∑Index

Equation index

238 equations across 39 documents. Each entry is automatically paired with the prose paragraph immediately preceding it, so the index reads as a brief catalogue rather than a wall of TeX. Click any document to see its full equation list.

29 equations· Notes
Chapter 17 — End-to-End Mobile Manipulator
${r a w se n sor} \to W_{t} \to r_{t} \to g_{t}^{{} {t a s k}} \to ξ_{t} \to u_{t} .$
The aim is not to replace Chapters 1--16, but to provide an embodiment-level map:
16 equations· Notes
Chapter 0 — Formal State Space
$Top := {(n, E) : n \in N_{> 0}, E \subseteq {1, \dots, n}^{2} ∖ diag} .$
The collection of all admissible topologies is denoted:
16 equations· ONN
ONN vs Transformer — Experiment Plan
$CSR = \frac{# constraints satisfied in G ^}{# total constraints}$
16 equations· Notes
SCC Canonical Spec — Part 2: Axiomatic Groups & Proto-Cohesion
$Cl_{t} (u) (x) \geq u (x) whenever u (x) \leq c^{*} and (P_{t} u) (x) \geq u (x)$
A1'. Conditional Extensivity (Self-Regulation). For all and all ,
13 equations· Notes
Chapter 12 — Three Computational Axes
$W_{t} \Rightarrow F_{t} \Rightarrow Σ_{t} \Rightarrow Axis 1 \Rightarrow Axis 2 \Rightarrow Axis 3 .$
The sequential protocol extends:
13 equations· Notes
Fruit
$cut_{t} (F, \overset{ˉ}{F}) = cut_{t} (F_{1}, \overset{ˉ}{F}) + cut_{t} (F_{2}, \overset{ˉ}{F})$
Step 1. Since there are no internal edges between and :
13 equations· Notes
SCC Canonical Spec — Part 3: Energy Principle & Provisional Operators
$Σ_{m} = {u \in [0, 1]^{n} : x \in X_{t} \sum u_{t} (x) = m}$
The energy is minimized on the constraint manifold
11 equations· Notes
Main Theorems A–H
$P_{F} (i, j) := \frac{W _{t} ( i , j )}{d _{t} ( i )} (i, j \in F), \tilde{P}_{F} := \frac{1}{2} (I ∣_{F} + P_{F}) .$
Step 1 (Restricted chain). Define the restricted sub-stochastic matrix on :
11 equations· Notes
SCC Canonical Spec — Part 1: Foundations & Formal Universe
$C^{soft} = (T, {X_{t}}_{t \in T}, {u_{t}}_{t \in T}, {Cl_{t}}_{t \in T}, {N_{t}, D_{t}}_{t \in T}, {M_{t \to s}}_{t, s \in T})$
The formal universe of the soft theory is a structured tuple
9 equations· Notes
Chapter 10 — Worked Examples
$g_{t}^{h} (i, j) = e^{i (α_{t} (i, j) + φ_{i} - φ_{j})} .$
8 equations· Notes
Existence
$E^{'} (h) = \sum W_{t} (i, j) d_{G} (g^{' h} (i, j), e)^{2} = \sum W_{t} (i, j) d_{G} (g^{h_{0} \cdot h} (i, j), e)^{2} = E (h_{0} \cdot h) .$
(ii) Starting from a different representative :
7 equations· Notes
Appendix B — Prerequisites
$d_{G} (k g k^{'}, k h k^{'}) = d_{G} (g, h) \forall g, h, k, k^{'} \in G .$
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:
6 equations· Notes
Appendix D — Hybrid Dynamical Systems Background
$\frac{d h _{i}}{d t} = - j : (i, j) \in E \sum \nabla_{h_{i}} E_{F^{\circ}} (h)$
Within a fixed topology , the continuous dynamics are driven by Yang--Mills gradient flow:
5 equations· Notes
Chapter 11 — Čech Cohomology Framework
$g_{ij} \cdot g_{j k} \cdot g_{k i} = e on U_{i} \cap U_{j} \cap U_{k} .$
A principal -bundle is defined by an open cover with transition functions satisfying the Cech cocycle condition:
5 equations· Notes
Chapter 16 — Open Problems
$ρ_{F^{\circ}}^{(1)} (i) = ρ_{F^{\circ}}^{(2)} (i) \forall i \in F^{\circ} .$
If and are two global minima of with , then:
5 equations· ONN
ONN Canonical Specification
$O = (V_{O}, E_{O}, w_{O})$
Ontology graph. The target ontology is a weighted directed graph
5 equations· Notes
Theorem D — Metastability
$P_{F} (i, j) := \frac{W _{t} ( i , j )}{d _{t} ( i )} (i, j \in F), \tilde{P}_{F} := \frac{1}{2} (I ∣_{F} + P_{F}) .$
Step 1 (Restricted chain). Define the restricted sub-stochastic matrix on :
5 equations· Notes
Theorem E — Curvature Localisation
$\tilde{α} = (I - B^{T} L^{†} B diag (W)) α =: Π α .$
(ii) Case . The optimal gauge satisfies the linear system (Theorem 7.11). The residual angles are . These solve:
4 equations· Notes
Chapter 13 — Yang–Mills Flow
$Ω_{△} (A) := g_{ij} \cdot g_{j k} \cdot g_{k i} \in G .$
Let be a graph with triangle set (all 3-cliques). For a -valued connection , the holonomy of triangle is:
4 equations· Notes
Chapter 14 — Time Evolution
$\exists F \in F_{t^{*} - ϵ} ∖ F_{t^{*} + ϵ} or \exists F \in F_{t^{*} + ϵ} ∖ F_{t^{*} - ϵ} .$
A topological phase transition occurs at time if the combinatorial topology of the fruit structure changes discontinuously:
4 equations· Notes
Chapter 15 — Applications
$CS (A) := △ \in T \sum d_{G} (Ω_{△} (A), e)^{2} .$
For a -connection on a finite graph , define the discrete Chern--Simons-type invariant:
4 equations· Notes
T-L1-F — Hard-Bar / Active-Count Bridge under L1-J Regime
$U (u) = j = 1 \sum K_{field} u^{(j)}, A^{ε} (u) = {j : m_{j} (u) > ε}, K_{act}^{ε} = ∣ A^{ε} ∣,$
Let be a finite graph and a shared-pool multi-formation state. Let
4 equations· Notes
T8-Core — Phase Transition (Spectral Universality)
$c \in (\frac{3 - 3}{6}, \frac{3 + 3}{6}) \approx (0.211, 0.789),$
Let be a finite connected graph with Fiedler eigenvalue (algebraic connectivity). Let with volume fraction
3 equations· Notes
T-V5b-T — Pre-Objective Goldstone on Translation-Invariant Graphs (W4-Extended Capstone)
$ζ_{*} (2 D torus, c) \approx 0.40 + (c - 0.10) \cdot 0.30, c \in [0.10, 0.30],$
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).
3 equations· Notes
World
$∣ Δ_{cut} ∣ = i \in F \sum j \in / F \sum (W_{t}^{'} (i, j) - W_{t} (i, j)) \leq ∣ F ∣ \cdot ∣ \overset{ˉ}{F} ∣ \cdot ∥ δ W ∥_{\infty} .$
(i) Track the perturbation of numerator and denominator separately.
2 equations· Notes
Integrated Architecture — SCC × ONN as one cognitive-reasoning system
$Relational substrate Cheeger Fruits ONN Semantic constraints ORTSF Embodied behaviour$
Forward flow is the bottom-up emergence of meaning:
2 equations· Notes
SCC Canonical Spec — Part 5: Proved Results Registry & Closing Notes
$μ_{joint} \geq k min μ_{k} - (K - 1) λ_{rep} \cdot ω_{soft}, ω_{soft} := j \neq = k max ∥ P_{O_{j k}} ψ_{k}^{soft} ∥^{2} .$
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:
2 equations· Notes
T11 — Sharp-Interface Γ-Convergence
$E_{bd}^{ε} (u) = ε x, y \sum N_{t} (x, y) (u (x) - u (y))^{2} + \frac{1}{ε} x \sum W (u (x))$
Let be the smoothness-to-double-well ratio. As , the rescaled boundary-morphology energy
2 equations· Notes
T14 — Gradient Flow Convergence (Łojasiewicz)
$\overset{u}{˙} = - Π_{Σ_{m}} \nabla E (u),$
The projected gradient flow on the constraint manifold ,
2 equations· Notes
T6 — Closure Fixed Point (Banach Contraction)
$Cl_{t} (u) (x) = σ (a_{cl} ((1 - η_{cl}) u (x) + η_{cl} (P_{t} u) (x) - τ_{cl})) .$
Let be the sigmoid closure operator
1 equation· Notes
Door
$τ \cdot ∣ Σ_{τ} ∣ \leq i \in Σ_{τ} \sum b_{F, t} (i) \leq i \in F \sum b_{F, t} (i) = cut_{t} (F, \overset{ˉ}{F}) \leq θ \cdot vol_{t} (F) .$
(i) Each has , so:
1 equation· Journal
First entry on the new site
$H^{n} (X; R) ≅ Hom_{R} (H_{n} (X; R), R) .$
A brief taste of the typography pipeline. Inline math like $\nabla \cdot \mathbf{F} = \rho$ should typeset cleanly; block math too:
1 equation· Notes
Part II · Main Theorems A–H (summary)
$\frac{\sum _{i, j \in F} W _{t} ( i , j )}{vol _{t} ( F )} \geq 1 - θ .$
For any fruit ,
1 equation· Journal
Perception · W5 — Multi-Formation Count Bridge
$∣ K_{soft}^{ϕ} (U (u)) - K_{act}^{ε} (u)∣ \leq ε_{sub}^{ϕ} (τ) N_{sub} (U; τ) + ε_{dom}^{ϕ} (τ) K_{act}^{ε} (u) .$
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:
1 equation· Journal
Perception · W7 — Six Sealed Versions, Compositional Closure, First Archive
$K_{act}^{ε} (u_{1} \circ_{K} u_{2}) = f (K_{act}^{ε} (u_{1}), K_{act}^{ε} (u_{2}))$
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
1 equation· Notes
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
1 equation· Notes
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
1 equation· Notes
T1 — Existence of Minimizers
$Σ_{m} = {u \in [0, 1]^{n} : x \in X_{t} \sum u_{t} (x) = m},$
On the constraint manifold
1 equation· Notes
Theorem F — Spectral Stability
$∣ ϕ^{'} - ϕ ∣ = \frac{cut + Δ _{c}}{vol + Δ _{v}} - \frac{cut}{vol} = \frac{Δ _{c} \cdot vol - cut \cdot Δ _{v}}{( vol + Δ _{v} ) \cdot vol} .$
Denominator perturbation: .

Chapter 17 — End-to-End Mobile Manipulator

Chapter 0 — Formal State Space

ONN vs Transformer — Experiment Plan

SCC Canonical Spec — Part 2: Axiomatic Groups & Proto-Cohesion

Chapter 12 — Three Computational Axes

Fruit

SCC Canonical Spec — Part 3: Energy Principle & Provisional Operators

Main Theorems A–H

SCC Canonical Spec — Part 1: Foundations & Formal Universe

Chapter 10 — Worked Examples

Existence

Appendix B — Prerequisites

Appendix D — Hybrid Dynamical Systems Background

Chapter 11 — Čech Cohomology Framework

Chapter 16 — Open Problems

ONN Canonical Specification

Theorem D — Metastability

Theorem E — Curvature Localisation

Chapter 13 — Yang–Mills Flow

Chapter 14 — Time Evolution

Chapter 15 — Applications

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

T8-Core — Phase Transition (Spectral Universality)

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

World

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

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

T11 — Sharp-Interface Γ-Convergence

T14 — Gradient Flow Convergence (Łojasiewicz)

T6 — Closure Fixed Point (Banach Contraction)

Door

First entry on the new site

Part II · Main Theorems A–H (summary)

Perception · W5 — Multi-Formation Count Bridge

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

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

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

T1 — Existence of Minimizers

Theorem F — Spectral Stability