Simulations

The Quantum Family Tree conjecture makes testable predictions about the structure of mutual information in recursive quantum branching networks. The simulations below test those predictions at increasing scale, culminating in depth-12 GPU results (4,096 leaves, 8.4 million pairs, 20-tree ensembles).

All source code: github.com/kevin-nothing-matters/QFT

Method (v6 — Verified)

Each internal node implements a branching CPTP map expressed as two Kraus operators A₀, A₁ of shape (4×4). The distance metric is d_G = 2 × (depth − k) where k is the depth of the lowest common ancestor — the true graph distance through the tree. Verified against depth-4 ground truth to machine precision (max_err = 1.11×10⁻¹⁵).

Simulations ran on NVIDIA H100 SXM5 GPUs (RunPod) using CuPy for batched computation. Depth-12 results parallelized across 10 GPU pods (2 trees each), completing in approximately 2 hours.

System Sizes

Result 1: First Law of Entanglement Entropy

The most significant result. The Faulkner–Van Raamsdonk theorem derives linearized Einstein equations from a single structural assumption: the first law of entanglement entropy δS(A) = Tr(δρ_A · H_A) holds for all boundary regions A. Our discrete branching model reproduces this relation numerically.

The convergence is systematic: as ε halves, the deviation from 1.0 halves, confirming first-order scaling. At ε = 0.001, the ratio converges to 1.001–1.006 across all tested pairs. This is the algebraic signature of a genuine linearized identity rather than a numerical coincidence.

The model does not satisfy all conditions of the Faulkner–Van Raamsdonk theorem (it lacks a continuum QFT, geometric modular Hamiltonians, and AdS asymptotics). But it satisfies the discrete analogue of the key structural input. This is the result most likely to interest quantum gravity researchers.

Result 2: Divergent Fluctuation Law (Depth 12, 20 Trees)

Combined fluctuation spectrum averaged across 20 independent depth-12 trees (168 million pair measurements).

The near-field universality is the key observation. At d_G = 2, the std of σ/μ across 20 independent random trees is 0.004. Twenty different universes with twenty different random unitary sequences produce the same coefficient of variation at short range to 3 significant figures. This is a law. The far-field divergence — σ/μ = 18.6 at d_G = 24 — means some maximally separated pairs retain entanglement orders of magnitude above the mean. Those outliers are the physics.

Result 3: Ultrametricity

The information distance d_I(i,j) = −log MI(i,j) satisfies the ultrametric strong triangle inequality for 98.3% of triples. Median isosceles ratio R = 0.000; R < 0.05 for 97.7% of triples. This places the emergent geometry in the same mathematical class as p-adic spaces and Bruhat–Tits trees, the structures used in p-adic holography (Gubser et al. 2017).

Figure: Two-dimensional MDS embedding of information distance d_ij = −log MI_ij between 256 leaves at depth 8 (single tree). Left: leaves colored by top-level subtree cluster. Right: leaves colored by distance from leaf 0. Negative eigenvalues (82 of 256) confirm non-Euclidean geometry. Variance explained by 2D projection: 17% (expected for high-dimensional tree metric).

MDS eigenvalue spectrum. The negative eigenvalues are the signature of non-Euclidean (hyperbolic) geometry — the tree metric cannot be embedded in flat space.

Result 4: Boundary Entropy Scaling

Each doubling of interval length l increases boundary entropy by approximately 0.65 nats, consistent with (c/3)·log(2) = 0.655. This logarithmic scaling emerged from Haar-random branching dynamics without being assumed — it is the Ryu-Takayanagi prediction for a hyperbolic bulk.

Result 5: Mean Decay Law

Monotonic MI decay (strictly decreasing with d_G) observed in 100% of trials at all depths. The decay exponent α ≈ 1.2 is stable across depths 4–12. No theoretical derivation of this value exists; deriving α from the branching structure and Haar unitary ensemble is the central open problem.