Fractal Dimension of Value: Lightning Network as Percolation

The Lightning Network introduces a fractal structure to the BTF (Bitcoin Topological Fabric):

Lemma 10.1 (Channel Graph Percolation). Payment channels form a percolation network. At critical liquidity density, a giant connected component emerges—a spanning cluster where any two nodes can transact off-chain.

The fractal dimension of this network determines routing efficiency. Below the percolation threshold, the network fragments into isolated islands.

The "Homorupt" phase at 100% velocity now has a deeper interpretation: it's the percolation phase transition where the Lightning network achieves ∞ connectivity—every economic agent is reachable via off-chain paths, hence value flows become instantaneous and frictionless, approaching the hyperbolic limit.