The Model
A precise, minimal setup.
Science requires more than a beautiful idea. It requires a precise model that makes testable predictions. Here is ours.
The tree
Take a binary tree: a root node that splits into two children, each of which splits into two more, and so on. This is the skeleton of the theory. Each node represents a quantum branching event — a moment when one quantum system becomes two. The leaves of the tree are particles.
The depth of the tree is a stand-in for time. The more branchings since the origin, the more “evolved” the particle. In a tree of depth 12, there are 212 = 4,096 leaf particles, connected by up to 24 branch points separating any two.
The branching rule
At each branching, the quantum state of the parent gets distributed to its two children via a random quantum operation — technically, a Haar-random isometry. “Haar-random” means the branching is drawn uniformly from all possible ways to split a quantum state, with no preference for any direction. It is the quantum equivalent of a fair coin flip.
This randomness is not a flaw. It is the model. We are not assuming any specific quantum dynamics. We are asking: if the only structure is the tree, and the branchings are as random as possible, what geometry emerges? The answer turns out to be highly constrained, because the structure of the tree imposes powerful correlations between what the branchings can be.
The distance measure
The genealogical distance dG between two leaf particles is defined as the number of branch points on the path between them through the tree. Two siblings (sharing a parent) have dG = 2. First cousins have dG = 4. And so on. This is the proposed physical distance — not the Euclidean distance between points in a pre-existing space, but the tree distance itself.
The entanglement measure
For each pair of leaf particles, we compute their mutual information — the standard quantum information measure of how correlated two systems are. Mutual information is zero for completely independent systems and maximal for perfectly entangled ones. It is the quantum analogue of knowing one twin’s mood tells you something about the other’s.
The question the theory answers: how does mutual information depend on dG?