WHITE PAPER · WP-01SPEC / TRIAD-MESH / K₃
The Three-Body Mesh
A Formal Specification of the Minimum Complete Junction Network
CLASSIFICATION: formal specification · normative reference
SUBJECT: three diodes, end-to-end, fully bridged · K₃ topology
SERIES E · June 2026
Abstract
This document specifies the
three-body mesh: three directional elements (diodes) connected pairwise such that every node is bridged to every other. The resulting structure is the complete graph K₃ — the minimum fully-connected network in which every node reaches every other node directly, with no intermediary. The specification defines its element count (3 nodes, 3 bridges, 6 ports, 9 total elements), its graph-theoretic properties (diameter 1, uniform degree 2), and its two canonical configurations (the directed cycle and the bidirectional mesh).
1 Scope
This specification defines the topology, element inventory, and configurations of a network of three directional two-terminal elements connected end-to-end with complete pairwise bridging. It is normative with respect to counts and properties; it is descriptive with respect to applications.
2 Definitions
2.1 Node
A connection point at which two or more elements meet. The mesh has three nodes, designated A, B, C.
2.2 Bridge
An undirected connection between an ordered pair of nodes. The mesh has three bridges: AB, AC, BC.
2.3 Port
A directional terminal of a bridge. Each bridge possesses two ports (an inbound and an outbound). The mesh has six ports.
2.4 Element
Any node or port. The total element count is the sum of nodes and ports: 3 + 6 = 9.
3 Element Inventory
| Quantity | Class | Members | Count |
|---|
| Nodes | connection points | A, B, C | 3 |
| Bridges | undirected edges | AB, AC, BC | 3 |
| Ports | directional terminals | 2 per bridge | 6 |
| Total elements | nodes + ports | — | 9 |
The figure 9 is the ported element count: three nodes plus six directional ports (three inbound, three outbound). It is distinct from the undirected edge count (3) and the port count (6); all three figures describe the same object at different granularities.
4 Graph-Theoretic Properties
Topology : K₃ (complete graph on 3 nodes)
Edges : n(n−1)/2 = 3(2)/2 = 3
Degree (each) : n−1 = 2
Diameter : 1 (every node reaches every other directly)
Bottleneck node : none (no traffic routes through a third)
Minimality : the smallest complete mesh
K₃ is the minimum complete mesh. At two nodes, a network has a single link and no redundancy; the failure of that link partitions the network. At three nodes fully bridged, the network is the smallest in which every node retains a direct path to every other, and in which the loss of any single bridge does not partition the whole. Three is therefore the minimum node count at which a network is both complete and fault-tolerant.
5 Canonical Configurations
5.1 Directed Cycle
All three elements oriented in the same rotational sense (A→B→C→A). Current circulates in one direction. The structure is a directed 3-cycle: three nodes, three one-way edges, unidirectional circulation.
5.2 Bidirectional Mesh
Each bridge provides both an inbound and an outbound port. Every node communicates with every other in both directions. This configuration realizes the full nine-element inventory and is the minimum complete two-way mesh.
6 Normative Claims
CLAIM 6.1. A three-node complete mesh contains exactly three undirected bridges. Proof: edges of Kₙ = n(n−1)/2; for n=3, this is 3. ∎
CLAIM 6.2. The ported element count of the bidirectional three-body mesh is nine. Proof: 3 nodes + (3 bridges × 2 ports) = 3 + 6 = 9. ∎
CLAIM 6.3. Three is the minimum node count for a complete fault-tolerant mesh. Proof: at n=2 a single edge is a cut-edge; its removal partitions the graph. At n=3 complete, each pair retains a direct edge and a two-hop alternate, so no single edge is a cut-edge. ∎