Fabricon Neighborhood

Status Boundary

This work is part of the authorial PHYSICA / Science of Fabric Reality corpus. It is presented as a research framework, formalization target, computational model, or theoretical synthesis unless explicitly marked otherwise. It is not presented as accepted physics, external consensus, or experimentally confirmed science.

1. Scientific Purpose

Defines the visual and structural requirements for representing the Fabricon Neighborhood, localized relational discrete cells, and bidirectional conjugate couplings within the formal framework.

2. Concepts Visualized

Fabricon Relational Quanta · Fundamental discrete nodes $v_{i} \in V$ with bidirectional conjugate couplings $F (x, y)$ .
Reflexive Inversion & Reciprocity · Double-conjugation identity $\overset{―}{\overset{―}{F}} = F$ ensuring exact relational conservation.
Relational Neighborhood Shells ( $N_{k} (v)$ ) · Concentric 1-hop, 2-hop, and $k$ -hop boundary zones governing interaction bounds.

3. Mathematical Symbols & Grounding

$F_{i j}$ : Relational coupling amplitude from fabricon $i$ to fabricon $j$ .
${\overset{―}{F}}_{j i}$ : Conjugate reciprocal coupling satisfying ${\overset{―}{\overset{―}{F}}}_{i j} = F_{i j}$ .
$N (v)$ : 1-hop local neighborhood ${u \in V ∣ (v, u) \in E}$ .
$Γ_{rel}$ : Relational holonomy groupoid element over closed loop $\partial N$ .

4. Required SVG Layers & Coordinate Geometry

Coordinate System: viewBox="0 0 800 500", responsive container scaling down to 320px mobile viewport.
Layer 1: base-topology:
- Central fabricon node $v_{0}$ surrounded by radial 1-hop neighborhood nodes ${v_{1}, \dots, v_{6}}$ .
- Bounding neighborhood shell boundary drawn with dashed stroke bound to var(--vp-c-brand-soft).
Layer 2: interaction-vectors:
- Dual bidirectional curved arc arrows representing conjugate coupling pair $(F_{0 k}, {\overset{―}{F}}_{k 0})$ .
- Gradient colors transitioning from var(--vp-c-brand-1) to var(--vp-c-brand-2).
Layer 3: labels-and-nodes:
- Node identifiers, edge weights $w_{i j}$ , and mathematical labels bound to var(--vp-c-text-1).
- Accessible title and description metadata embedded inside <svg> header.

5. Caption & Interpretation Boundary

Figure: Schematic representation of a discrete fabricon neighborhood, demonstrating bidirectional conjugate couplings and neighborhood boundary shells.

NOTE

This figure serves as a formalization target and visual aid for the authorial model. It must not be interpreted as empirical proof or accepted physics.

6. Canonical Continuations

Direction	Target Resource	Purpose	Formal Code Link
Foundational Theory	Fabricon	Discrete topological quanta & neighborhood definitions	`Fabrica.Fabricon`
Formal Lean Module	Formalization Roadmap	Machine-verified theorems in `Fabrica.Fabricon`	`Fabrica.Fabricon`
Relational Calculus	Principia Fabrica	Mathematical axioms of relational fabrics	`Fabrica.Realica`
Figures Atlas	Visual Cartography Atlas	Figure taxonomy and vector registry	Visual Cartography
Formalization Gateway	Formalization Gateway	Lean 4 Interactive Proof Suite	`Fabrica.Fabricon`

EXTERNAL REFERENCE

Fabricon Neighborhood ​

1. Scientific Purpose ​

2. Concepts Visualized ​

3. Mathematical Symbols & Grounding ​

4. Required SVG Layers & Coordinate Geometry ​

5. Caption & Interpretation Boundary ​

6. Canonical Continuations ​