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title: "Fabric Graph — Schematic Specification" description: "Visual schematic and layer topology specifications for the relational cellular complex and boundary-free 1-cycle topology." layout: doc status: "canonical-figure-spec" claim_boundary: "Visual schematic specification only. Demarcates relational graph topology from physical structure validation."

Fabric Graph

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.

Scientific Purpose

Defines the visual and structural requirements for representing the Fabric Graph concept within the formal framework.

Concepts Visualized

• Relational Cellular Graph · Discrete metric graph $G=(V,E)$ with edge weight conductance matrix $W_{ij}$.
• Boundary-Free 1-Cycle Subspace · Simplicial 1-cycle elements $c\in C_1(G,\mathbb{Z})$ satisfying ${\partial }_{1}c=0$.
• Graph Laplacian Spectrum · Discrete combinatorial Laplacian ${\mathrm{\Delta }}_G=D-W$ and spectral gap dynamics.

Symbols Visualized

• $G=(V,E)$, ${\mathrm{\Delta }}_G$, $C_1(G,\mathbb{Z})$, ${\partial }_1$, $\ker ({\partial }_1)$, $W_{ij}$

Equations Referenced

• Science of Fabric Reality — Relational fabric ontology and network topology.
• Mathematical Invariants — 1-cycle linear combination closure (thm_one_cycle_linear_combination, thm_one_cycle_zero_boundary).

Required SVG Layers

• base-topology: Vertex distribution and weighted relational edges.
• interaction-vectors: Boundary-free oriented 1-cycle flows and circulation currents.
• labels-and-nodes: Node indices, edge weights, and algebraic boundary annotations.

Caption

Figure: Schematic representation of the Fabric Graph cellular complex, highlighting closed boundary-free 1-cycles and discrete Laplacian connectivity.

Construction Notes

Do not use generic raster images. The final SVG must strictly adhere to the mathematical and topological constraints defined in the corresponding route.

Route Bindings

• Primary Theory: Science of Fabric Reality
• Formal Foundation: Principia Fabrica
• Mathematical Invariants: Mathematical Invariants & Admissibility

Interpretation Boundary

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.