Theorem index

… 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 …

… 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 …

… ted | 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 …

… 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 …

… 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 …

… 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 conve …

… ed | 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 d …

… 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 m …

… 05-OBS partially resolved. Alongside these promotions, T-Temporal-Identity and T-σ-Inherit were developed as working Cat B candidates with numerical anchors (exp83 4/4, e …

… 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 …

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… rily on $T_{u^*} \Sigma_m$. *Status:* **Proved**, Cat A. *(C-0712, P-0712.)* **T-σ-Lemma-2. σ-Framework Nodal Count Properties.** *(New, W5 Day 1 G0, 2026-04-27.)* For $\ …

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… ved (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 c …

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… 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 …

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… emma-1/2/3 + T-σ-Theorem-3** (W5 σ-framework), **T-Commitment-14-Multi-Static / T-σ-multi-A/D-Static** (CV-1.5.1 Multi-Static cluster), **V5b-T-zero**, **T-L1-F** (CV-1.5. …

… \circ}_M$ per Option A; corner-saturated regime open per OP-0008 + OP-0009). **T-σ-multi-A-Static. Multi-Formation σ-A Component (Static).** *(New, CV-1.5.1, 2026-04-29.)* σ-A c …

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… gime (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 c …

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… 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 cl …

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… (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). Furt …

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… from $\max \sigma' = 1/4$. A1 failure from $\sigma^{-1}(0.9)/z > 4$. *(R4.)* **T-A2. Monotonicity of Sigmoid Closure.** $u \leq v$ pointwise $\implies \mathrm{Cl}_ …

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… (2026-05-13/14, W7 D4) — SEAL** intermediate audits of CV-1.14 reserved state; T-ACT-KERNEL-COMP→REL registered with explicit forward-reference (conditional lift queued). - **C …

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… r $\mathbf{T}_t$, which appeared in the formal universe and carried two axioms (T-Ax1, T-Ax2), had zero realizations, zero theorems, zero energy terms, and zero pred …

… hbf{T}_t$, which appeared in the formal universe and carried two axioms (T-Ax1, T-Ax2), had zero realizations, zero theorems, zero energy terms, and zero predicate r …

… neral graph case incomplete; requires Cheeger/spectral clustering analysis.) **T-Beyond-Weyl. Structured Spectral Perturbation Bound for Multi-Formations.** The joint K-for …

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… i-formation Cat A), and the 2026-05-04 audit (test count 175 → 215 + 1 xfailed, T-Bind Phase 13 confirmation, NQ-187 continuum-vs-discrete caveat, `open_problems.md` …

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… 1)** per Phase 13 Erratum 2026-04-07 + W6 G2 audit reconciliation 2026-05-04. **T-Bind-Full** is **Cat A as a corollary of T-Bind-Proj**. Both entries were physically pres …

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… cordingly. ### OP-0010 status update (CV-1.5.2 / W6 G2 audit, 2026-05-04) - **T-Bind-Proj** is **Cat A for all τ_cl ∈ (0,1)** per Phase 13 Erratum 2026-04-07 + W6 G2 aud …

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… pprox 0.89$ moved to Category B — empirical fit, no analytical derivation.)* **T-Birth-Parametric. Supercritical Pitchfork Bifurcation in Formation Stability.** At the critical …

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… oved**. The CV-1.12–1.17 additions (T-Temporal-Identity (a) Cat A via > H-SINK; T-CC-StableK-Kernel Cat B, H-COMP-KERNEL closed; H-MORSE partially > closed) are recorded in the > …

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… e 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.)* Exte …

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… s configuration-specific. See Category B section.)* **Moved to Category B.** **T-d_min-Formula. Analytical Formula for Critical Inter-Formation Distance.** The branch-conditi …

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… ll graphs (spectral universality) - T8-Full: Phase transition for full energy - T-FORMATION-BIRTH: General graph (Phase 14 upgrade) **Group 3: Stability & Dynamics (8)** - T14: …

… ly post](/journal/2026-05-08-perception-week-6/). --- ## The 10 heroes | # | T-ID | Name | Group | Cat | |---|------|------|-------|-----| | 1 | **T1** | Existen …

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… T-PF-A1-AR/SDE/GI/PE) + 9 Cat B (T-P-F-ε0-K, T-ST-5b, D-ST-1..5, T-K-Select-PF, T-K-Select-OBS) = 17 new claims. **As sealed in this snapshot: 54A/14B/5C/5R = 78 claims, ~69% …

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… 6-B, T-P-F-ε0, T-PF-A1-AR/SDE/GI/PE) + 9 Cat B (T-P-F-ε0-K, T-ST-5b, D-ST-1..5, T-K-Select-PF, T-K-Select-OBS) = 17 new claims. **As sealed in this snapshot: 54A/14B/5C/5R = …

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… re-activation rider; CV-1.5.2 promotes the first multi-formation Cat A theorem T-L1-F under the L1-J regime — see notes below.)* - **F-1 (K=2 vacuity) — SPLIT-RESOL …

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… ckage I (CV-1.9, complete). - **L1-M-AUDIT:** Close R-1/R-2/R-3 audit items for T-L1-M canonical promotion. - **OMS formal rows:** Register Appendix OMS §A–§M theorem …

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… Note, not a fixed Commitment). The static global minimum on pure E_bd (K=1, by T-Merge (b)) and the dynamic protocol-endpoint observables under full SCC (K-hat, F) ar …

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… at a K-jump event is not determined by pre-merger $\sigma^A$ alone. This blocks T-MF-Synthesis and requires the Wigner-projection computation (W9+). - **T-Temporal-Identity / …

… *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 …

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… eObj-1, T-V5b-T), W5 multi-formation bridge (T-L1-F), W6 stochastic foundation (T-P-F-ε₀, T-PF-A1-AR/SDE/GI/PE), W6 boundary + selection (T-OP6-B, T-K-Select-PF/OBS) …

… *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, …

… n-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 diagnost …

… des (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 …

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… 5-08):** +8 Cat A (T-ST-5a, T-OP6-B, T-P-F-ε0, T-PF-A1-AR/SDE/GI/PE) + 9 Cat B (T-P-F-ε0-K, T-ST-5b, D-ST-1..5, T-K-Select-PF, T-K-Select-OBS) = 17 new claims. **As seale …

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… ext{core-to-core inheritance via } \mathbf{M}^k$, applying the single-formation T-Persist directly to each $(u^k_t, u^k_s)$ pair. When formations have disjoint supports …

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… * Each formation $u^k_t$ satisfies the single-formation hypotheses (H1)-(H4) of T-Persist-1. - **(WS)** Well-separated: $D_{\text{sep}} \geq 3$ with $4\varepsilon_1/\min_k …

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… 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 ellips …

… ase 11 completion — all 5 T-Persist-1 conditions now proved or unconditional.)* T-Persist-Full ([Part 5 · Registry](/notes/part-0/canonical-spec-scc-5-results-registry/)) syn …

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… eshold $d_{\min}^*$ depends on the minimum spectral gap: well-separated (d > 3, T-Persist-K-Sep proved), weakly-interacting (boundary overlap allowed, T-Persist-K-Weak conditi …

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… k|)$), strongly-interacting (significant bulk overlap, barrier-crossing regime, T-Persist-K-Strong local stability proved). Closed transport plans ensure persistence across time …

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… sites. - **Multi-formation regimes unified by $\Lambda_{\mathrm{coupling}}$** (T-Persist-K-Unified, Corollary III): Persist degradation rates parametrized by soft overlap measure …

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… mation branch; branch identity, tie-breaker, and history are external data.)* *(T-PERSIST-K-UNIFIED.md, PHASE-AB-SYNTHESIS.md.)* *Implementation:* `scc/multi.py` functions `coupli …

… (d > 3, T-Persist-K-Sep proved), weakly-interacting (boundary overlap allowed, T-Persist-K-Weak conditionally proved, $|O_{jk}| \leq 0.2 \min(|\text{Core}_j|, |\text{Core}_k|) …

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… 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 …

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… CC energy has a well-defined stochastic dynamics on the constraint manifold. **T-PF-A1-GI. Unique Gibbs Invariant Measure $\pi_{T_*}$.** *(New, CV-1.9, 2026-05-06.)* The …

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… temperature $T_*$ canonically selects a distribution over formation states. **T-PF-A1-PE. Poincaré Inequality + Exponential Ergodicity.** *(New, CV-1.9, 2026-05-06.)* T …

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… :* **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 Lan …

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… rough exp97**, plus the W4-extended **E-0090 through E-0097** cluster anchoring T-PreObj / T-V5b-T / σ supporting structures, plus the W5 numerical work that does not c …

… athcal{F} \geq 2$ as the default ground state — is **mathematically grounded by T-PreObj-1 + T-PreObj-1G** (Cat A graph-class independent), not merely an ontological decl …

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… q 2$ as the default ground state — is **mathematically grounded by T-PreObj-1 + T-PreObj-1G** (Cat A graph-class independent), not merely an ontological declaration. Pre-o …

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… te-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 …

… , 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 (W …

… etric 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 …

… , 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- …

… reObj-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-L …

… 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 (sm …

… 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- …

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… Cat A (T-ST-5a, T-OP6-B, T-P-F-ε0, T-PF-A1-AR/SDE/GI/PE) + 9 Cat B (T-P-F-ε0-K, T-ST-5b, D-ST-1..5, T-K-Select-PF, T-K-Select-OBS) = 17 new claims. **As sealed in this …

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… ubstantially 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_*(gra …

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… a 1-fold Goldstone branch; the **Goldstone nodal count = 2 universal**. Source: T-V5b-T (after 8 V5b iterations V1 → V5b''). Numerical verification: E-0095 (NQ-170b ζ- …

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… rpose:** 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 …

… 0 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-emp …

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