Papers
One man's attempt to find a simple law that overlays the quantum universe.
The Quantum Family Tree trilogy develops a single framework in three steps: first, that spatial distance on a quantum genealogy tree grows at an exact rate derived from Haar-random two-qubit gates; second, that the same structure extends to general tree geometries and produces four structural theorems of an emergent gravitational setting; third, that when the tree is allowed to grow, its dynamics and the dynamics of the emergent spacetime are the same object described in two languages.
All three papers are posted on Zenodo with permanent DOIs.
The trilogy
Paper 1 — Emergence of Distance from a Quantum Family Tree
Binary Bruhat–Tits tree with Haar-random U(4) gates on every bond. The quenched von Neumann entanglement entropy grows linearly with tree distance at rate cVN = (9/10) log2(5/2) ≈ 1.1897 bits per generation. Confirmed numerically at 10.34σ and on a 156-qubit IBM Heron quantum run.
Paper 2 — The Emergence of Geometry from a Quantum Family Tree
Extends the entropy-rate theorem from binary qubits (p=2, d=2) to general p-ary trees with d-dimensional bonds. Proves four structural theorems: fixed-point convergence, the first law at all scales, the geometric precondition, and boundary-mode decomposition. The factor w2 = (1−η2)/2 is derived unconditionally from two-endpoint symmetry.
Paper 3 — Spacetime and the Quantum Family Tree
The dynamic case: what happens when the tree grows. Tree ancestry is shown to be an exact causal structure; tree growth is cosmological expansion with de Sitter scale factor; gate perturbations produce a retarded Green’s-function response; the holographic RG flow has a monotone c-function. The Conjoined Theorem identifies circuit dynamics and spacetime structure as one object in two languages.
Companion: Lemma D
The trilogy reduces to a single remaining scalar identity, stated as Lemma D. It is numerically supported at 10.34σ and confirmed on quantum hardware, but remains analytically open. The companion document below states it formally and catalogues the eighteen attacks that have already been tried.
Lemma D — An Open Problem in the Quantum Family Tree Program
A layman’s explanation, the formal mathematical challenge, a catalogue of dead routes, and an invitation to the field. Roughly nine pages. If you see an attack route the author and his advisor have missed, the problem is yours.
Code and data
Reproducibility code and results: github.com/kevin-nothing-matters/nothingmatters
Citation
K. Donahue, “Emergence of Distance from a Quantum Family Tree,” Zenodo, April 2026. doi:10.5281/zenodo.19464058
K. Donahue, “The Emergence of Geometry from a Quantum Family Tree,” Zenodo, April 2026. doi:10.5281/zenodo.19629206
K. Donahue, “Spacetime and the Quantum Family Tree,” Zenodo, April 2026. doi:10.5281/zenodo.19629257
Version history
| April 2026 | Trilogy posted to Zenodo. Paper 1 submitted to Communications in Mathematical Physics. Lemma D companion document released. |
| March 2026 | v30 conjecture (superseded). Reframed as conjecture with five testable predictions. |
| Feb 2026 | v20. GPU simulations at depth 12–14. Weingarten k=4 exact. |
| Jan 2026 | v2. Original public release. |