SCC Theorem Registry

Current canonical: CV-1.17 (68 Cat A / 19 B / 6 C / 5 Retracted = 98 claims, ~70% proved, sealed 2026-05-15). This registry is the CV-1.11 (W6) snapshot; the row-by-row CV-1.12–1.17 additions (T-Temporal-Identity, T-CC-StableK-Kernel, …) live in the status page and are pending reseal here.

Purpose: Register all claims (C-xxxx), proofs (P-xxxx), and canonical theorems (T-xxxx, CV-x.y). This is the authoritative index of what has been proved.

Structure: Rows are organized by canonical version (CV-1.0 → CV-1.11; this registry seals CV-1.11 — current canonical is CV-1.17, see the status page).

Directed acyclic graph of 18 SCC hero theorems across 6 bands: Foundation, Phase+Stability, W4 Capstone, W5 Bridge, W6 Stochastic Foundation, W6 Boundary+Selection. — The 16 SCC hero theorems among the 54 Cat A entries below, organized across 6 bands (Foundation, Phase+Stability, W4 Capstone, W5 Bridge, W6 Stochastic Foundation, W6 Boundary+Selection). Updated to CV-1.11. See /notes/scc-heroes/ for the curated index.

Canonical Theorems (Accepted into Canonical Spec)

Canonical Spec v1.11 (2026-05-06) — Current Version (W6 Close: 54A / 14B / 5C / 5R = 78 claims)

Additions over v1.5.2 (W6, 2026-05-04 → 2026-05-06 — CV-1.6 through CV-1.11):

CV-1.6 — Stereo-SCC branch (D1, 2026-05-04)

T-ID	Name	Status	Category	Notes
T-ST-5a	Hard-Depth Topological Locking	accepted	A	On a depth-disconnected stereo graph (objects at depths $z_1 \ll z_2$ ), the energy barrier between the two formation basins is $\Delta E = +\infty$ (state-space disconnection via the Goldstone mechanism). System forced to $K=2$ stable formations — one per depth-connected component. First theorem connecting SCC to stereo geometry.

v1.5.2 → v1.6 release notes (2026-05-04):

Added: 1 Cat A theorem (T-ST-5a, Stereo-SCC branch).
Counts: 46A → 47A, 61 → 62 claims.

CV-1.7 — Boundary Precision + Gibbs Continuity (D3, 2026-05-05/06)

T-ID	Name	Status	Category	Notes
T-OP6-B	PersRidge Boundary Equivalence	accepted	A	In phase-separation regime (H1–H5): $d_H(\mathrm{PersRidge}(u^), \partial\Omega^) \leq 2\sqrt{\alpha/\beta}$ . OP-0006 RESOLVED — computationally accessible PersRidge is a provably faithful proxy for the theoretical boundary.
T-P-F-ε0	Gibbs Measure Continuity ( $\varepsilon \to 0$ )	accepted	A	The Gibbs measure $\pi_T$ on $\widetilde\Sigma_M$ converges weakly to $\pi_T^0$ (deterministic formation-selector) as $\varepsilon \to 0$ . Geometric foundation for the low-temperature limit.
T-P-F-ε0-K	Kramers Exponent Stability	accepted	B	Kramers escape rate exponent is stable under small $\varepsilon$ perturbations; exponential-rate corrections sub-leading. Cat B (explicit spectral-gap condition).

v1.6 → v1.7 release notes (2026-05-05/06):

Added: 2 Cat A (T-OP6-B, T-P-F-ε0) + 1 Cat B (T-P-F-ε0-K).
OP-0006 Boundary Precision: RESOLVED via T-OP6-B.
Counts: 47A → 49A, 4 Cat B total, 62 → 65 claims.

CV-1.8 / CV-1.9 — P-F-A1 Package I: Reflected SDE + Stochastic Foundation (D4, 2026-05-06)

T-ID	Name	Status	Category	Notes
T-PF-A1-AR	Affine Reduction — Bounded Convex Polytope	accepted	A	The feasibility polytope $\mathcal{F}_M(\mathcal{P})$ is a bounded compact convex polytope; Lions-Sznitman domain condition satisfied. Geometric foundation for well-posed SDE.
T-PF-A1-SDE	Lions-Sznitman Reflected SDE	accepted	A	Reflected Langevin SDE on $\widetilde\Sigma_M$ is well-posed: existence via Lions-Sznitman Theorem 1 (convex domain), uniqueness via Tanaka pathwise argument.
T-PF-A1-GI	Unique Gibbs Invariant Measure $\pi_{T_*}$	accepted	A	$\pi_{T_} \propto e^{-\tilde{\mathcal{E}}/T_}$ is the unique invariant measure of the reflected SDE via zero-current + $L^2(\pi)$ kernel argument (Aronson 1968 heat kernel).
T-PF-A1-PE	Poincaré Inequality + Exponential Ergodicity	accepted	A	$\lambda_1 \geq (\pi^2/n)\exp(-\mathrm{osc}(\tilde{\mathcal{E}})/T_*)$ (Payne-Weinberger on bounded convex polytope + Holley-Stroock perturbation). Exponential TV convergence with explicit rate.

v1.7 → v1.8/v1.9 release notes (2026-05-06):

Added: 4 Cat A theorems — P-F-A1 Package I (AR → SDE → GI → PE chain), all Cat A.
Stochastic dynamics on $\widetilde\Sigma_M$ : well-posed reflected SDE + unique Gibbs invariant measure + exponential ergodicity.
Counts: 49A → 53A, 65 → 69 claims.

CV-1.10 — K-Selection Partial (EQ + OBS branches) + D-ST Diagnostic Bridge (D4, 2026-05-06/07)

T-ID	Name	Status	Category	Notes
T-K-Select-PF	Equilibrium K-Selection via Gibbs Measure	accepted	B	Under $\pi_{T_}$ , formation count $K_{\mathrm{act}}$ concentrates near the Gibbs-weighted modal value. Partially resolves OP-0005-EQ* (equilibrium branch). Does NOT solve OP-0005-DYN (Eyring-Kramers, W9+).
T-K-Select-OBS	Observation-Conditioned K-Selection	accepted	B	Under observation conditioning $\pi_{T_}(\cdot \mid \mathcal{O})$ , the posterior over $K_{\mathrm{act}}$ is computable. Partially resolves OP-0005-OBS* (observation branch).
D-ST-1..5	Stereo-SCC Diagnostic Bridge Entries	accepted	B	Five diagnostic bridge entries connecting stereo-graph depth structure to the SCC formation diagnostics $\mathbf{d} = (\mathrm{Bind}, \mathrm{Sep}, \mathrm{Inside}, \mathrm{Persist})$ . Categorical bridge between depth topology and cohesion field.

v1.9 → v1.10 release notes (2026-05-06/07):

OP-0005 K-Selection: 3-WAY SPLIT registered — EQ partially resolved via T-K-Select-PF (Cat B), OBS partially resolved via T-K-Select-OBS (Cat B), DYN (Eyring-Kramers) remains open (W9+).
Added: 7 Cat B entries (T-K-Select-PF, T-K-Select-OBS, D-ST-1..5).
Counts: 53A, 69 → 76 claims, 11 Cat B total.

CV-1.11 — Audit + Declarations + OMS-2.0 Full (D5–D6, 2026-05-07/08)

T-ID	Name	Status	Category	Notes
T-ST-5b	Smooth-Depth Barrier Raising	accepted	B	On smooth-depth stereo graphs (objects at depths $z_1 < z_2$ not sharply separated), the merge barrier is raised monotonically in $\Delta z = z_2 - z_1$ . Cat B (explicit depth-gap lower bound condition).
OMS-2.0	Observer Moduli Space — Accepted Full	accepted	infrastructure	The space of symmetry-inequivalent critical formations is proved finite and stratified: static branch decomposition ( $G_{\mathrm{cw}} = \{e\}$ generic symmetry, codim-1 strata) + temporal extension ( $G_{\mathrm{cw}}^{\mathrm{temp}} = \{e\}$ , 19-branch $\Delta^3$ map). Appendix OMS (§A–§M) added to canonical.md.

v1.10 → v1.11 release notes (2026-05-07/08):

Added: 1 Cat B (T-ST-5b); OMS-2.0 Accepted — Full (Appendix OMS §A–§M in canonical.md).
Infrastructure: DECLARATION.md (DECL-1.0) + hypothesis_tree.md (HT-3.0) created.
Final W6 counts: 54A / 14B / 5C / 5R = 78 claims, ~69% fully proved.
Counts: 53A → 54A, 11 → 14 Cat B (net; T-σ-Theorem-4 re-promotion deferred), 76 → 78 claims.

Full W6 (CV-1.5.2 → CV-1.11) summary:

Added: 8 Cat A + 9 Cat B = 17 new claims.
Counts: 46A → 54A, 5B → 14B, 61 → 78 claims, 75% → ~69% fully proved (denominator grows faster — more partial results).
OP-0006 RESOLVED (T-OP6-B). OP-0005 3-way split (EQ/OBS partial, DYN open). OMS-2.0 Full.

Canonical Spec v1.5.2 (2026-05-02) — Previous Version (W5 Day 6: L1-F Hard-Bar / Active-Count Bridge Conditional Cat-A)

Additions over v1.5.1 (W6, 2026-05-02):

T-ID	Name	Status	Category	Source	Proof	Experiments	Notes
T-L1-F	Hard-Bar / Active-Count Bridge under L1-J Regime	accepted	A (conditional under L1-J package)	C-0721	P-0721	(theoretical via L1-A..L1-L chain; numerical L1-I 439/1920 FEASIBLE_WITH_BUDGET on $T^2_{20}$ ; L1-H2 stress tests 5/5; L1-J PO-1 decay-to-cut 6/6; L1-K external audit passed)	First multi-formation canonical Cat A theorem. Conditional theorem on finite shared-pool multi-formation states under hypothesis package $(P0)$ – $(P11)$ : $K_{\mathrm{bar}}^{\ell_{\min}}(U(\mathbf u);G) = K_{\mathrm{act}}^{\varepsilon}(\mathbf u)$ AND labeled bijection $\mathcal{A}_{\mathrm{bar}}: A^{\varepsilon} \to \mathrm{Bars}_0^{\mathrm{term}}(U;G)$ via primary representative $q_j^U = \arg\max_{x \in N_j^r}^{\prec} U(x)$ . L1-A through L1-L 13-step working chain + L1-K external audit + L1-K-REPAIR (R-1, R-2, R-3, R-4 proof-hygiene repairs). Explicit non-claims: does NOT solve OP-0005 (K-Selection) or OP-0008 ( $\sigma^A$ K-jump non-determinism); does NOT establish $K_{\mathrm{soft}}^\phi = K_{\mathrm{act}}$ ; reservoir theory not promoted to canonical; P7 not generally derived from all SCC states.

v1.5.1 → v1.5.2 release notes (2026-05-02):

Added: 1 Cat A conditional (T-L1-F) synthesizing the L1-A..L1-L chain.
Counts: 45A → 46A, 60 → 61 claims, 75% → 75% fully proved (unchanged %).
T-L1-F is the first multi-formation canonical Cat A theorem.

Canonical Spec v1.5.1 (2026-04-29) — Previous Version (W5 Day 3 EOD: D-6a Multi-Static + Ontological Depth + Critic reinforcement)

Additions over v1.5 (W5 Day 3 EOD, 2026-04-29):

T-ID	Name	Status	Category	Source	Proof	Experiments	Notes
T-Commitment-14-Multi-Static	Multi-Formation σ-Signature on K-field (Static)	accepted	A definitional	C-0717	P-0717	(theoretical extension of Commitment 14)	Defined on $\widetilde{\Sigma}^{K_{\mathrm{field}},\circ}_M$ interior (Option A pragmatic; corners deferred to NQ-248 W7+). $\sigma_{\mathrm{multi}} = (\sigma^A, \sigma^D)$ joint invariant under wreath-product $\mathrm{Aut}(G) \wr S_{K_{\mathrm{act}}}$ . From D-6a Multi-Static.
T-σ-multi-A-Static	Within-Formation σ-Tuple Multi-Set Invariance	accepted	A (well-separated regime)	C-0718	P-0718	(theoretical via Coupling Bound Lemma + T-σ-Lemma-1)	Multi-set $\{\sigma_j\}_{j=1}^{K_{\mathrm{act}}}$ under $S_{K_{\mathrm{act}}}$ permutation; reduces to Commitment 14 σ at $K_{\mathrm{act}} = 1$ . From D-6a Multi-Static.
T-σ-multi-D-Static	Between-Formation Cohomology Pull-Back	accepted	A definitional	C-0719	P-0719	(wreath-product representation theory, Specht 1935 + James-Kerber 1981)	Conjugacy-class label in $H^1(\mathrm{Aut}(G) \wr S_{K_{\mathrm{act}}}; \mathrm{Stab}(\mathbf{u}^*))$ . From D-6a Multi-Static.
T-σ-Multi-1	Multi-Formation Goldstone-Pair Instability (Phase 4 Static)	tentative	B target	C-0720	(sketch)	(Phase 4 D-6a numerical, Phase 6 Q1 box-clipping)	Goldstone-pair separation $\Delta\lambda \approx O(\lambda_{\mathrm{rep}} e^{-c_0 D_{\mathrm{sep}}})$ under V5b-T per-formation regime. Cat A pending NQ-242 numerical anchor.

Sub-statement additions to T-V5b-T:

Sub-ID	Name	Status	Category	Notes
(V5b-T-zero)	Sub-Spinodal Translation-Invariant Regime	accepted	A definitional (sub-statement)	$\mu_{\mathrm{Gold}}^{\mathrm{V5b-T-zero}} = 0$ exactly (discrete translation orbit on $\mathbb{Z}_L^d$ ). Replaces V5b-T' WITHDRAWN (NQ-198f phantom on torus). Empirical anchor NQ-198f $T^2_{20}/T^2_{28}$ $
(V5b-F-empirical)	V5b-F Goldstone Mass Scaling	accepted	B target (sub-statement)	$\mu_{\mathrm{Gold}}^{\mathrm{V5b-F}} \approx C(\beta) \cdot \\|\partial S\\|/n$ , $C(\beta=4, \xi_0=0.5) \approx 13.2 \pm 0.4$ (NQ-198a 6 corner-sat, SNR 35). Refutes Phase 3 heuristic + Day 3 §4 derivation.

Status changes to existing entries:

C-ID	Name	Old Status	New Status	Reason
C-0716	T-σ-Theorem-4 σ at First Pitchfork	Cat A in $\epsilon$ -small regime	Cat B in $\epsilon$ -small regime	Retroactive Critic 7-agent verdict 2026-04-29: Errata Round 1 caught Morse-index inconsistency at original merge time. Cat A re-promotion deferred to CV-1.7+.

v1.5 → v1.5.1 release notes (2026-04-29):

Added (4 new C-IDs): 3 Cat A definitional (Multi-Static + multi-A-Static + multi-D-Static) + 1 Cat B target (Multi-1).
Added (sub-statements within T-V5b-T entry): V5b-T-zero (Cat A def, replaces V5b-T' WITHDRAWN) + V5b-F-empirical (Cat B target, μ_Gold ≈ C(β)·‖∂S‖/n NQ-198a).
Status revision: T-σ-Theorem-4 retroactive Cat A → Cat B per Critic 7-agent verdict + Errata Round 1 Morse-index inconsistency.
Withdrawn: D-5 V5b-T' new entry candidate (NQ-198f phantom finding).
Counts: 43A → 45A (net +2: +3 D-6a Cat A − 1 Theorem-4 격하), 4B → 5B (+1 Theorem-4 + 1 Multi-1), 57 → 60 claims, 75% → 75% fully proved.
Commitment 16 K_field/K_act two-tier decomposition added (resolves OP-0009-K).
Open problems: OP-0008 σ^A K-jump non-determinism + OP-0009 Multi-Formation Ontological Foundations registered HIGH; OP-0003 MO-1 re-activation rider added.

Canonical Spec v1.5 (2026-04-27) — Previous Version (W5 Day 1 G0: σ-Framework Supporting Structures)

Additions over v1.4 (W5 Day 1 G0, 2026-04-27):

T-ID	Name	Status	Category	Source	Proof	Experiments	Notes
T-σ-Lemma-1	σ-Framework Irrep Decomposition Well-Defined	accepted	A	C-0712	P-0712	(theoretical, finite-group rep theory)	Hessian commutes with $G_u$ -action; isotypic decomposition via Maschke + Schur; finite-graph hypothesis essential; trivial-stabilizer case vacuous
T-σ-Lemma-2	σ-Framework Nodal Count Properties	accepted	A (i,ii,iii,iv) + C (v,vi conditional)	C-0713	P-0713	NQ-141 (W4-04-25, R23 32×32 numerical)	Graph-intrinsic + Aut(G)-equivariance + lower bound $\mathcal{N} \geq 2$ + sign-flip Cat A; Courant upper bound + $G_u$ -orbit divisibility Cat C riders
T-σ-Lemma-3	Goldstone–ℓ=1 Angular Saturation	accepted	A (continuum, dimension-general)	C-0714	P-0714	NQ-170c (2D torus L=20 ζ=0.5 overlap 0.97)	IBP identity $\mathcal{P}_{\ell=1}[\delta u_x] = (-c_d \int u^*(r)\,dr,\ 0)$ with $c_1=2,\,c_2=\pi,\,c_3=4\pi$ ; Goldstone basis automatically ℓ=1; nodal count = 2 universal (anchors T-V5b-T-(e))
T-σ-Theorem-3	σ at Uniform on $D_4$ Free-BC Grid (Closed Form)	accepted	A	C-0715	P-0715	exp_hessian_uniform_v2 (NQ-141, $L = 4$ to $32$ , $< 10^{-9}$ precision)	$\mu_k = 4\alpha\lambda_k^{\mathrm{Lap}} + \beta W''(c)$ ; full $D_4$ irrep table on cosine basis; $\mathcal{N}(\phi_{(p,q)}) = (p+1)(q+1)$
T-σ-Theorem-4	σ at First Pitchfork on $D_4$ Free-BC Grid (Leading Order, continuum-limit claim)	accepted	B (was A → B retroactive at CV-1.5.1, 2026-04-29)	C-0716	P-0716	(theoretical via T-Birth-Parametric + R22 axis-aligned, continuum-limit; numerical NQ-187 on discrete grid)	σ at First Pitchfork on $D_4$ Free-BC Grid (Leading Order, continuum-limit claim). Cat A → Cat B retroactive at CV-1.5.1 per Critic 7-agent verdict + Errata Round 1. Continuum vs discrete grid caveat (W6 NQ-187 audit 2026-05-04): canonical statement (ii) is a continuum-limit prediction (R22 §3.3 unit-square Lebesgue integral) NOT realized on finite discrete $L \le 16$ grids. NQ-187 measured $\mu_0/\epsilon \approx 1$ , $\mu_1/\epsilon \approx 2$ , ratio $\mu_1/\mu_0 \approx 2$ (not $1$ degeneracy), exponent $p \approx 1.03$ (not $p = 2$ predicted). Three reconciliation hypotheses (α continuum extrapolation, β R22 re-derivation, γ $\Sigma_m$ -Hessian convention) under audit. Cat A re-promotion deferred to CV-1.7+.

v1.4 → v1.5 release notes (2026-04-27):

Added: 5 Cat A theorems (T-σ-Lemma-1/2/3 + T-σ-Theorem-3/4) — σ-framework supporting structures grounding Commitment 14.
Decision: Option α (5 separate §13 entries) — mathematically independent statements deserve individual canonical visibility.
Counts: 38A → 43A, 52 claims → 57 claims, 73% → 75% fully proved.
T1 explosion: 3 → 8 (Lemma 1/2/3 + Theorem 3/4 each individually T1).
Two erratum/refinement rounds (2026-04-27 evening + night) caught and fixed substantive math errors and structural refinements; theorem status unchanged (all 5 remain Cat A).
Sub-statement riders canonically registered: T-σ-Lemma-2 (v) Courant + (vi) orbit divisibility Cat C bundled within parent Cat A entry.
Carry: NQ-176..NQ-190 (15 spawn questions across the five structures + Round-2 audit spawns).
Pending user decision: Commitment 14 (O5') multi-irrep eigenspace convention, Commitment 14 (O7) tie-breaking convention.

Canonical Spec v1.4 (2026-04-26) — Previous Version (W4-extended close)

Additions over v1.3 (W4-extended close, 2026-04-26):

T-ID	Name	Status	Category	Source	Proof	Experiments	Notes
T-V5b-T	Pre-Objective Goldstone on Translation-Invariant Graphs	accepted	A	C-0710	P-0710	E-0095, E-0096, E-0097	Sub/super-lattice spectral dichotomy on torus T^d / cycle C_n; 2D commensurability split; 1D Goldstone; nodal count = 2 universal. After 8 V5b iterations (V1 → V5b'').

v1.3 → v1.4 release notes (2026-04-26):

Added: 1 Cat A theorem (T-V5b-T) — V5b verification cycle (8 iterations) result.
Counts: 37A → 38A, 51 claims → 52 claims, 73% → 73% fully proved.
New finding registered: V5b-F (Partial Goldstone on Boundary-Modified Graphs) — Cat C, NQ-173 carry.
Reproducibility crisis identified+resolved: NQ-172 (mode-indexing artifact in NQ-170 analysis script).

Canonical Spec v1.3 (2026-04-25) — Previous Version (Frozen)

Additions over v1.2 (W4 merge, 2026-04-19 ~ 2026-04-25):

T-ID	Name	Status	Category	Source	Proof	Experiments	Notes
T-PreObj-1	Pre-Objective Multi-Peak Formation Mechanism	accepted	A	C-0700	P-0700	E-0090, E-0091	F=1 single-disk minimizer non-critical under full SCC; gradient flow attracts to multi-peak F≥2; IC-protocol dichotomy adaptive bounded vs random ~ $L^{2.8}$
T-PreObj-1G	Pre-Objective Mechanism Graph-Class Independent	accepted	A	C-0701	P-0701	(theoretical)	Conclusions (i),(ii) of T-PreObj-1 hold on any finite connected graph under (G1)–(G4) hypotheses
Lemma 4	Quadratic Form Positive Definite (M matrix)	accepted	A	C-0702	P-0702	E-0090	$M \in \mathbb{R}^{2 \times 2}$ of $(g_{cl}, g_{sep})$ gradients, PD under linear independence; destabilization magnitude $\Lambda^T M \Lambda > 0$
F-1 Resolution Corollary	F-1 SPLIT-RESOLVED via T-Merge(b) + T-PreObj-1	accepted	A	(corollary)	(corollary)	—	Pure $\mathcal{E}_{bd}$ portion via T-Merge(b); full SCC portion via T-PreObj-1 (i)

v1.2 → v1.3 release notes (2026-04-25):

Added: 2 Cat A theorems (T-PreObj-1, T-PreObj-1G), 1 Cat A lemma (Lemma 4), 1 Cat A corollary (F-1 split-resolution).
Status changes: C-0550 (F-1), C-0551 (M-1), C-0552 (MO-1) — challenged → resolved/clarified/sidestepped.
Counts: 35A → 37A, 49 claims → 51 claims, 71% → 73% fully proved.
Critical blockers: 3 (F-1, M-1, MO-1) → 0 (all resolved/clarified/sidestepped).

Canonical Spec v1.2 (2026-04-12) — Previous Version (Frozen)

T-ID	Name	Status	Category	Source	Proof	Experiments	Notes
T-1	Existence of Minimizers	accepted	A	C-0001	P-0001	E-0001, E-0002	SCC minimizer always exists on Σ_m
T-3	Stability of Interior Minimizers	accepted	A	C-0003	P-0003	E-0003	Hessian positive on interior; local stability
T-6a	Closure Fixed Point (Existence)	accepted	A	C-0006a	P-0006a	E-0005	u* = Cl_t(u*) ∃ for all parameters
T-6b	Closure Fixed Point (Stability)	accepted	A	C-0006b	P-0006b	E-0006	Closure FP is attracting in stability metric
T-6-Stability	Stability of Closure FP	accepted	A	C-0006c	P-0006c	E-0007	Full spectral analysis
T-7	Enhanced Metastability	accepted	A	C-0007	P-0007	E-0008, E-0009	Residence time > expected near saddle
T-8-Core	Phase Transition (Core Dominance)	accepted	A	C-0008	P-0008	E-0010, E-0011	Binuclear → mononuclear at critical β
T-8-Full	Phase Transition (Global)	accepted	A	C-0009	P-0009	E-0012, E-0013	Full energy landscape bifurcation
T-11	Γ-Convergence	accepted	A	C-0011	P-0011	E-0014	Variational convergence under scaling
T-14	Gradient Flow	accepted	A	C-0014	P-0014	E-0020:E-0022	Gradient descent converges to minimizer
T-20	Axiom Consistency	accepted	A	C-0020	P-0020	E-0025	Axioms A1–E mutually consistent
C-Axioms	Cohomology-Resolvent Alignment	accepted	A	C-0101	P-0101	E-0030:E-0032	C3'' symmetrization complete (upgraded 04-03)
QM-1	Quantum Mechanical Analogy (Eigenvalue)	accepted	A	C-0110	P-0110	E-0040	Fiedler eigenvalue is binding edge
QM-2	QM-2 (Spectral Gap)	accepted	A	C-0111	P-0111	E-0041	Spectral gap related to phase transition
QM-3	QM-3 (Perturbation)	accepted	A	C-0112	P-0112	E-0042	Perturbations stay confined
QM-4	QM-4 (Commutation)	accepted	A	C-0113	P-0113	E-0043	Operator commutation holds generically
T-Bind-Proj	Tangential Residual Bound at Constrained Minimizers	accepted	A (for all τ_cl ∈ (0,1))	C-0200	P-0200	E-0050:E-0052	Tangential Residual Bound at Constrained Minimizers; KKT projection + Banach inversion of restricted operator with $\sigma_{\min} \ge 1 - a_{\mathrm{cl}}/4$ ; general τ via binary mass-balance formula $\Phi(\tau; a_{\mathrm{cl}}, c)$ . Phase 13 upgrade applied 2026-04-07 (Erratum). Reconfirmed W6 G2 audit 2026-05-04.
T-Bind-Full	Bind Lower Bound at Constrained Minimizers	accepted	A	C-0201	P-0201	E-0053	Bind Lower Bound at Constrained Minimizers; $\mathsf{Bind}(\hat u) \ge 1 - f(\text{params})$ , $n$ -independent for $O(1)$ params. Follows from T-Bind-Proj + universal gradient bounds. Phase 13 upgrade applied 2026-04-07 (Erratum).
Predicate-Energy Bridge	Energy ↔ Diagnostic Alignment	accepted	A	C-0300	P-0300	E-0060:E-0063	Energy minimization ↔ diagonal optimization (upgraded 04-03)
Deep Core Dom. 2b	Deep Core Dominance	accepted	A	C-0301	P-0301	E-0064, E-0065	Core is always dominant in asymmetric regime (upgraded 04-03)
T-Persist-1(a)	Transport Persistence (base)	accepted	C	C-0400	P-0400	E-0070	Conditional: assumes generic parameters
T-Persist-1(b)	Transport Persistence (basin unconditional)	accepted	A	C-0401	P-0401	E-0071, E-0072	Unconditional: genericity automatic (upgraded 04-03)
T-Persist-1(d)	Transport Persistence (fixed stratum)	accepted	C	C-0402	P-0402	E-0073	Conditional: on fixed active stratum
T-Persist-1(e)	Transport Persistence (confinement)	accepted	A	C-0403	P-0403	E-0074	Tight confinement bound (upgraded 04-03)
T-Persist-Full	Transport Persistence (full composition)	accepted	C	C-0404	P-0404	E-0075	Conditional on multiple regime conditions
T-Persist-K-Sep	K-field Persistence (well-separated)	accepted	C	C-0500	P-0500	E-0076, E-0077	Conditional: on well-separated regime + per-formation persist. Category C per canonical Erratum 2026-04-07 (regime-conditional WS/SR).
T-Persist-K-Weak	K-field Persistence (weak coupling)	accepted	C	C-0501	P-0501	E-0078	Conditional: on weakly-interacting regime
T-Persist-K-Unified	K-field Persistence (parametric)	accepted	C	C-0502	P-0502	E-0046, E-0047	Parametric family (Sep/Weak/Strong); 100% validation (new v1.2). Category C per canonical Erratum 2026-04-07 (selected-branch + 5 structural hypotheses).

Active Claims (Not Yet Canonical) / Resolved Claims

C-ID	Name	Status	Category	Proof (P-ID)	Experiments	Notes
C-0550	F-1: K=2 Vacuity Problem	✅ SPLIT-RESOLVED (2026-04-24)	A	P-0700 + T-Merge(b)	E-0090, E-0091	Pure $\mathcal{E}_{bd}$ portion: T-Merge(b) Cat A pre-existing. Full SCC portion: T-PreObj-1 (i) Cat A.
C-0551	M-1: K=1 Always Preferred	✅ LAYER-CLARIFIED (2026-04-24)	A	T-Merge(b)	none	Proved theorem (T-Merge(b)) misframed as problem. Pure $\mathcal{E}_{bd}$ : theorem holds. Full SCC: comparison not framed (T-PreObj-1 makes F=1 non-critical).
C-0552	MO-1: Morse Theory Invalid	⚪ SIDESTEPPED (2026-04-24)	A (single-formation)	(sidestep)	none	Single-formation σ-framework operates on Σ_m (no corners). Multi-formation extension to $\Sigma^K_M$ still open (Phase 5).
C-0553	Type A/B Classification	challenged	OP	exp65	E-0065	exp65 invalidates; Type B never observed
C-0600	K-field Model Selection	tentative (partially addressed)	pending	none	exp66:exp73	W4 σ-framework + Static/Dynamic Separation (CN15) provides partial answer; full mechanism still open
C-0700	T-PreObj-1 Pre-Objective Mechanism	✅ accepted Cat A	A	P-0700	E-0090, E-0091	New 2026-04-24. F=1 disk non-criticality + multi-peak attractor + IC-protocol dichotomy
C-0701	T-PreObj-1G Graph-Class Independent	✅ accepted Cat A	A	P-0701	(theoretical)	New 2026-04-24. Conclusions (i),(ii) hold on any finite connected graph under (G1)–(G4)
C-0702	Lemma 4 Quadratic Form PD	✅ accepted Cat A	A	P-0702	E-0090	New 2026-04-24. M positive definite under $g_{cl}, g_{sep}$ linear independence
C-0710	T-V5b-T Pre-Objective Goldstone on Translation-Invariant Graphs	✅ accepted Cat A	A	P-0710	E-0095, E-0096, E-0097	New 2026-04-26 (W4 extended). After 8 V5b iterations
C-0711	V5b-F Partial Goldstone on Boundary-Modified Graphs	tentative	C	P-0711	E-0096 (free BC partial)	New 2026-04-26 (W4 extended). Cat C new finding. NQ-173 carry.

Counterexamples & Challenges (X-xxxx)

X-ID	Refutes	Status	Description	Impact
X-0001	~~C-0550 (F-1 Validity)~~ — superseded 2026-04-24	superseded	Originally: K=2 energy 4.66 vs K=1 energy 2.25. W4 reframing: this evidence is the correct T-Merge(b) statement, not a refutation. F-1 was misframed as problem.	F-1 SPLIT-RESOLVED via Option D (premise dissolution)
X-0002	Type A/B Classification	validated	exp65: all configs Type A (0 Type B observed); breaks 04-07 interpretation	Type classification rejected as non-real phenomenon

Canonical Spec Version History

CV-1.0 (2026-04-01)

Theorems: T-1, T-3, T-6a, T-6b, T-6-Stability, T-7, T-8-Core, T-8-Full, T-11, T-14, T-20
QM Results: QM-1, QM-2, QM-3, QM-4 (11 Category A)
Provisional: Predicate-Energy Bridge, Deep Core Dom. 2b (Category B at the time)
K-field: T-Persist-K-Sep (Category B), T-Persist-K-Weak (Category C)
Notes: Initial comprehensive spec; 35+ theorems claimed

CV-1.1 (2026-04-03) — PLAN_0403 Tier 1 Complete

Upgraded to Category A:
- C-Axioms (C3'' symmetrization gap closed)
- Predicate-Energy Bridge (formalized)
- Deep Core Dominance 2b (strengthened)
New Unconditional Results:
- T-Persist-1(b): Basin unconditional via genericity argument
- T-Persist-1(e): Confinement tight bounds (2.4-3.5×)
New: T-Bind-Full (Category A, τ=1/2 only)
Count: 35 → 38 Category A (3 upgraded)

CV-1.2 (2026-04-12) — Frozen, with Audit Clarifications

Added: T-Persist-K-Unified (Category B; parametric coverage of Sep/Weak/Strong regimes)
Explicit Assumptions: All K-field theorems now state "fixed K, fixed m" constraint
Status Clarifications:
- F-1, M-1, MO-1 documented as unresolved (not silently ignored) — resolved in CV-1.3
- Type A/B classification retracted (exp65 invalidated)
- Morse theory MO-1 vulnerability flagged — sidestepped in CV-1.3
Count: 35A/4B/5C/5R (49 claims, 71% fully proved)
Retracted: K-Saddle Conjecture, r̄₀ general τ, T-Merge (c)(d)(e)

Top-down split diagram: F-1 box decomposes into pure E_bd (T-Merge b, Cat A) and full SCC (T-PreObj-1, Cat A) layers, with dichotomy dissolved at the bottom. — F-1 split-resolution (W4, 2026-04-24): both layers are Category A. The Critical-3 OPs were dissolved by Option D (premise dissolution), not by an external constraint or K-selection mechanism.

Open Problems (OP-xxxx) — Updated 2026-05-04 (post-W6 G2 audit; full set unified into theorem_status.md)

OP-ID	Problem	Severity	Status
OP-0001	F-1: K=2 vacuous	(was Critical) Resolved	SPLIT-RESOLVED (2026-04-24) via T-PreObj-1 (i) + T-Merge (b)
OP-0002	M-1: K=1 preferred	(was Critical) Resolved	LAYER-CLARIFIED (2026-04-24) — proved theorem misframed
OP-0003	MO-1: Morse fails	(was High) Sidestepped	SIDESTEPPED (2026-04-24) for single-formation σ scope; re-activation rider on D-6b approval or NQ-248 multi-formation Morse work (added CV-1.5.1, 2026-04-29)
OP-0004	Type A/B Classification Invalidation	High	RETRACTED (empirically invalidated; exp65 0/4 Type B observed)
OP-0005	K Selection Mechanism (Missing)	High	3-WAY SPLIT (CV-1.10, 2026-05-06): EQ partially resolved via T-K-Select-PF (Cat B); OBS partially resolved via T-K-Select-OBS (Cat B); DYN (Eyring-Kramers) open (W9+)
OP-0006	Boundary Definition Precision	High	RESOLVED (CV-1.7, 2026-05-06) via T-OP6-B (PersRidge Boundary Equivalence, Cat A, phase-separation regime): $d_H(\mathrm{PersRidge}(u^), \partial\Omega^) \leq 2\sqrt{\alpha/\beta}$
OP-0008	σ^A K-jump Inheritance Non-Determinism	High	OPEN (CV-1.5.1, W5 Day 4); Path B σ-rich + Φ-rich Cat B target; CV-1.7 Commitment 18 candidate
OP-0009	Multi-Formation Ontological Foundations	High	OPEN (CV-1.5.1, W5 Day 4; 7 sub-items); 1/7 RESOLVED via Commitment 16 (OP-0009-K), 6/7 PARTIALLY
OP-0010	Bind Generalization	Medium	OPEN (T-Bind-Proj/Full now Cat A per Phase 13 — see W6 G2 audit; OP-0010 retains for any further generalization questions)
OP-0011	Transport kernel exact form	Medium	TENTATIVE
OP-0012	Persistence composition	Medium	OPEN
OP-0013	Closure operator convergence rate	Medium	OPEN
OP-0020	Dynamic Topology (Out of Scope)	Low	seed (formerly listed as OP-0007 in this file pre-2026-05-04 unification)
OP-0021	Stochastic Dynamics	Low	SUBSTANTIALLY RESOLVED — P-F-A1 Package I fully Cat A (T-PF-A1-AR/SDE/GI/PE, CV-1.8–CV-1.9, 2026-05-06): reflected SDE well-posed, unique Gibbs invariant measure, exponential ergodicity. P-F-A1 Package II OPEN.
OP-0022	Continuous-Time Limit	Low	seed

W4 changes (2026-04-25): Critical blockers 3 → 0. F-1/M-1/MO-1 all resolved/clarified/sidestepped via T-PreObj-1 family + T-Merge(b) + σ-framework single-formation scope.

W5 changes (2026-04-29 CV-1.5.1 + 2026-04-30 W5 Day 4): OP-0008 σ^A K-jump non-determinism + OP-0009 Multi-Formation Ontological Foundations registered High. OP-0009-K resolved via Commitment 16. OP-0003 MO-1 re-activation rider added.

W6 changes (CV-1.6 → CV-1.11, 2026-05-04 to 2026-05-08): OP-0006 RESOLVED via T-OP6-B (Cat A, phase-separation regime). OP-0005 3-way split registered: EQ partially resolved (T-K-Select-PF Cat B), OBS partially resolved (T-K-Select-OBS Cat B), DYN open (Eyring-Kramers, W9+). OP-0021 Stochastic Dynamics substantially resolved via P-F-A1 Package I (T-PF-A1-AR/SDE/GI/PE all Cat A). High active set now: 3 (OP-0005 EQ/OBS partial + DYN open; OP-0008 σ^A K-jump; OP-0009 Multi-Formation Foundations).

Proof Status Summary (Updated 2026-05-09 — CV-1.11 final)

Status	Count	Examples
Category A (Fully Proved)	54 (46 at CV-1.5.2 + 8 W6: T-ST-5a, T-OP6-B, T-P-F-e0, T-PF-A1-AR/SDE/GI/PE)	T-1, T-20, QM-1:4, C-Axioms, Predicate-Energy Bridge, T-PreObj-1, T-PreObj-1G, Lemma 4 (W4), T-V5b-T (W4 extended), T-sigma-Lemma-1/2/3, T-sigma-Theorem-3 (W5 G0); T-Commitment-14-Multi-Static, T-sigma-multi-A/D-Static (CV-1.5.1); T-L1-F (CV-1.5.2); W6: T-ST-5a, T-OP6-B, T-P-F-e0, T-PF-A1-AR, T-PF-A1-SDE, T-PF-A1-GI, T-PF-A1-PE
Category B (Conditional)	14 (5 pre-W6 + 9 W6 additions)	Pre-W6: gamma_eff, T-Birth-Parametric General, T-d_min-Formula, T-Beyond-Weyl, T-sigma-Theorem-4 (retroactive), T-sigma-Multi-1; W6: T-P-F-e0-K (Kramers stability), T-ST-5b (smooth-depth barrier), D-ST-1..5 (stereo diagnostic bridge), T-K-Select-PF (partial OP-0005-EQ), T-K-Select-OBS (partial OP-0005-OBS)
Category C (Very Conditional)	5 + riders	T-Persist-1(a/d), T-Persist-Full, T-Persist-K-Sep, T-Persist-K-Weak, T-Persist-K-Unified; V5b-F (NQ-173 carry); T-sigma-Lemma-2 (v)+(vi) riders bundled in Cat A parent
Resolved/Clarified/Sidestepped (W4)	3	C-0550 (F-1 split-resolved), C-0551 (M-1 layer-clarified), C-0552 (MO-1 sidestepped)
Challenged	1	C-0553 (Type A/B)
Retracted	5	K-Saddle Conjecture, r_0-bar general tau, T-Merge (c)(d)(e)
Open (active)	High: 3 (OP-0005 partial + DYN open, OP-0008, OP-0009) + Medium: 4 + Low: 2	OP-0006 RESOLVED (T-OP6-B W6); OP-0005 3-way split: EQ/OBS partial (Cat B), DYN open (W9+); OP-0008 sigma^A K-jump open; OP-0009 7 sub-items partially resolved; OP-0021 substantially resolved (P-F-A1 Package I Cat A)

Cross-Reference by Topic

Single-Formation Theory

Existence: T-1
Stability: T-3, T-6-Stability, T-7
Phase Transition: T-8-Core, T-8-Full
Convergence: T-11, T-14
Diagnostics: T-Bind-Proj/Full, Predicate-Energy Bridge

Multi-Formation (K-field)

Temporal Persistence: T-Persist-K-Sep, T-Persist-K-Weak, T-Persist-K-Unified
Global Stability: Deep Core Dominance 2b (conditional)
Open: F-1, M-1, OP-0006

Foundational

Consistency: T-20, C-Axioms
Quantum Analogy: QM-1:4

Maintenance

Owned by: Lead + Archivist
Updated: When new C-xxxx promoted to canonical or P-xxxx completed
Validation: build_dependency_graph.py checks for consistency

Last updated: 2026-05-09 (CV-1.11 final; W6 additions CV-1.6 through CV-1.11)
Total canonical theorems: 68 = 54 Cat A + 14 Cat B + 5 Cat C — 5 retracted (78 claims, ~69% fully proved)
Open problems: see Open Problems table above (3 High + 4 Medium + 2 Low active; 3 Critical resolved W4; OP-0006 RESOLVED W6; OP-0021 substantially resolved W6)
W4 additions (2026-04-25): T-PreObj-1, T-PreObj-1G, Lemma 4
W4 extended (2026-04-26): T-V5b-T (Goldstone on Translation-Invariant Graphs)
W5 Day 1 G0 (CV-1.5, 2026-04-27): T-sigma-Lemma-1/2/3 + T-sigma-Theorem-3/4 sigma-framework supporting structures
CV-1.5.1 (2026-04-29): T-Commitment-14-Multi-Static, T-sigma-multi-A/D-Static (Cat A); T-sigma-Multi-1 (Cat B); Commitment 16 K_field/K_act decomposition
CV-1.5.2 (2026-05-02): T-L1-F (Hard-Bar / Active-Count Bridge, first multi-formation canonical Cat A)
W6 — CV-1.6 (2026-05-04): T-ST-5a Cat A (Stereo-SCC Hard-Depth Topological Locking)
W6 — CV-1.7 (2026-05-05/06): T-OP6-B Cat A (OP-0006 RESOLVED), T-P-F-e0 Cat A (Gibbs Continuity), T-P-F-e0-K Cat B
W6 — CV-1.8/CV-1.9 (2026-05-06): P-F-A1 Package I fully Cat A: T-PF-A1-AR, T-PF-A1-SDE, T-PF-A1-GI, T-PF-A1-PE
W6 — CV-1.10 (2026-05-06/07): T-K-Select-PF Cat B (partial OP-0005-EQ), T-K-Select-OBS Cat B (partial OP-0005-OBS), D-ST-1..5 Cat B; OP-0005 3-way split registered
W6 — CV-1.11 (2026-05-07/08): T-ST-5b Cat B; OMS-2.0 Accepted Full; DECLARATION.md (DECL-1.0); hypothesis_tree.md (HT-3.0)
Pending (W7+): T-sigma-Theorem-4 Cat A re-promotion via gamma-path audit; OP-0005-DYN (Eyring-Kramers); P-F-A1 Package II; H-SINK -> CV-1.12 (T-Temporal-Identity a/b/d)
See also: Canonical Spec Part 5 (Results Registry), Open Problems (Part 4 §12)

Last updated 2026-05-09 (CV-1.11 final; 54 Cat A / 14 Cat B / 5 Cat C / 5 Retracted = 78 claims, ~69% proved).

Canonical Theorems (Accepted into Canonical Spec)#

Canonical Spec v1.11 (2026-05-06) — Current Version (W6 Close: 54A / 14B / 5C / 5R = 78 claims)#

CV-1.6 — Stereo-SCC branch (D1, 2026-05-04)#

CV-1.7 — Boundary Precision + Gibbs Continuity (D3, 2026-05-05/06)#

CV-1.8 / CV-1.9 — P-F-A1 Package I: Reflected SDE + Stochastic Foundation (D4, 2026-05-06)#

CV-1.10 — K-Selection Partial (EQ + OBS branches) + D-ST Diagnostic Bridge (D4, 2026-05-06/07)#

CV-1.11 — Audit + Declarations + OMS-2.0 Full (D5–D6, 2026-05-07/08)#

Canonical Spec v1.5.2 (2026-05-02) — Previous Version (W5 Day 6: L1-F Hard-Bar / Active-Count Bridge Conditional Cat-A)#

Canonical Spec v1.5.1 (2026-04-29) — Previous Version (W5 Day 3 EOD: D-6a Multi-Static + Ontological Depth + Critic reinforcement)#

Canonical Spec v1.5 (2026-04-27) — Previous Version (W5 Day 1 G0: σ-Framework Supporting Structures)#

Canonical Spec v1.4 (2026-04-26) — Previous Version (W4-extended close)#

Canonical Spec v1.3 (2026-04-25) — Previous Version (Frozen)#

Canonical Spec v1.2 (2026-04-12) — Previous Version (Frozen)#

Active Claims (Not Yet Canonical) / Resolved Claims#

Counterexamples & Challenges (X-xxxx)#

Canonical Spec Version History#

CV-1.0 (2026-04-01)#

CV-1.1 (2026-04-03) — PLAN_0403 Tier 1 Complete#

CV-1.2 (2026-04-12) — Frozen, with Audit Clarifications#

Open Problems (OP-xxxx) — Updated 2026-05-04 (post-W6 G2 audit; full set unified into theorem_status.md)#

Proof Status Summary (Updated 2026-05-09 — CV-1.11 final)#

Cross-Reference by Topic#

Single-Formation Theory#

Multi-Formation (K-field)#

Foundational#

Maintenance#