The Geometry
What falls out of the model without being put in.
Hyperbolic space
Define the “information distance” between two particles as dI = −log MI(A,B). This converts entanglement strength into a geometric distance: strongly entangled particles are close, weakly entangled ones are far.
The geometry defined by dI is hyperbolic. Specifically, the Gromov boundary of the infinite binary tree — the set of all infinite paths from the root — maps to the boundary of the hyperbolic plane H². This is not an assumption. It is a mathematical theorem about the structure of infinite trees.
Hyperbolic geometry is the natural geometry of anti-de Sitter (AdS) space — the setting in which the most developed theories of quantum gravity (the AdS/CFT correspondence) operate. The Quantum Family Tree model reproduces AdS geometry from nothing but a binary branching rule.
The holographic entropy formula
In 2006, Shinsei Ryu and Tadashi Takayanagi proved that the entropy of a region of space is proportional to the area of the minimal surface separating it from the rest. This is the Ryu-Takayanagi (RT) formula, and it is the centerpiece of modern holography.
In the Quantum Family Tree model, the RT formula holds: the expected mutual information between a region of leaves and its complement equals the number of edges cut by the minimal tree cut. The formula falls out of the tree structure without any additional assumptions.
Ultrametricity
Real physical space obeys the triangle inequality: the distance from A to C is at most the distance from A to B plus B to C. But the information distance in our model obeys something stronger: the ultrametric inequality, which says the distance from A to C is at most the maximum of the A-to-B and B-to-C distances.
We tested this on 1 million random triples of particles in a depth-8 tree. Ultrametricity holds for 98.3% of all triples, with a standard deviation of 0.006 across trees.
Ultrametric spaces are the natural home of p-adic numbers — a mathematical structure that has appeared repeatedly in string theory and quantum gravity. The model independently lands in the same mathematical class.