries above) and per-block Hessian decomposition in the weakly-interacting regime. Status: Category B target.

Barrier Exponent $\gamma_{\mathrm{eff}} \approx 0.89$. Merge barrier scaling is branch/path/manifold conditioned. Asymptotic analysis for a specified source branch, target manifold, endpoint, and path class may give a two-term form $\Delta E_B = A_B\beta + B_B\sqrt{\beta}$, predicting local effective exponents tending to $1$ as $\beta \to \infty$. The quoted exponent 0.89 is an empirical fit from exp38 (R² = 0.997) over $\beta \in [20,100]$ for a particular protocol, not a branch-free invariant. (Erratum 2026-04-10: Branch-free notation such as $F''(M/2)$ or "the merge barrier" is ambiguous unless the selected K=2 branch, relaxed/constrained target manifold, endpoint, and admissible path class are specified.) Status: Category B. The EXISTENCE of a positive barrier may be theorem-level for a valid specified branch/path; the EXPONENT 0.89 is empirical and branch/path/manifold conditioned.

T-Birth-Parametric — General (Non-D₄) Graphs. Supercriticality on general non-symmetric graphs: requires Cheeger/spectral clustering analysis not yet completed. D₄-symmetric case proved (Cat A above). General graph case: supercriticality coefficient $A > 0$ requires eigenmode $\Phi_4 = \int \psi^4 > 0$, guaranteed by D₄ symmetry but not proved for arbitrary graphs. Validated on 32 graphs experimentally. Status: Category B. Experimental validation is not a proof.

T-d_min-Formula. Critical Inter-Formation Distance. (Erratum 2026-04-07: Downgraded from Cat A.) The branch-conditioned threshold should be written $d_{\min}^*(B,\mathrm{rule},\mathrm{params},G)$. The earlier scalar fit $d_{\min}^* = 4.8 + 0.31\sqrt{\beta/\alpha} - 0.018\beta/\alpha$ has strong empirical fit (R² = 0.987) on 20+ configurations, but the coefficients are not analytically derived and depend on branch/selection protocol and graph geometry. Analytical bounds from Sobolev trace inequality give scaling but not specific coefficients. The single-site Gram boost analysis predicts 0.3% reduction vs 30% observed — "collective Gram boost" scaling argument is dimensionally incorrect (DMIN-FORMULA.md). (Erratum 2026-04-10: $d_{\min}^*$ is not a universal branch-free scalar; any quantitative formula must specify branch, tie-breaker, parameters, and graph class.) Status: Category B. Qualitative result (closure reduces the observed critical distance in tested protocols) is retained; quantitative coefficient formula is Cat B.

T-Beyond-Weyl. Structured Spectral Perturbation Bound. (Erratum 2026-04-07: Downgraded from Cat A.) $\mu_{\mathrm{joint}} \geq \min_k \mu_k - (K-1)\lambda_{\mathrm{rep}} \cdot \omega_{\mathrm{soft}}$, where $\omega_{\mathrm{soft}} = \max_{j \neq k} \|\mathcal{P}_{O_{jk}}\psi_k^{\mathrm{soft}}\|^2$ is branch/overlap-geometry dependent. The mathematical structured perturbation bound (Davis-Kahan + variational, under the relevant second-gap/localization hypotheses) is rigorous. The "33× improvement" is the special value $1/\omega_{\mathrm{soft}} \approx 33$ when $\omega_{\mathrm{soft}} \approx 0.03$ in a limited 12×12 well-separated configuration (exp46-47); it is not a universal constant. (Erratum 2026-04-10: The improvement factor should be reported as $1/\omega_{\mathrm{soft}}(B,O)$ or as a branch/geometric interval, not as a universal 33× factor.) Status: Category B. The structured bound formula is theorem-level under explicit conditions; the QUANTITATIVE CLAIM (33× improvement, extended coexistence window) is Cat B — verified on limited configurations.

Category C: Conditional (5 theorems)#

(Erratum 2026-04-07: T-Persist-K-Sep moved here from Category B — regime conditions WS, SR are non-removable structural hypotheses. T-Persist-1(d) and T-Persist-Full listed here as conditional. Total 5 Cat C: T-Persist-1(d), T-Persist-Full, T-Persist-K-Sep, T-Persist-K-Weak, T-Persist-K-Unified.) (Erratum 2026-04-10: These Category C entries should be read as regime/branch-conditioned theorems, not merely incomplete proofs. Shifted-threshold persistence is robust; exact-threshold persistence is a deep-core theorem requiring positive interior gap; K-formation persistence requires a selected branch and its regime hypotheses.)

T-Persist-K-Sep. Multi-Formation Temporal Persistence (Well-Separated Regime). (Moved from Category B, 2026-04-07.) Let $(u^1_t, \ldots, u^K_t)$ be a well-separated joint minimizer with $d_{\min}(j,k) \geq D_{\text{sep}} \geq 3$ for all $j \neq k$. Under per-formation hypotheses (H1-K), well-separation (WS), and spectral-repulsion compatibility (SR: $\min_k \mu_k > (K-1)\lambda_{\text{rep}}$), an $\varepsilon$-gentle transition preserves per-formation minimizers, separation, core inclusion, transport concentration, and simplex constraint. Relies on Coupling Bound Lemma. Proof: Coupling Bound Lemma provides positive joint spectral gap via Weyl bound under (SR); exponentially small gradient perturbation at core sites ensures IFT displacement matches single-formation results. Full statement in Part 4 · §12. Status: Category C — conditional on per-formation T-Persist-1 hypotheses, well-separation WS, and spectral-repulsion SR. These are non-removable regime definitions. Implementation: scc/multi.py function transport_k_formations.

T-Persist-1. Temporal Persistence (Core Inheritance under Transport). Let $\hat{u}_t$ be a formation at time $t$ and $\hat{u}_s$ the minimizer at time $s$ with $|t - s|$ small. The persistence theorem has five components:

(a) Minimizer Persistence via IFT. If the constrained Hessian at $\hat{u}_t$ is non-degenerate, the IFT guarantees a smooth family of minimizers $\hat{u}_s(\delta)$ with $\|\hat{u}_s - \hat{u}_t\| = O(\delta)$ where $\delta$ parametrizes the perturbation. Status: Proved (Gap 2 closed). (I7, GAP-CLOSURES.md §G2.)

(b) Gradient Flow Convergence to New Minimizer. The gradient flow at time $s$, initialized from transported time-$t$ data, converges to a minimizer that inherits the formation structure. Status: Proved (Category A, Phase 12 upgrade via Theorem BC' + Theorem PSM). (Erratum 2026-04-03: Unconditional upgrade via Kupka-Smale genericity (NB removal) + Sard's theorem (GT absorption) — see T-PERSIST-1B-UNCONDITIONAL.md. Directional basin containment (Theorem BC', BC-PRIME-THEOREM.md) has ellipsoidal radius $r_{\mathrm{eff}} = \sqrt{2\Delta_{\mathrm{bdy}}/(f_1^2\mu + (1-f_1^2)\mu_2)}$ providing 2.5-4.3× improvement over isotropic bound $r_{\mathrm{iso}}$, extending persistence to any μ > 0. Soft-mode fraction analytically proved: $f_1^{\mathrm{grad}} \leq \sqrt{n_{\mathrm{bdy}}/n_F}$ via four-lemma chain (Lemma HDG, BMD, TC-DIR, volume orthogonality) — Theorem PSM (F1-BOUND-CATA-UPGRADE.md, Cat A). All basin components now Category A: Proposition BMD (boundary-mode dominance, BASIN-ESCAPE-ANALYSIS.md), Theorem BC' (directional radius), Theorem PSM (soft-mode fraction). Basin containment condition quantified: $2\varepsilon_2 + 2\varepsilon_1/\mu < r_{\mathrm{eff}}$ under gentleness $\varepsilon < r_{\mathrm{eff}}/(4+2/\mu)$, valid for all μ > 0.) (Erratum 2026-03-31: basin radius analysis corrected — see below.) Basin radius via energy sublevel set: $r_{\mathrm{basin}} = \sqrt{2\Delta_{\min}/\lambda_{\max}}$ where $\Delta_{\min} = \min(\Delta_{\mathrm{core}}, \Delta_{\mathrm{ext}}, \Delta_{\mathrm{bdy}})$. Core escape barrier: $\Delta_{\mathrm{core}} \geq 0.0441\beta$ ($\beta$-independent ratio $r_{\mathrm{core}} \geq 0.210$). Exterior escape barrier: $\Delta_{\mathrm{ext}} = O(\beta)$ per node (Proposition E1, proved). Boundary escape barrier: $\Delta_{\mathrm{bdy}}$ is formation-shape-dependent; $r_{\mathrm{bdy}} \geq 0.210$ away from shape bifurcation but can be small near bifurcation ($\mu \to 0$). Soft modes are boundary-dominated (~90% weight on boundary nodes, exp19/exp21-23); (Erratum 2026-04-01: boundary-mode dominance now analytically proved — Proposition BMD in BASIN-ESCAPE-ANALYSIS.md §8. The Hessian diagonal at core sites is ≥ 4α + 0.92β while boundary/spinodal sites can be as low as 4αd_max − β, forcing the minimum eigenvector onto the boundary subspace with core fraction O(1/β).) Barrier stability under gentle transition: $\Delta_s \geq \Delta_t - 2\varepsilon_1$ (Proposition 4). Basin containment holds when $\varepsilon_1 < \Delta_t/4$ and $2\varepsilon_2 + 2\varepsilon_1/\mu < r_{\mathrm{basin}}$ (Proposition 5). (Erratum 2026-04-01: quantitative $\Delta_{\mathrm{bdy}}$ formula derived — Taylor normal form along soft mode gives $\Delta_{\mathrm{bdy}} = \mu/2 \cdot (t^*)^2 + L_3/6 \cdot (t^*)^3 + L_4/24 \cdot (t^*)^4$ where $t^*$ solves $\mu + L_3 t/2 + L_4 t^2/6 = 0$; verified $<1\%$ error. Empirical basins are 3–12$\times$ larger than sublevel-set estimate (exp24), confirming conservativeness of the sublevel bound.) (I13, PERSIST-MORSE-ANALYSIS.md, BASIN-ESCAPE-ANALYSIS.md.)

(c) Core Inclusion with Shifted Threshold. The core of the transported formation $\{x : \hat{u}_s(x) \geq \theta - \epsilon\}$ contains the transported core $\{x : (\mathbf{M}_{t \to s} \hat{u}_t)(x) \geq \theta\}$ for perturbations of size $\epsilon$. Status: Proved. (I7.)

(d) Exact Threshold Preservation. The core at exact threshold $\theta$ (not $\theta - \epsilon$) is preserved. Status: Proved (Category C — requires structural condition $\beta > 7\alpha$; Phase 10 upgrade via H3 analytical bound). (Erratum 2026-04-10: Exact-threshold preservation is a protected deep-core statement, not a universal all-core statement. Boundary core sites retain the shifted-threshold fallback; near-bifurcation regimes retain only shrinking-window/remnant persistence.) (Erratum 2026-03-31: H2 upgraded to H2' — deep core existence proved.) The Interior Gap Lower Bound establishes that for core sites at graph distance $\delta(x) \geq 2$ from $\partial\text{Core}$: $\hat{u}(x) - \theta_{\text{core}} \geq (1 - \theta_{\text{core}}) - 2\exp(-c_0\,\delta(x)) - C_{\text{op}} R/\beta$, where $c_0 = \text{arccosh}(1 + \kappa^2/d_{\min})$, $\kappa = \sqrt{\beta/(2\alpha)}$, $C_{\text{op}} \leq 2(1 + a_{\text{cl}}/4) \cdot R$ with $R = (\lambda_{\text{cl}} + \lambda_{\text{sep}})/\lambda_{\text{bd}}$, giving $C_2 \leq 2.875$ at default parameters. Requires: (H2') Deep core existence: $\text{Core}^2(\hat{u}) \neq \emptyset$ — proved for $|\text{Core}| \geq 25$ via $\Gamma$-convergence isoperimetric analysis + discrete maximum principle contraction (Theorem 1, CORE-DEPTH-ISOPERIMETRIC.md). (Erratum 2026-04-01: threshold tightened from $\beta \geq 58\alpha$ to $\beta \geq 20\alpha$ (source-free: $8\alpha$; with cl/sep source: config-dependent, $\beta_{\mathrm{crit}} = 15\text{--}33\alpha$ depending on $\lambda_{\mathrm{cl}}/\lambda_{\mathrm{bd}}$ ratio after normalization). Verified universally at $\beta \geq 20$ in exp28/exp31.) Deep Core Dominance: unconditionally $|\text{Core}^2| = |\text{Core}| - |\partial_V\text{Core}|$ (Theorem 2a, identity); conditionally $|\text{Core}^2|/|\text{Core}| \geq 1 - 4C/\sqrt{m}$ under iso_ratio $\leq C$ (Theorem 2b). (H3—Cat A) Lagrange multiplier interior gap condition: β > 7α analytically proved via KKT + formation-conditioned Jacobian analysis (Phase 11 completion). 8 critical gaps closed via formal derivations: screened Poisson equation, W''' Taylor expansion (W''(1)=2 correction), C₂^eff weighting formula, mean-subtracted source cancellation, S_x bound, ν_eff sign resolution, β > 7α threshold justification, Poisson derivation formalization. Experimental validation: 490 configs, R² > 0.93. Audit score: 9/10 (all gaps verified closed, no new gaps introduced). (Phase 11: H3-ANALYTICAL-BOUND-FINAL.md, H3-FINAL-AUDIT-REPORT.md, CATEGORY-A-CERTIFICATION-FINAL.md. Phase 10: H3-KKT-ANALYSIS.md, H3-JACOBIAN-ANALYSIS.md, H3-PROOF-OUTLINE.md.)

(e) Two-Tier Transport Concentration. Entropy-regularized OT with self-referential cost concentrates transport mass on core-to-core mappings, with exponential concentration controlled by the fingerprint gap $\Delta_\varphi^2$. Status: Proved (Category A). (Erratum 2026-04-03: Upgraded via Theorem TC-TIGHT-CONFINEMENT — Sinkhorn-Lipschitz analysis with formation-aware cost decomposition. The tight a priori bound $\|\tilde{u} - \hat{u}_t\| \leq \sqrt{(\sigma_{\mathrm{eff}}^2/2)} \|\Delta u\| + e^{-\gamma\Delta_\varphi^2/\varepsilon_{\text{OT}}}$ achieves 4.5–10× safety margin at $\varepsilon_{\text{OT}} \leq 0.01$. Core E_self exponentially small ($O(e^{-4\sqrt{\beta/(2\alpha)}\cdot 2})$), boundary error ∝ $\sqrt{\varepsilon_{\text{OT}}}$. All components (decomposition, core bound, diffusion, Gibbs bound, composition) Cat A. Extended to K-formations (exp45). See TIGHT-CONFINEMENT-FINAL.md.) (Erratum 2026-04-02: formation-conditioned bound tightens from TC' by 2.4–3.5×. Erratum 2026-03-31: fixed-point existence upgraded from conditional to proved; fingerprint reduced to 3 components; contraction constants tightened.)

Fixed-point existence (Schauder): For any $\varepsilon_{\text{OT}} > 0$, the self-referential transport map $T : \Sigma_m \to \Sigma_m$ has a fixed point. Proof: entropic regularization makes the inner OT problem strictly convex $\Rightarrow$ $M^*(c)$ is unique and continuous in $c$ (Berge's maximum theorem) $\Rightarrow$ define $T_\tau$ via finite-time projected gradient flow ($\Phi_\tau$, continuous by Picard-Lindelöf on compact $\Sigma_m$) $\Rightarrow$ $T_\tau : \Sigma_m \to \Sigma_m$ is continuous $\Rightarrow$ Schauder gives fixed point for each $\tau$ $\Rightarrow$ compactness + Łojasiewicz gives accumulation point that is both a transport fixed point and an energy critical point. (Erratum 2026-04-01: Step 7 replaced — IFT-based minimizer continuity replaced with finite-time flow truncation, avoiding the μ > 0 requirement entirely.) Uniqueness near $\hat{u}_t$ guaranteed when (WR') holds. (TRANSPORT-CONCENTRATION-STRENGTHENED.md §4.)

Canonical fingerprint: $\varphi = (u, \mathrm{Cl}(u), D(x;1\!-\!u)) \in [0,1]^3$ (3 components; resolvent $C(x,x)$ demoted to optional diagnostic — contributes $<0.4\%$ of fingerprint gap but $\|\partial C/\partial u\| \approx 9300$ dominates Jacobian). With 3-component fingerprint: $\|\partial\varphi/\partial u\|_{\text{op}} \leq 2.83$ (conservative analytical), $\leq 1.75$ (formation-conditional bound exploiting free-set restriction and boundary subgraph spectral radius, TRANSPORT-CONCENTRATION-STRENGTHENED.md §7), measured $\approx 1.43$ at formations.

Formal result: Under conditions (TC1) $\Delta_\phi^2(\delta) > 0$, (TC2) $\sigma^2 \geq \text{diam}(X)^2/2$, (TC3) $\gamma\Delta_\phi^2(\delta)/\varepsilon_{\text{OT}} > \log n + \text{diam}(X)^2/\sigma^2$:

• Deep core ($\delta \geq 2$): $\sum_{y \in \text{Core}_s} M^*(x,y) / \sum_y M^*(x,y) \geq 1 - n \cdot \exp(-(\gamma\Delta_\phi^2(\delta) - \text{diam}(X)^2/\sigma^2)/\varepsilon_{\text{OT}})$.
• Shallow core ($\delta = 1$): Weak concentration only; shifted-threshold fallback (part (c)) applies.
• Boundary thinness: $\text{Core} \setminus \text{Core}^{\delta \geq 2} = \partial\text{Core}$ (set-theoretic identity, no Γ-convergence needed).

Proof structure (5 steps): (1) Fingerprint gap lower bound via Fingerprint Amplification Lemma + Interior Gap Proposition; (2) Cost comparison argument showing $c(x,z) - c(x,y^*) \geq \gamma\Delta_\phi^2(\delta) - \text{diam}(X)^2/\sigma^2$; (3) Sinkhorn factor analysis showing $b(z)/b(y^*) \leq 1$ (target marginal capacity favors concentration); (4) Summation over non-core targets with union bound; (5) Boundary thinness identity.

Contraction-concentration compatibility (tightened): Both regimes are simultaneously satisfiable iff $\mu \cdot \Delta_\phi^2(\delta) > (\log n + C) \cdot \lambda_{\text{tr}} \cdot \|\partial\phi/\partial u\|_{\text{op}}$. With 3-component fingerprint ($\|\partial\varphi/\partial u\|_{\text{op}} \leq 1.43$): for deep core ($\Delta_\phi^2 \approx 2.38$): $\mu_0 \approx 3.4$, satisfied at non-bifurcation parameters ($\mu \approx 70$–$130$). For boundary core ($\Delta_\phi^2 \approx 0.05$): requires $\mu > 170$, never observed — hence the two-tier structure.

Numerical verification (Experiments 10–11):

• Fingerprint gap $\Delta_\varphi^2(\text{deep core}) = 2.44$ (theory: 2.87, same order of magnitude)
• Core-to-core transport fraction $> 99.99\%$ at $\gamma/\varepsilon_{\mathrm{OT}} > 5$
• Weak-regime fixed-point convergence in 2–3 iterations
• Transport-based Persist $\approx 0.90$–$0.97$ across perturbation scenarios

Implementation: scc/transport.py implements the full pipeline: 3-component cohesion fingerprint $\varphi = (u, \mathrm{Cl}(u), D(x;1{-}u))$ (erratum 2026-04-01: resolvent $C(x,x)$ demoted — see §7.1), self-referential cost, log-domain Sinkhorn partial OT, transport field application, and self-referential fixed-point iteration. 175 tests pass (28 transport-specific). Strong-regime selection resolved: no multiplicity observed across $\lambda_{\mathrm{tr}} \in [0.01, 10]$ (exp29); transport confinement bound proved (ISOPERIMETRIC-TRANSPORT-PROOFS.md).