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

Part 5 of 5 — Soft Cognitive Cohesion canonical specification, current release CV-1.11 (2026-05-06). 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. Current: 54A/14B/5C/5R = 78 claims, ~69% fully proved (CV-1.11, 2026-05-06).)

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 $\le a_{\mathrm{cl}}/4<1$. (R4.)

T20. Axiom Consistency (Parameter Admissibility). The sigmoid closure satisfies A1' (conditional extensivity with threshold $c^{\ast }=\sigma (a_{\mathrm{cl}}(c^{\ast }-{\tau }_{\mathrm{cl}}))$), 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 $[0,c^{\ast }]$. A2 from monotonicity of $\sigma$ and $P_t$. A3 from $\max {\sigma }^{\prime }=1/4$. A1 failure from ${\sigma }^{-1}(0.9)/z>4$. (R4.)

T-A2. Monotonicity of Sigmoid Closure. $u\le v$ pointwise $⟹{\mathrm{Cl}}_t(u)\le {\mathrm{Cl}}_t(v)$ 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 $\beta /\alpha >4{\lambda }_2/|W^{\prime \prime }(c)|$, the global minimizer of ${\mathcal{E}}_{\mathrm{bd}}{|}_{{\Sigma }_m}$ is non-uniform. Proof: The ordered-pair summation gives smoothness functional $2\alpha v^{T}Lv$ with Hessian $4\alpha L$. Second variation at $u\equiv c$ has eigenvalue $4\alpha {\lambda }_2+\beta W^{\prime \prime }(c)<0$ when $\beta /\alpha >4{\lambda }_2/|W^{\prime \prime }(c)|$. 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 ${\beta }_{\mathrm{crit}}=4\alpha {\lambda }_2/|W^{\prime \prime }(c)|$ 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 ${\beta }_{\mathrm{crit}}=4{\alpha }_0{k}^2{\lambda }_2/|W^{\prime \prime }(c)|\approx 4{\alpha }_0{\pi }^2/|W^{\prime \prime }(c)|=O(1)$, 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 $c\ne 0$ 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 $\Vert J_{\mathrm{Cl}}{\Vert }_{\mathrm{op}}<1$, 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 ($\le (n-k)/n$ 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 $\epsilon =\alpha /\beta \to 0$, 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 $\Vert \hat{u}-{\hat{u}}_0\Vert =O({\lambda }_{\mathrm{cl}}+{\lambda }_{\mathrm{sep}})$. 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). $\mathsf{Bind}\ge 1-\sqrt{{\mathcal{E}}_{\mathrm{cl}}/n}$ (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.) $|{\mathrm{Core}}^2|/|\mathrm{Core}|\ge 1-4C/\sqrt{m}$ 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 $|{\partial }_V\mathrm{Core}|/|\mathrm{Core}|\le C/\sqrt{|\mathrm{Core}|}$ 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 ${\Sigma }_M^K$. 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 ${\Sigma }_M^K$. On the per-formation mass-constrained manifold ${\Sigma }_M^K$, merge endpoint $(u_{\mathrm{merged}},0)\notin {\Sigma }_M^K$ 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 ${\Sigma }_M^K$, 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 ${\beta }_{\mathrm{crit}}=4\alpha {\lambda }_2/|W^{\prime \prime }(c)|$, 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 ${\mu }_{\mathrm{joint}}\ge {\min }_k{\mu }_k-(K-1){\lambda }_{\mathrm{rep}}$ 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 $d_{\min }^{\ast }(B,\mathrm{rule},\mathrm{params},G)$. 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 $d_{\min }^{\ast }\approx 5$ nodes at $\beta =30,\alpha =1$; without closure (Allen-Cahn), $d_{\min }^{\ast }\approx 7$ nodes. (Erratum 2026-04-10: $d_{\min }^{\ast }$ 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 ${\bar{r}}_0=|1^{T}r|/n$ the per-site mean residual. Then: $\Vert r_T{\Vert }_2\le \frac{{\lambda }_{\mathrm{sep}}{G}_{\mathrm{sep}}+{\lambda }_{\mathrm{bd}}{G}_{\mathrm{bd}}}{2{\lambda }_{\mathrm{cl}}(1-a_{\mathrm{cl}}/4)}+\frac{(1+a_{\mathrm{cl}}/4)\sqrt{n}{\bar{r}}_0}{1-a_{\mathrm{cl}}/4}$. Proof: KKT projection + Banach inversion of restricted operator with ${\sigma }_{\min }\ge 1-a_{\mathrm{cl}}/4$. 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. $\mathsf{Bind}(\hat{u})\ge 1-f(\text{params})$, $n$-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: $\beta >\Gamma (a_{\mathrm{cl}},\tau ,\eta ,{\lambda }_{\mathrm{cl}},\theta )\cdot {\epsilon }_1^2\cdot \alpha$ 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 ${\epsilon }_1\le 0.1$. 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 ${\zeta }_{\ast }(G)$ 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 $\sigma (u^{\ast })=(\mathcal{F};\{(n_k,[{\rho }_k],{\lambda }_k)\})$ — 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 $H(u^{\ast })$ commutes with the stabilizer $G_u={\mathrm{Stab}}_{\mathrm{Aut}(G)}(u^{\ast })$ action on $T_{u^{\ast }}{\Sigma }_m$, 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 $T_{u^{\ast }}{\Sigma }_m$. 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 ${\varphi }_k\in 1^{\perp }$ a Hessian eigenvector, the nodal count $\mathcal{N}({\varphi }_k)$ satisfies: (i) graph-intrinsic (independent of basis choice); (ii) $\mathrm{Aut}(G)$-equivariant under $G_u$-orbit action; (iii) lower bound $\mathcal{N}({\varphi }_k)\ge 2$ on $1^{\perp }$ (from the constraint $\sum {\varphi }_k=0$ — corrected from the plan-template wording "$n_k=1$ iff constant", which was incorrect: a constant in $1^{\perp }$ requires ${\varphi }_k=0$); (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 ${\varphi }_k$). 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 ${\mathcal{P}}_{\ell =1}[\delta u_x]=(-c_d{\int }_0^{\infty }{u}^{\ast }(r)dr,0)$ (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 $\int u^{\ast }(r)dr\approx r_0$, 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 $u^{\ast }\equiv c$ on a $D_4$-symmetric free-BC grid, the Hessian eigenvalues are ${\mu }_k=4\alpha {\lambda }_k^{\mathrm{Lap}}+\beta W^{\prime \prime }(c)$ where ${\lambda }_k^{\mathrm{Lap}}$ are the Laplacian eigenvalues. Full $D_4$ irrep table on the cosine basis ${\varphi }_{(p,q)}=\cos (p\pi i/L)\cos (q\pi j/L)$ with nodal count $\mathcal{N}({\varphi }_{(p,q)})=(p+1)(q+1)$. 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 $\approx 1\cdot |W^{\prime \prime }(c)|$ not $4\cdot |W^{\prime \prime }(c)|$). 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 ${\tilde{\Sigma }}_M^{K_{\mathrm{field}},\circ }={\tilde{\Sigma }}_M^{K_{\mathrm{field}}}\setminus \partial \tilde{\Sigma }$ 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 ${\tilde{\Sigma }}_M^{K_{\mathrm{field}},\circ }$. For a K-field minimizer ${\mathbf{u}}^{\ast }=(u^{(1)\ast },\dots ,u^{(K_{\mathrm{field}})\ast })\in {\tilde{\Sigma }}_M^{K_{\mathrm{field}},\circ }$ with $K_{\mathrm{act}}({\mathbf{u}}^{\ast })\le K_{\mathrm{field}}$, 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 ${\tilde{\Sigma }}_M^{K_{\mathrm{field}},\circ }$ 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 $\mathbf{u}\in {\tilde{\Sigma }}_M^{K_{\mathrm{field}}}(G)$ a shared-pool multi-formation state. Let $U(\mathbf{u})={\sum }_{j=1}^{K_{\mathrm{field}}}{u}^{(j)}$, $A^{\epsilon }(\mathbf{u})=\{j:m_j(\mathbf{u})>\epsilon \}$, $K_{\mathrm{act}}^{\epsilon }=|A^{\epsilon }|$, 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 $N_j^r\cap N_k^r=\varnothing$, 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

$K_{\mathrm{bar}}^{{\ell }_{\min }}(U(\mathbf{u});G)=K_{\mathrm{act}}^{\epsilon }(\mathbf{u}),$

AND a labelled bijection ${\mathcal{A}}_{\mathrm{bar}}:A^{\epsilon }\to {\mathrm{Bars}}_0^{\mathrm{term}}$ 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 $\varphi \in {\Phi }_{\mathrm{res}}$ 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 $T_{20}^2$ 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 ${\Delta }_{\mathrm{depth}}\ge {\Delta }_{\min }^{\mathrm{ST}}$), 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 $\partial {\Omega }^{\ast }$ satisfy: $d_H(\mathrm{PersRidge}(u^{\ast }),\partial {\Omega }^{\ast })\le 2\sqrt{\alpha /\beta }.$ 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 ($\epsilon \to 0$). (New, CV-1.7, 2026-05-06.) The Gibbs measure ${\pi }_T^{\epsilon }$ on ${\tilde{\Sigma }}_M(\epsilon )$ converges weakly to ${\pi }_T^0$ (a probability measure concentrated on the set of energy minimizers of $\tilde{\mathcal{E}}$ on $\partial {\tilde{\Sigma }}_M$) as $\epsilon \to 0$. 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 ${\mathcal{F}}_M(\mathcal{P})=\{\mathbf{u}\in {\tilde{\Sigma }}_M:\mathcal{P}(\mathbf{u})\le 0\}$ (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 ${\tilde{\Sigma }}_M$, $d{\mathbf{u}}_t=-\nabla \tilde{\mathcal{E}}({\mathbf{u}}_t)dt+\sqrt{2T_{\ast }}d{W}_t+d{\mathbf{k}}_t,$ 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 ${\pi }_{T_{\ast }}$. (New, CV-1.9, 2026-05-06.) The Gibbs measure ${\pi }_{T_{\ast }}\propto \exp (-\tilde{\mathcal{E}}(\mathbf{u})/T_{\ast })$ on ${\tilde{\Sigma }}_M$ 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) $L^2({\pi }_{T_{\ast }})$ 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 $p(t,x,y)\ge c{t}^{-n/2}$ 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 $T_{\ast }$ 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 ${\pi }_{T_{\ast }}$ (T-PF-A1-GI) satisfies a Poincaré inequality with explicit constant: ${\lambda }_1\ge \frac{{\pi }^2}{n}\exp \left(-\frac{\mathrm{osc}(\tilde{\mathcal{E}})}{T_{\ast }}\right)$ (Payne-Weinberger on bounded convex polytope + Holley-Stroock perturbation lemma). Consequently, the law ${\mu }_t$ of ${\mathbf{u}}_t$ converges to ${\pi }_{T_{\ast }}$ exponentially in total variation: $\Vert {\mu }_t-{\pi }_{T_{\ast }}{\Vert }_{\mathrm{TV}}\le C{e}^{-{\lambda }_{1}t}$. 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 $\mathrm{osc}(\tilde{\mathcal{E}})/T_{\ast }$; 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 ${\pi }_{T_{\ast }}$ 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 $\epsilon$ 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 ${\lambda }_1\ge {\lambda }_{\min }^{\mathrm{Kr}}$. 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 $\Delta z\ge {\Delta }_{\min }$. 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 ${\pi }_{T_{\ast }}$ (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 ${\pi }_{T_{\ast }}(\cdot \mid \mathcal{O})$ (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 $\approx 1\cdot |W^{\prime \prime }(c)|$ not $4\cdot |W^{\prime \prime }(c)|$. 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 $F^{\prime \prime }(M/2)$ 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 $d_{\min }^{\ast }(B,\mathrm{rule},\mathrm{params},G)$. The earlier scalar fit $d_{\min }^{\ast }=4.8+0.31\sqrt{\beta /\alpha }-0.018\beta /\alpha$ 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: $d_{\min }^{\ast }$ 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.) ${\mu }_{\mathrm{joint}}\ge {\min }_k{\mu }_k-(K-1){\lambda }_{\mathrm{rep}}\cdot {\omega }_{\mathrm{soft}}$, where ${\omega }_{\mathrm{soft}}={\max }_{j\ne k}\Vert {\mathcal{P}}_{O_{jk}}{\psi }_k^{\mathrm{soft}}{\Vert }^2$ 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 $(u_t^1,\dots ,u_t^K)$ be a well-separated joint minimizer with $d_{\min }(j,k)\ge D_{\text{sep}}\ge 3$ for all $j\ne k$. Under per-formation hypotheses (H1-K), well-separation (WS), and spectral-repulsion compatibility (SR: ${\min }_k{\mu }_k>(K-1){\lambda }_{\text{rep}}$), an $\epsilon$-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 $\Vert {\hat{u}}_s-{\hat{u}}_t\Vert =O(\delta )$ 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: $f_1^{\mathrm{grad}}\le \sqrt{n_{\mathrm{bdy}}/n_F}$ 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: $2{\epsilon }_2+2{\epsilon }_1/\mu <r_{\mathrm{eff}}$ under gentleness $\epsilon <r_{\mathrm{eff}}/(4+2/\mu )$, 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: ${\Delta }_{\mathrm{core}}\ge 0.0441\beta$ ($\beta$-independent ratio $r_{\mathrm{core}}\ge 0.210$). Exterior escape barrier: ${\Delta }_{\mathrm{ext}}=O(\beta )$ per node (Proposition E1, proved). Boundary escape barrier: ${\Delta }_{\mathrm{bdy}}$ is formation-shape-dependent; $r_{\mathrm{bdy}}\ge 0.210$ 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: ${\Delta }_s\ge {\Delta }_t-2{\epsilon }_1$ (Proposition 4). Basin containment holds when ${\epsilon }_1<{\Delta }_t/4$ and $2{\epsilon }_2+2{\epsilon }_1/\mu <r_{\mathrm{basin}}$ (Proposition 5). (Erratum 2026-04-01: quantitative ${\Delta }_{\mathrm{bdy}}$ formula derived — Taylor normal form along soft mode gives ${\Delta }_{\mathrm{bdy}}=\mu /2\cdot (t^{\ast }{)}^2+L_3/6\cdot (t^{\ast }{)}^3+L_4/24\cdot (t^{\ast }{)}^4$ where $t^{\ast }$ 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 $\{x:{\hat{u}}_s(x)\ge \theta -ϵ\}$ contains the transported core $\{x:({\mathbf{M}}_{t\to s}{\hat{u}}_t)(x)\ge \theta \}$ for perturbations of size $ϵ$. Status: Proved. (I7.)

(d) Exact Threshold Preservation. The core at exact threshold $\theta$ (not $\theta -ϵ$) 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 $\delta (x)\ge 2$ from $\partial \text{Core}$: $\hat{u}(x)-{\theta }_{\text{core}}\ge (1-{\theta }_{\text{core}})-2\exp (-c_0\delta (x))-C_{\text{op}}R/\beta$, where $c_0=\text{arccosh}(1+{\kappa }^2/d_{\min })$, $\kappa =\sqrt{\beta /(2\alpha )}$, $C_{\text{op}}\le 2(1+a_{\text{cl}}/4)\cdot R$ with $R=({\lambda }_{\text{cl}}+{\lambda }_{\text{sep}})/{\lambda }_{\text{bd}}$, giving $C_2\le 2.875$ at default parameters. Requires: (H2') Deep core existence: ${\text{Core}}^2(\hat{u})\ne \varnothing$ — proved for $|\text{Core}|\ge 25$ via $\Gamma$-convergence isoperimetric analysis + discrete maximum principle contraction (Theorem 1, CORE-DEPTH-ISOPERIMETRIC.md). (Erratum 2026-04-01: threshold tightened from $\beta \ge 58\alpha$ to $\beta \ge 20\alpha$ (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 $\beta \ge 20$ in exp28/exp31.) Deep Core Dominance: unconditionally $|{\text{Core}}^2|=|\text{Core}|-|{\partial }_V\text{Core}|$ (Theorem 2a, identity); conditionally $|{\text{Core}}^2|/|\text{Core}|\ge 1-4C/\sqrt{m}$ under iso_ratio $\le C$ (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 ${\Delta }_{\phi }^2$. 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 $\Vert \tilde{u}-{\hat{u}}_t\Vert \le \sqrt{({\sigma }_{\mathrm{eff}}^2/2)}\Vert \Delta u\Vert +e^{-\gamma {\Delta }_{\phi }^2/{\epsilon }_{\text{OT}}}$ achieves 4.5–10× safety margin at ${\epsilon }_{\text{OT}}\le 0.01$. Core E_self exponentially small ($O(e^{-4\sqrt{\beta /(2\alpha )}\cdot 2})$), boundary error ∝ $\sqrt{{\epsilon }_{\text{OT}}}$. 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 ${\epsilon }_{\text{OT}}>0$, 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 $\Rightarrow$ $M^{\ast }(c)$ 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: $\phi =(u,\mathrm{Cl}(u),D(x;1-u))\in [0,1{]}^3$ (3 components; resolvent $C(x,x)$ demoted to optional diagnostic — contributes $<0.4\%$ of fingerprint gap but $\Vert \partial C/\partial u\Vert \approx 9300$ dominates Jacobian). With 3-component fingerprint: $\Vert \partial \phi /\partial u{\Vert }_{\text{op}}\le 2.83$ (conservative analytical), $\le 1.75$ (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) ${\Delta }_{\varphi }^2(\delta )>0$, (TC2) ${\sigma }^2\ge \text{diam}(X{)}^2/2$, (TC3) $\gamma {\Delta }_{\varphi }^2(\delta )/{\epsilon }_{\text{OT}}>\log n+\text{diam}(X{)}^2/{\sigma }^2$:

• Deep core ($\delta \ge 2$): ${\sum }_{y\in {\text{Core}}_s}{M}^{\ast }(x,y)/{\sum }_y{M}^{\ast }(x,y)\ge 1-n\cdot \exp (-(\gamma {\Delta }_{\varphi }^2(\delta )-\text{diam}(X{)}^2/{\sigma }^2)/{\epsilon }_{\text{OT}})$.
• Shallow core ($\delta =1$): Weak concentration only; shifted-threshold fallback (part (c)) applies.
• Boundary thinness: $\text{Core}\setminus {\text{Core}}^{\delta \ge 2}=\partial \text{Core}$ (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 $c(x,z)-c(x,y^{\ast })\ge \gamma {\Delta }_{\varphi }^2(\delta )-\text{diam}(X{)}^2/{\sigma }^2$; (3) Sinkhorn factor analysis showing $b(z)/b(y^{\ast })\le 1$ (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 $\mu \cdot {\Delta }_{\varphi }^2(\delta )>(\log n+C)\cdot {\lambda }_{\text{tr}}\cdot \Vert \partial \varphi /\partial u{\Vert }_{\text{op}}$. With 3-component fingerprint ($\Vert \partial \phi /\partial u{\Vert }_{\text{op}}\le 1.43$): for deep core (${\Delta }_{\varphi }^2\approx 2.38$): ${\mu }_0\approx 3.4$, satisfied at non-bifurcation parameters ($\mu \approx 70$–$130$). For boundary core (${\Delta }_{\varphi }^2\approx 0.05$): requires $\mu >170$, never observed — hence the two-tier structure.

Numerical verification (Experiments 10–11):

• Fingerprint gap ${\Delta }_{\phi }^2(\text{deep core})=2.44$ (theory: 2.87, same order of magnitude)
• Core-to-core transport fraction $>99.99\%$ at $\gamma /{\epsilon }_{\mathrm{OT}}>5$
• 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 $\phi =(u,\mathrm{Cl}(u),D(x;1-u))$ (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).

T-Persist-K-Weak. Multi-Formation Temporal Persistence (Weakly-Interacting Regime). Extends T-Persist-K-Sep to formations with boundary overlap ($|O_{jk}|=O(|\partial \text{Core}|)$). Joint Hessian has off-diagonal blocks $V_{jk}={\lambda }_{\text{rep}}I$ (global coupling); spectral gap via Weyl bound: ${\mu }_{\text{joint}}\ge {\min }_k{\mu }_k-(K-1){\lambda }_{\text{rep}}$. Under (H1-K), (WI: $|O_{jk}|\le 0.2\cdot \min (|{\text{Core}}_j|,|{\text{Core}}_k|)$), (SR: ${\min }_k{\mu }_k>(K-1){\lambda }_{\text{rep}}$), (NB-K: ${\mu }_{\text{joint}}>{\mu }_0$): (a) joint minimizer persistence, (b) deep core unaffected by coupling, (c) boundary overlap sites have shifted-threshold fallback, (d) post-hoc simplex correction within basin radius, (e) deep core fingerprint gap preserved. Status: Category C — conditionally proved under (H1-K), (WI), (SR), (NB-K), plus per-formation T-Persist-1 conditions. WI and SR are structural regime definitions. Full statement in Part 4 · §12. Implementation: scc/multi.py coupled transport modes.

T-Persist-K-Unified. Unified Multi-Formation Persistence Theorem. (New in v2.1.) A single parametric theorem covering Sep/Weak/Strong persistence regimes as corollaries, parametrized by the coupling measure ${\Lambda }_{\mathrm{coupling}}={\max }_{j\ne k}{\lambda }_{\mathrm{rep}}\cdot {\omega }_{jk}/\min ({\mu }_j,{\mu }_k)$. Under five universal hypotheses:

• (PS) Phase separation: $\beta /\alpha >4{\lambda }_2/|W^{\prime \prime }(c)|$.
• (ND) Non-degeneracy: ${\mu }_k>0$ for all $k$.
• (BC'-K) Basin containment (K-formation): joint-formation directional basin radius exceeds transport displacement.
• (TC-K) Transport confinement (K-formation): $C_{\mathrm{conf}}\sqrt{m}<r_{\mathrm{basin}}^{(K)}$ for each formation.
• (SR-Λ) Spectral-repulsion compatibility: ${\Lambda }_{\mathrm{coupling}}<1/(K-1)$.

The following hold under $\epsilon$-gentle transition:

1. Joint minimizer persistence: $\Vert u_s^k-u_t^k\Vert \le 2{\epsilon }_1/{\mu }_{\mathrm{joint}}$.
2. Per-formation deep core inheritance (for $\delta \ge 2$).
3. Transport concentration at rate $\exp (-\gamma {\Delta }_{\phi }^2/{\epsilon }_{\mathrm{OT}})$.
4. ${\mathsf{Persist}}^k\ge 1-C\cdot {\Lambda }^2$ (persist degradation continuous in $\Lambda$).

Corollaries:

• Corollary I (Sep): $\Lambda \approx 0$ (well-separated) — reduces to T-Persist-K-Sep. (Proved.)
• Corollary II (Weak): $\Lambda \ll 1$ — reduces to T-Persist-K-Weak. (Conditional.)
• Corollary III-a (Strong-Coexist): $\Lambda <1/(K-1)$ — formations persist with degraded Persist. (Conditional.)
• Beyond $\Lambda =1/(K-1)$: merge bifurcation (not covered; see T-Persist-K-Strong).

Key finding: isoperimetric ordering is NOT needed for persistence (only for metastability characterization); (TC) is strictly weaker than (WR') — persistence holds in 3/6 configs where (WR') fails.

Status: Category C — conditional on 5 structural hypotheses (PS, ND, BC'-K, TC-K, SR-Λ). Sep corollary proved. 100% geometric-Lambda agreement in 69 experimental configurations (exp46-47). (Erratum 2026-04-10: ${\Lambda }_{\mathrm{coupling}}$ is a spectral-coupling parameter, not a complete branch selector. T-Persist-K-Unified assumes a selected K-formation branch; branch identity, tie-breaker, and history are external data.) (T-PERSIST-K-UNIFIED.md, PHASE-AB-SYNTHESIS.md.) Implementation: scc/multi.py functions coupling_strength(), classify_regime(method='lambda').

T-Persist-Full. Unified Temporal Persistence. (Erratum 2026-03-31: Multiple upgrades — H2 closed, fixed-point existence proved, WR relaxed to selection condition, fingerprint tightened, basin radius corrected.) Synthesizes all components of T-Persist-1 into a single conditional theorem. Under hypotheses (WR', PS, ND, NB, H2', H3, GT). Proof: Chains T-Persist-1(a) (IFT), Deep Core Existence, Interior Gap, Basin Radius, Barrier Stability, Basin Containment, Fixed-Point Existence (Schauder), and Transport Concentration (two-tier with 3-component fingerprint). Status: Category C — composition whose weakest component is T-Persist-1(d) Cat C. Most components are Category A. End-to-end chain verified: exp27 (5/5 × 5/5 = 100% pass); exp28 stress test (84/100, all failures at $n<64$ or $\beta <20$). (PERSIST-MORSE-ANALYSIS.md, PERSIST-SYNTHESIS.md, BASIN-ESCAPE-ANALYSIS.md, CORE-DEPTH-ISOPERIMETRIC.md, TRANSPORT-CONCENTRATION-STRENGTHENED.md.)

W4-extended new finding (2026-04-26)#

V5b-F. Partial Goldstone on Boundary-Modified Graphs. (New finding, W4-extended close, 2026-04-26.) On graphs whose translation symmetry is broken at the boundary (free / mixed / Dirichlet boundary conditions), a partial Goldstone mode survives: the boundary lifting mechanism qualitatively shifts and partially gaps the Goldstone branch, but a residual approximate Goldstone signature remains observable. Quantitative characterization is in progress (NQ-173 carry). Status: Category C — qualitative observation only; full mechanism and quantitative scaling carried as NQ-173 to W5+. Source: E-0096 (free BC partial). (C-0711, P-0711.)

Retracted (5 claims)#

(Erratum 2026-04-07: Added T-Merge (c)(d)(e) and K-Saddle Conjecture. Total 5 retracted claims.)

R1. Theorem 3.3 (${\bar{r}}_0=O(n^{-1/d})$ for general $\tau$). (Retracted 2026-04-02.) The claim that ${\bar{r}}_0=O(n^{-1/d})$ for general ${\tau }_{\mathrm{cl}}$ (not just $\tau =1/2$) was experimentally falsified: ${\bar{r}}_0=0.169$ at $\tau =0.3$ on large grids, confirming ${\bar{r}}_0$ is genuinely $O(1)$ for $\tau \ne 1/2$.

R2. T-Merge (c) — Barrier Existence via Mountain Pass on ${\Sigma }_M^K$. (Retracted 2026-04-07.) Merge endpoint $(u_{\mathrm{merged}},0)\notin {\Sigma }_M^K$. Mountain Pass theorem requires connected path between two points on the manifold. The merge target doesn't exist on the domain. (MERGE-CRITIQUE.md, Flaw #1, CRITICAL.)

R3. T-Merge (d) — Barrier Lower Bound. (Retracted 2026-04-07.) Depends on merge path existence (R2). No valid merge path $\Rightarrow$ no barrier to bound.

R4. T-Merge (e) — Transition State Regularity. (Retracted 2026-04-07.) Conditional on Mountain Pass + Kupka-Smale, which requires the path that doesn't exist (R2).

R5. K-Saddle Conjecture. (Retracted pre-v2.1.) Previously retracted.

Remark on H2 (Experiment 13, 240 parameter combinations). The literal condition ${\delta }_{\min }\ge 2$ (every core site at graph distance $\ge 2$ from non-core) fails universally on finite grids: core boundary sites always neighbor non-core sites, giving ${\delta }_{\min }=1$. However, the transport concentration result (2) only requires $\delta (x)\ge 2$ per site, not globally. The operationally correct hypothesis is deep core non-emptiness: $\{x\in \text{Core}:\delta (x)\ge 2\}\ne \varnothing$. Experiment 13 verifies this across 4 grid sizes ($8^2$–$20^2$), 5 $\beta$ values (5–100), 4 volume fractions, 4 closure strengths: deep core exists in 208/219 formations with non-empty core (95.0%). All 11 failures occur at weak phase separation ($\beta \le 10$) with low closure ($a_{\mathrm{cl}}\le 3.0$), producing small cores ($\le 27$ sites). At $\beta \ge 20$, deep core existence is universal (144/144). The deep core contains a median 70.6% of core mass (u-weighted). H2 should therefore be read as "the deep core is non-empty," which holds whenever phase separation is sufficient to produce a bulk core region.

14. Commitment Notes#

The following Commitment Notes record foundational decisions that constrain interpretation and future development. They are not axioms; they are meta-theoretical commitments that explain why certain design choices were made and how they should be understood.

CN1. Contraction, Not Projection; Trajectory Matters (at the Energy Level). The closure operator is a contraction ($a_{\mathrm{cl}}<4$), not a projection. The trajectory of closure iterates carries structural information. However, the trajectory leads to a unique destination (the fixed point). Path-dependence and metastability arise at the energy landscape level, where different initial conditions converge to different metastable formations.

CN2. $\tau$ (Within-Time) Is Not a Primitive. The within-time evolution parameter $\tau$ (controlling gradient flow within a single time step) is not part of the formal universe. It is an implementation detail of the optimization procedure.

CN3. Definition Graph Acyclic; Computation Graph Cyclic. The definition graph of the theory is acyclic: operators are defined from the field, predicates from operators, energy from predicates. The computation/optimization graph is cyclic: the field is updated using the operators it defines. This distinction is essential and must not be conflated.

CN4. Group F Architecturally Distinct from A–E. The crisp recovery interface (Group F, superlevel filtrations) is architecturally distinct from the soft theory's axiomatic groups. It operates at a different layer (soft-to-crisp interface) and should not be mixed with the soft axioms.

CN5. Four-Term Independence Is Conceptual, Not Mathematical. The four energy terms address four logically independent structural requirements. This does not imply they are mathematically uncorrelated. In practice, they interact strongly.

CN6. $K$ Is Kinetically Determined, Not Thermodynamically Selected. The relational kernel $K_t$ emerges from initial conditions and spatial structure through the dynamics of formation nucleation and metastable persistence, not from energy minimization. On a single soft field, K>1 well-separated formations are metastable local minima when inter-formation distance exceeds $d_{\min }^{\ast }\approx 5$ nodes (at $\beta =30$ with closure). The $K$-field architecture guarantees K>1 by construction via per-field optimization with inter-field repulsion. Closure reduces $d_{\min }^{\ast }$ by ~30% compared to pure Allen-Cahn, realizing T7-Enhanced metastability through self-referential strengthening of attraction basins. CN6 resolved (2026-04-02 audit): K emerges kinetically and is constrained by spectral structure and barrier heights, not thermodynamic optimization.

CN7. Operator Pair, Not Generic Self-Referentiality. The theory's distinctive self-referentiality is the dual-mode operator pair (self-completion via closure, self-contrast via distinction), not generic nonlinear self-dependence. A third mode (self-integration via co-belonging) is available as a derived diagnostic but does not enter the energy or predicates. The claim is about the specific structure, not about self-referentiality per se.

CN8. Formations Are Metastable, Not Globally Optimal. Proto-cohesive formations are metastable critical points of the energy, not necessarily global minimizers. The global minimizer on ${\Sigma }_m$ exists (T1) and is non-trivial (T8-Core), but real formations may occupy local minima with enhanced Hessian curvature (T7-Enhanced).

CN9. Two-Landscape Structure. The closure operator has a unique fixed point (contraction landscape). The energy functional has multiple critical points (energy landscape). "Trajectory matters" applies to the energy landscape, not to closure in isolation. These are distinct mathematical objects and must not be conflated.

CN10. Contrastive Comparison Permitted; Reductive Identification Prohibited. Comparing SCC's operators or structures to those of other frameworks (Allen-Cahn, optimal transport, Gestalt psychology) for the purpose of illuminating similarities and differences is permitted and encouraged. Reducing SCC to any of these frameworks — claiming it "is just" Allen-Cahn, or "is just" clustering, or "is just" phase-field theory — is prohibited. The comparison is contrastive, not reductive.

CN11. Resolvent, Not Cesàro. The co-belonging operator uses the resolvent $(I-\alpha W_{\mathrm{sym}}{)}^{-1}$, not Cesàro averaging. Cesàro averaging degenerates to the stationary distribution, destroying pairwise structural information. The resolvent preserves the full geometric series of weighted paths.

CN12. ${\mathcal{Q}}_{\mathrm{morph}}$ Is Persistence-Based (Filtration Commitment). The morphological quality measure is defined through the superlevel-set filtration and persistence diagrams. This commits the theory to a topological data analysis interface for assessing morphological structure.

CN13. Separation Contributes to Instability (Preliminary). The separation energy ${\mathcal{E}}_{\mathrm{sep}}$ has a nonzero Hessian at uniform states and participates in the instability structure. The quantitative claim of $10^5\times$ dominance (R10) is parameter-dependent and methodologically unverified (parameter normalization, volume constraint status, sign conventions). The qualitative claim (separation participates in instability) survives all caveats; the quantitative claim requires verification.

CN14. Closure Expands Multi-Formation Stability. On a single soft cohesion field, K > 1 well-separated formations are metastable local minima when inter-formation distance exceeds $d_{\min }^{\ast }$. The self-referential closure operator reduces $d_{\min }^{\ast }$ compared to pure Allen-Cahn by approximately 30% (from $\approx 7$ nodes without closure to $\approx 5$ nodes with SCC closure at $\beta =30$; exp57 final). This distance reduction enables formation coexistence at closer separations. The physical mechanism: closure's self-reinforcement structure creates larger attraction basins for each formation, strengthening internal cohesion and raising the merge-direction barrier height. This is the multi-formation realization of T7-Enhanced metastability, where closure participation in the energy landscape (beyond its role as a fixed-point operator) directly enhances formation stability through increased Hessian curvature perpendicular to formation axes. Without closure (pure Allen-Cahn): $d_{\min }^{\ast }\approx 7$ nodes, barrier height $O({\beta }^{0.85})$. With closure (SCC, $a_{\mathrm{cl}}=3.0$): $d_{\min }^{\ast }\approx 5$ nodes, barrier height $O({\beta }^{0.89})$ (exp38, exp57). K-field architecture independence: The $K$-field decomposition with per-field optimization and global inter-field repulsion ensures K>1 by construction, regardless of $d_{\min }^{\ast }$ thresholds. The two mechanisms are complementary: single-field SCC extends the metastability range; K-field architecture guarantees structural K>1 even at the merge bifurcation.

CN15. Static/Dynamic Separation. (New, W4 close, 2026-04-24.) The static global minimum on pure ${\mathcal{E}}_{\mathrm{bd}}$ (constraint ${\Sigma }_m$, isoperimetric ordering, T-Merge (b)) is K=1 — this is a proved theorem, not an open problem. The dynamic observables — what a protocol actually delivers under full SCC gradient flow — are protocol-endpoint quantities $\hat{K},\mathcal{F}$ that need not equal the static minimum, because under full SCC parameters the F=1 single-disk minimizer of pure ${\mathcal{E}}_{\mathrm{bd}}$ is non-critical (T-PreObj-1). This separation dissolves the apparent paradox "K=1 cheaper static minimum vs empirically observed K>1": the two are quantities of different layers (static ${\mathcal{E}}_{\mathrm{bd}}$ vs dynamic full SCC) and there is no requirement that they match. CN15 is the conceptual key that resolved F-1 (OP-0001) and clarified M-1 (OP-0002) without requiring a K-selection mechanism, an external mass constraint, or a kinetic barrier argument. Source: T-Merge (b), T-PreObj-1, T-PreObj-1G, Lemma 4 (W4 cluster). Part 4 · §11.1 item 14 and §12 W4 Resolution Banner for the operational consequences.

CN16. Protocol-Parameterized Observables (W4 added 2026-04-25). SCC observables decompose into two classes:

• Protocol-invariant: depend only on $u^{\ast }$ and graph structure, not on the optimization protocol that produced $u^{\ast }$. Examples: $\mathcal{F}(u^{\ast })$ (local-maxima count), $\mathcal{E}(u^{\ast })$ (energy at minimizer), $\sigma (u^{\ast })$ (cohesion signature, Part 4 · Commitment 14), Hessian spectrum.
• Protocol-dependent: depend on the gradient flow trajectory, IC distribution, or optimizer choice. Examples: ${\hat{K}}_{\mathrm{step}}(\pi )$ (selector outcome), ${\mathcal{F}}_{\ast }(L)$ (extreme-value statistics over a sample), basin probabilities.

Canonical theorems must specify which class each observable belongs to. The IC-protocol dichotomy of T-PreObj-1 (v) — adaptive bounded (${\mathcal{F}}_{\ast }\le F^{\mathrm{first}-\mathrm{pitchfork}}+O(1)$) vs random (${\mathcal{F}}_{\ast }^{\mathrm{random}}\sim L^{2.8}$) — is the canonical example: ${\mathcal{F}}_{\ast }(L)$ values are protocol-dependent, but the dichotomy structure itself is protocol-invariant Cat A. Failure to distinguish these classes is a category error that produces apparent paradoxes (e.g., the K=1 vs K>1 conflict resolved by CN15). (W4 04-23 P-2026-04-23-03; W4 04-24 T-PreObj-1 (v).)

CN17. σ-Labeled Formation Quantization (W4 added 2026-04-25). Formation Quantization is characterized by σ-signature (Part 4 · Commitment 14), refining the integer count $K_{\mathrm{step}}$ to a labeled tuple. The single-formation vs multi-formation distinction refines as follows:

• $\mathcal{F}=1$: single-mode (atomic-like; rare under full SCC by T-PreObj-1).
• $\mathcal{F}\ge 2$: multi-mode (molecular-like; default under full SCC).

$K_{\mathrm{step}}$ becomes a derived connectivity statistic over $\mathcal{F}$-counted peaks ($K_{\mathrm{step}}\le \mathcal{F}$ always; bilobed $K_{\mathrm{step}}=1,\mathcal{F}=2$ configurations are admissible). The duality $\mathcal{F}$ (multi-peak structure) vs $K_{\mathrm{step}}$ (connectivity) — both relevant observables — is canonical. Empirical confirmation (W4 04-25, NQ-141 Cat A): R23 56 minimizers × 324 mode-ℓ pairs across 6 orbital letters (p, d, f, g, h, i) show 0 exceptions in the σ-irrep correspondence via $\ell \mathrm{mod}4\to D_4$ irrep table. (Supersedes the previously-implicit "single formation = single disk" reading of §13 single-formation theorems; W4 04-23 R-2026-04-23-01 retraction; W4 04-23 P-2026-04-23-04 supersedes earlier CN18 candidate; W4 04-25 NQ-141 empirical anchor.)

15. Closing Summary#

This canonical specification (CV-1.11, W6 close, 2026-05-08) establishes the formal structure of the theory of Soft Cognitive Cohesion in its current state. The theory is grounded in a single foundational commitment: that the structural properties of coherent formations — binding, separation, morphological articulation, and persistence — are decomposable into independently graded dimensions and evaluable through self-referential operators. This formation-level description is formally richer than and not reducible to discrete object-level description, though the theory's equilibrium formations may themselves be object-like.

From this commitment, the theory develops a formal universe in which closure captures relational self-support (A1'–A4, with proved contraction at $a_{\mathrm{cl}}<4$), distinction captures exterior asymmetry (D-Ax1–3), morphological structure captures the articulation of core, boundary, and exterior (${\mathcal{Q}}_{\mathrm{morph}}=\frac{{\ell }_{\max }-c}{1-c}\cdot \mathrm{Artic}$, with proved axiom satisfaction QM1–4), and temporal transport captures the structural inheritance that constitutes persistence through time (E1–E4, with E3 reclassified as a solution constraint). Co-belonging (C1–C4, with proved resolvent realization) serves as a derived diagnostic for non-local structural integration but does not enter any predicate or energy term.

The proto-cohesion diagnostic vector $\mathbf{d}\in [0,1{]}^4$ unifies these four requirements into a graded formal assessment of formation quality, and the minimal energy principle — with mandatory volume constraint on the manifold ${\Sigma }_m$ — provides a variational characterization of formations. The theory has 54 fully proved theorems (Category A), 14 with structural parameters (Category B), 5 conditional results (Category C), and 5 retracted claims — totaling 78 formal claims, ~69% fully proved. Key results include non-trivial minimizer existence under a computable phase transition (T8-Core, T8-Full), gradient flow convergence (T14), stability advantage for non-idempotent closure (T3/T6), exact predicate-energy bridge (Sep = 1 - E_sep/m), closure residual bound for all τ (T-Bind-Proj), formation-birth on D₄-symmetric graphs (T-Birth-Parametric), the unified multi-formation persistence theorem (T-Persist-K-Unified) parametrized by the coupling measure ${\Lambda }_{\mathrm{coupling}}$, the W4 Pre-Objective Mechanism cluster (T-PreObj-1, T-PreObj-1G, Lemma 4), the W4-extended Goldstone dichotomy on translation-invariant graphs (T-V5b-T), the W5 Day 1 G0 σ-framework supporting structures (T-σ-Lemma-1/2/3 + T-σ-Theorem-3) that ground Commitment 14, the CV-1.5.1 D-6a Multi-Static cluster (T-Commitment-14-Multi-Static + T-σ-multi-A-Static + T-σ-multi-D-Static) that extends the σ-framework to the multi-formation interior, the CV-1.5.2 multi-formation bridge T-L1-F (Hard-Bar / Active-Count Bridge under L1-J Regime), and the W6 additions (CV-1.6–CV-1.11): hard-depth topological locking (T-ST-5a, Cat A), boundary precision (T-OP6-B, Cat A, OP-0006 RESOLVED), Gibbs measure continuity (T-P-F-ε0, Cat A), and the full P-F-A1 stochastic foundation Package I (T-PF-A1-AR + T-PF-A1-SDE + T-PF-A1-GI + T-PF-A1-PE, all Cat A).

W4 Critical resolution (2026-04-24). The three Critical open problems that gated v2.0 release for nearly a year — F-1 (K=2 vacuity), M-1 (K=1 preference), MO-1 (Morse inapplicability) — are all resolved. F-1 is SPLIT-RESOLVED (T-Merge (b) on pure ${\mathcal{E}}_{\mathrm{bd}}$, T-PreObj-1 (i) on full SCC). M-1 is LAYER-CLARIFIED as the proved theorem T-Merge (b) misframed as a problem. MO-1 is SIDESTEPPED by the σ-framework operating on the smooth single-formation manifold ${\Sigma }_m$ (no corners). The conceptual key is CN15 Static/Dynamic Separation: the static global minimum on pure ${\mathcal{E}}_{\mathrm{bd}}$ and the dynamic protocol-endpoint observables under full SCC are quantities of different layers that need not match. Critical blocker count: 3 → 0.

W4-extended addition (2026-04-26). The Pre-Objective Goldstone theorem T-V5b-T establishes a sub/super-lattice spectral dichotomy on translation-invariant graphs (torus $T^d$, cycle $C_n$): 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 result emerged from 8 V5b iterations (V1 → V5b''), in the course of which a reproducibility crisis (NQ-172, mode-indexing artifact in the analysis script) was identified and resolved with mode-agnostic detection. A new Cat C finding, V5b-F (partial Goldstone on boundary-modified graphs), is registered and carried as NQ-173.

W5 Day 1 G0 addition (2026-04-27). The σ-framework signature $\sigma (u^{\ast })=(\mathcal{F};\{(n_k,[{\rho }_k],{\lambda }_k)\})$ — declared as a Cat A definitional commitment in W4 (Commitment 14) — receives five Cat A supporting structures canonically merged: T-σ-Lemma-1 (irrep decomposition well-defined via Maschke + Schur orthogonality), T-σ-Lemma-2 (nodal count properties — sub-statements (i)-(iv) Cat A; (v)-(vi) Cat C riders within the parent entry), T-σ-Lemma-3 (Goldstone–ℓ=1 angular saturation, anchoring T-V5b-T-(e) universal nodal count = 2), T-σ-Theorem-3 (closed-form spectrum at uniform on $D_4$ free-BC grid), T-σ-Theorem-4 ($D_4\to {\mathbb{Z}}_2$ symmetry breaking at first pitchfork, $K_0=K_1$ leading-order degeneracy). Decision: Option α (5 separate §13 entries). Two erratum/refinement rounds (2026-04-27 evening + night) caught and fixed substantive math errors (IBP factor, $K_1$-vs-$K_0$ degeneracy, irrep-table off-diagonal entries, dimension-general extension); theorem status unchanged (all five remain Cat A).

Theory status (CV-1.11, W6 close, 2026-05-08): 54 Category A, 14 Category B, 5 Category C, 5 Retracted (78 claims, ~69% fully proved). W6 Cat A additions: T-ST-5a (Stereo-SCC Hard-Depth Topological Locking), T-OP6-B (PersRidge Boundary Equivalence, OP-0006 RESOLVED), T-P-F-ε0 (Gibbs Continuity), T-PF-A1-AR/SDE/GI/PE (P-F-A1 Package I stochastic foundation). W6 Cat B additions: T-P-F-ε0-K (Kramers stability), T-ST-5b (smooth-depth barrier raising), D-ST-1..5 (stereo diagnostic bridge), T-K-Select-PF (partial OP-0005-EQ), T-K-Select-OBS (partial OP-0005-OBS). Category B pre-W6: γ_eff ≈ 0.89 barrier exponent, general-graph birth supercriticality, d_min formula, Beyond-Weyl 33× quantification, T-σ-Theorem-4 (retroactive Cat B, re-promotion deferred to CV-1.12+ via γ-path Σ_m-Hessian audit), T-σ-Multi-1 (Goldstone-pair instability). Category C items: T-Persist-1(d) (β > 7α interior-gap condition), T-Persist-Full, T-Persist-K-Sep, T-Persist-K-Weak, T-Persist-K-Unified; plus V5b-F (W4-extended) and T-σ-Lemma-2 sub-statement riders (v)+(vi). Retracted: Thm 3.3 general τ, T-Merge (c)(d)(e), K-Saddle Conjecture. Infrastructure: OMS-2.0 Accepted — Full (Observer Moduli Space, Appendix OMS §A–§M in canonical.md, 2026-05-08). DECLARATION.md (DECL-1.0) + hypothesis_tree.md (HT-3.0) created.

Active open problems (post-CV-1.11): Critical 0; High 3 — OP-0005 K-Selection (3-way split: EQ partially resolved via T-K-Select-PF Cat B, OBS partially resolved via T-K-Select-OBS Cat B, DYN open — Eyring-Kramers, W9+), OP-0008 σ^A K-jump non-determinism (NOT solved by T-L1-F; Path B Cat B target), OP-0009 Multi-Formation Ontological Foundations (7 sub-items; OP-0009-K resolved via Commitment 16; 6/7 PARTIALLY resolved); OP-0006 RESOLVED (T-OP6-B, W6); Medium 4 — OP-0010..OP-0013 (Bind τ generalization, transport kernel uniqueness, persist composition, closure convergence rate); Low 2 — OP-0020 (dynamic topology), OP-0022 (continuous-time limit); OP-0021 (Stochastic Dynamics) substantially resolved via P-F-A1 Package I Cat A. Next: CV-1.12 via H-SINK (T-Temporal-Identity a/b/d, +3A target). Carry NQ: NQ-173 (V5b-F partial Goldstone), NQ-174 (ζ_*(graph) dependence), NQ-175 (3D extension), NQ-242 (dynamic σ_multi^A(t)), NQ-248 (multi-formation σ Phase 5).

The theory now stands as a mathematically structured ontology of pre-objective cohesion — with genuine theorems, honest gap accounting, and a clear distinction between what is proved, what is provisional, and what remains open. The W4 close moved the theory from "Critical-3-blocked" to "release-path-unblocked" via Option D (premise dissolution); the W4-extended Goldstone result extends the Pre-Objective Mechanism cluster to translation-invariant graph classes; the W5 Day 1 G0 σ-framework supporting structures complete the canonical grounding of Commitment 14; the W5 Day 6 T-L1-F bridge established the first multi-formation canonical Cat A theorem. W6 close (CV-1.11, 2026-05-08) adds stereo geometry (T-ST-5a), resolves the boundary precision problem (T-OP6-B, OP-0006 RESOLVED), and establishes a complete stochastic foundation for the constraint-manifold dynamics (P-F-A1 Package I: T-PF-A1-AR/SDE/GI/PE). The Observer Moduli Space is Accepted — Full (OMS-2.0, Appendix OMS §A–§M). The active frontier is CV-1.12 via H-SINK (T-Temporal-Identity a/b/d, +3A target) and OP-0005-DYN (Eyring-Kramers K-selection, W9+).

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