The Ruliad—the entangled limit of all computation—can be constructed with universal computation using nothing but pure, unlabelled 2D graphs and a single primitive: the NAND gate.
1. Empty Vertices as Pure Relations
Instead of labeled NAND input and output variables, the vertices are empty identifiers. Values are encoded purely as topological degree (leaf counts):
- Bit 0: 1 attached leaf
- Bit 1: 2 attached leaves
For example, $1 \text{ NAND } 1 = 0$ is mapped directly to a simple directed graph:
Graph[{1->3, 2->3, 3->4, 1->5, 1->6, 2->7, 2->8, 4->9}]
- Inputs (1 & 2): Each has 2 leaves attached $\to$ Bit 1
- Hub (3): Processes incoming edges to output node (4)
- Output (4): Has 1 leaf attached $\to$ Bit 0
2. Timeless Computation & Infinite Unrolling
In this framework, execution doesn't happen "over time." Iterative programs and infinite loops are unrolled spatially into infinite acyclic subgraphs:
- Time becomes graph depth.
- A loop isn't a state repeating in time—it is an endless forward path through the net.
- The entire execution history exists simultaneously as a static, geometric feature (a Block Universe).
3. The Whole Ruliad
Because NAND is functionally complete, a network of unlabelled NAND graphs can express any Turing machine or computational state. Unrolling all possible NAND configurations yields the complete Ruliad—achieving full computational universality on the simplest possible topological substrate.
