Equation index · Jaehong Oh

Fact B.1 (Bi-invariant metric). Every compact Lie group admits an -invariant inner product on its Lie algebra . The corresponding left-invariant Riemannian metric on is automatically bi-invariant. The geodesic distance satisfies:

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Appendix D — Hybrid Dynamical Systems Background

\boxed{\frac{dh_i}{dt} = -\sum_{j: (i,j)\in E} \nabla_{h_i} \mathcal{E}_{F^\circ}(h)}

Within a fixed topology , the continuous dynamics are driven by Yang--Mills gradient flow:

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Chapter 11 — Čech Cohomology Framework

g_{ij}\cdot g_{jk}\cdot g_{ki}=e\quad\text{on }U_i\cap U_j\cap U_k.

A principal -bundle is defined by an open cover with transition functions satisfying the Cech cocycle condition:

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Chapter 16 — Open Problems

\rho^{(1)}_{F^\circ}(i)=\rho^{(2)}_{F^\circ}(i)\quad\forall\,i\in F^\circ.

If and are two global minima of with , then:

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Theorem D — Metastability

P_F(i,j):=\frac{W_t(i,j)}{d_t(i)}\quad(i,j\in F),\qquad\tilde P_F:=\tfrac{1}{2}(I|_F+P_F).

Step 1 (Restricted chain). Define the restricted sub-stochastic matrix on :

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Theorem E — Curvature Localisation

\tilde\alpha = (I-B^T L^\dagger B\,\mathrm{diag}(W))\,\alpha =: \Pi\,\alpha.

(ii) Case . The optimal gauge satisfies the linear system (Theorem 7.11). The residual angles are . These solve:

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Chapter 13 — Yang–Mills Flow

\Omega_\triangle(A):=g_{ij}\cdot g_{jk}\cdot g_{ki}\in G.

Let be a graph with triangle set (all 3-cliques). For a -valued connection , the holonomy of triangle is:

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Chapter 14 — Time Evolution

\exists\,F\in\mathfrak{F}_{t^*-\epsilon}\setminus\mathfrak{F}_{t^*+\epsilon}\quad\text{or}\quad\exists\,F\in\mathfrak{F}_{t^*+\epsilon}\setminus\mathfrak{F}_{t^*-\epsilon}.

A topological phase transition occurs at time if the combinatorial topology of the fruit structure changes discontinuously:

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Chapter 15 — Applications

\mathrm{CS}(A):=\sum_{\triangle\in\mathcal{T}}d_G(\Omega_\triangle(A),e)^2.

For a -connection on a finite graph , define the discrete Chern--Simons-type invariant:

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T-L1-F — Hard-Bar / Active-Count Bridge under L1-J Regime

U(\mathbf u) = \sum_{j=1}^{K_{\mathrm{field}}} u^{(j)},\qquad A^\varepsilon(\mathbf u) = \{j : m_j(\mathbf u) > \varepsilon\},\qquad K_{\mathrm{act}}^\varepsilon = |A^\varepsilon|,

Let be a finite graph and a shared-pool multi-formation state. Let

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T8-Core — Phase Transition (Spectral Universality)

c \in \left(\frac{3 - \sqrt{3}}{6},\; \frac{3 + \sqrt{3}}{6}\right) \approx (0.211,\, 0.789),

Let be a finite connected graph with Fiedler eigenvalue (algebraic connectivity). Let with volume fraction

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T-V5b-T — Pre-Objective Goldstone on Translation-Invariant Graphs (W4-Extended Capstone)

\zeta_*(2D \text{ torus}, c) \approx 0.40 + (c - 0.10) \cdot 0.30,\quad c \in [0.10, 0.30],

Crossover scale — refined at CV-1.5.1. The transition scale between sub-lattice and super-lattice regimes is graph-class dependent and -dependent; theoretical derivation as an analytic function of dimensionality and lattice topology remains open (NQ-174).

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World

|\Delta_{\mathrm{cut}}| = \Bigl|\sum_{i\in F}\sum_{j\notin F}(W'_t(i,j)-W_t(i,j))\Bigr| \le |F|\cdot|\bar F|\cdot\|\delta W\|_\infty.

(i) Track the perturbation of numerator and denominator separately.

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Integrated Architecture — SCC × ONN as one cognitive-reasoning system

\text{Relational substrate} \;\xrightarrow{\text{Cheeger}}\; \text{Fruits} \;\xrightarrow{\text{ONN}}\; \text{Semantic constraints} \;\xrightarrow{\text{ORTSF}}\; \text{Embodied behaviour}

Forward flow is the bottom-up emergence of meaning:

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

\mu_{\mathrm{joint}} \geq \min_k \mu_k - (K-1)\lambda_{\mathrm{rep}} \cdot \omega_{\mathrm{soft}}, \qquad \omega_{\mathrm{soft}} := \max_{j \neq k} \|\mathcal{P}_{O_{jk}} \psi_k^{\mathrm{soft}}\|^2.

T-Beyond-Weyl. Structured Spectral Perturbation Bound for Multi-Formations. The joint K-formation Hessian spectral gap tightens from the standard Weyl bound to a formation-aware bound exploiting soft-mode localization:

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T11 — Sharp-Interface Γ-Convergence

\mathcal{E}^\varepsilon_{\mathrm{bd}}(u) = \varepsilon \sum_{x,y} \mathbf{N}_t(x,y)(u(x) - u(y))^2 + \frac{1}{\varepsilon} \sum_x W(u(x))

Let be the smoothness-to-double-well ratio. As , the rescaled boundary-morphology energy

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T14 — Gradient Flow Convergence (Łojasiewicz)

\dot u = -\Pi_{\Sigma_m} \nabla \mathcal{E}(u),

The projected gradient flow on the constraint manifold ,

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T6 — Closure Fixed Point (Banach Contraction)

\mathrm{Cl}_t(u)(x) = \sigma\!\big(a_{\mathrm{cl}}\,((1-\eta_{\mathrm{cl}})\,u(x) + \eta_{\mathrm{cl}}\,(P_t u)(x) - \tau_{\mathrm{cl}})\big).

Let be the sigmoid closure operator

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Door

\tau\cdot|\Sigma_\tau|\le\sum_{i\in\Sigma_\tau}b_{F,t}(i)\le\sum_{i\in F}b_{F,t}(i)=\mathrm{cut}_t(F,\bar F)\le\theta\cdot\mathrm{vol}_t(F).

(i) Each has , so:

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First entry on the new site

H^n(X; \mathbb{R}) \;\cong\; \mathrm{Hom}_{\mathbb{R}}\!\bigl(H_n(X; \mathbb{R}),\, \mathbb{R}\bigr).

A brief taste of the typography pipeline. Inline math like $\nabla \cdot \mathbf{F} = \rho$ should typeset cleanly; block math too:

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Part II · Main Theorems A–H (summary)

\frac{\sum_{i, j \in F} W_t(i, j)}{\mathrm{vol}_t(F)} \;\ge\; 1 - \theta.

For any fruit ,

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Perception · W5 — Multi-Formation Count Bridge

\lvert K_{\mathrm{soft}}^\phi(U(\mathbf u)) - K_{\mathrm{act}}^\varepsilon(\mathbf u)\rvert \le \varepsilon_{\mathrm{sub}}^\phi(\tau) N_{\mathrm{sub}}(U;\tau) + \varepsilon_{\mathrm{dom}}^\phi(\tau) K_{\mathrm{act}}^\varepsilon(\mathbf u).

L1-M was drafted as the controlled soft-count companion to T-L1-F. Under the same regime plus an envelope class , it bounds the soft count against the active-slot count:

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Perception · W7 — Six Sealed Versions, Compositional Closure, First Archive

K_{\mathrm{act}}^\varepsilon(\mathbf{u}_1 \circ_K \mathbf{u}_2) = f\big( K_{\mathrm{act}}^\varepsilon(\mathbf{u}_1),\ K_{\mathrm{act}}^\varepsilon(\mathbf{u}_2) \big)

The compositional consistency question is the following. Two formations are read out via the active-count diagnostic . If we compose them through a kernel-side composition operator (the precise form is the W6 kernel convolution carried into the multi-formation interior), does

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SCC — Current Research Status (May 2026, CV-1.17)

u_t : X_t \to [0, 1]

A mathematical theory of how coherent formations emerge prior to discrete objecthood. The primitive is a soft cohesion field

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SCC — Research Status Snapshot (April 2026, W5 Day 1 G0 close, CV-1.5) — HISTORICAL

u_t : X_t \to [0, 1]

A mathematical theory of how coherent formations emerge prior to discrete objecthood. The primitive is a soft cohesion field

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T1 — Existence of Minimizers

\Sigma_m = \Big\{u \in [0,1]^n : \sum_{x \in X_t} u_t(x) = m\Big\},

On the constraint manifold

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Theorem F — Spectral Stability

|\phi'-\phi|=\left|\frac{\mathrm{cut}+\Delta_c}{\mathrm{vol}+\Delta_v}-\frac{\mathrm{cut}}{\mathrm{vol}}\right|=\left|\frac{\Delta_c\cdot\mathrm{vol}-\mathrm{cut}\cdot\Delta_v}{(\mathrm{vol}+\Delta_v)\cdot\mathrm{vol}}\right|.

Denominator perturbation: .