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; current = CV-1.11).
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 ), the energy barrier between the two formation basins is (state-space disconnection via the Goldstone mechanism). System forced to 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): . OP-0006 RESOLVED — computationally accessible PersRidge is a provably faithful proxy for the theoretical boundary. |
| T-P-F-ε0 | Gibbs Measure Continuity () | accepted | A | The Gibbs measure on converges weakly to (deterministic formation-selector) as . 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 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 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 is well-posed: existence via Lions-Sznitman Theorem 1 (convex domain), uniqueness via Tanaka pathwise argument. |
| T-PF-A1-GI | Unique Gibbs Invariant Measure | accepted | A | is the unique invariant measure of the reflected SDE via zero-current + kernel argument (Aronson 1968 heat kernel). |
| T-PF-A1-PE | Poincaré Inequality + Exponential Ergodicity | accepted | A | (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 : 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 , formation count 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 , the posterior over 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 . 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 not sharply separated), the merge barrier is raised monotonically in . 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 ( generic symmetry, codim-1 strata) + temporal extension (, 19-branch 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 ; 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 –: AND labeled bijection via primary representative . 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 ( K-jump non-determinism); does NOT establish ; 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 보강)
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 interior (Option A pragmatic; corners deferred to NQ-248 W7+). joint invariant under wreath-product . 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 under permutation; reduces to Commitment 14 σ at . 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 . 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 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) | exactly (discrete translation orbit on ). Replaces V5b-T' WITHDRAWN (NQ-198f phantom on torus). Empirical anchor NQ-198f $ |
| (V5b-F-empirical) | V5b-F Goldstone Mass Scaling | accepted | B target (sub-statement) | , (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 -small regime | Cat B in -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 -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 + sign-flip Cat A; Courant upper bound + -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 with ; Goldstone basis automatically ℓ=1; nodal count = 2 universal (anchors T-V5b-T-(e)) |
| T-σ-Theorem-3 | σ at Uniform on Free-BC Grid (Closed Form) | accepted | A | C-0715 | P-0715 | exp_hessian_uniform_v2 (NQ-141, to , precision) | ; full irrep table on cosine basis; |
| T-σ-Theorem-4 | σ at First Pitchfork on 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 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 grids. NQ-187 measured , , ratio (not degeneracy), exponent (not predicted). Three reconciliation hypotheses (α continuum extrapolation, β R22 re-derivation, γ -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 ~ |
| 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 | of gradients, PD under linear independence; destabilization magnitude |
| F-1 Resolution Corollary | F-1 SPLIT-RESOLVED via T-Merge(b) + T-PreObj-1 | accepted | A | (corollary) | (corollary) | — | Pure 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 ; general τ via binary mass-balance formula . 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; , -independent for 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 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 : 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 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 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 | 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)
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): |
| 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).