Nothing Matters

The QFT

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The Papers

The Quantum Family Tree program is organized as a trilogy of papers. Paper 1 establishes that spatial distance emerges from entanglement structure. Paper 2 extends this to geometry. Paper 3 addresses spacetime dynamics. All three are structurally complete; the central open problem (Lemma D) is isolated in Paper 1 and referenced throughout.

Paper 1: Emergence of Distance from a Quantum Family Tree SUBMITTED

Kevin Donahue — April 2026 — Submitted to Communications in Mathematical Physics

A binary Bruhat–Tits tree of Haar-random U(4) unitaries produces a boundary MPDO whose entanglement encodes spatial distance. The purity eigenvalue η2 = 2/5 is proved via Weingarten calculus. The entropy-rate problem reduces exactly to a single scalar w2, with all structural components solved in closed form. The conjectured value w2 = 3/10 is confirmed to 10.34σ.

Key results: Exact purity recursion, rank-2 theorem, Standard Sector Projection Theorem, two-chain MPDO architecture, entropy-rate reduction to Lemma D.

PDF preprint LaTeX source

Paper 2: Emergence of Geometry from a Quantum Family Tree PREPRINT

Kevin Donahue — April 2026

Extends the distance result of Paper 1 to a full geometric framework. The first law of entanglement entropy is proved exactly at k = 1. The annealed Rényi-2 slope converges to 8/27 exactly. The fixed-point theorem establishes E[T†T] = η2 I3 with L2 rate η2 = 2/5. Geometric quantities (purity, entropy, modular Hamiltonian) are computed for general bond dimension (p,d).

Key results: First law exact, geometric precondition, annealed Rényi-2 asymptotics, boundary current decomposition, Reduction Theorem.

PDF preprint

Paper 3: Spacetime and the Quantum Family Tree PREPRINT

Kevin Donahue — April 2026

Develops the dynamical and gravitational aspects of the QFT program. Linear response, modular Hamiltonian structure, de Sitter interpretation (a(t) = pt, H = ln(p)), causal factorization, c-theorem (c(d) = η2d monotone), and the Conjoined Theorem (6-part formal duality). Validated on IBM Quantum hardware (ibm_fez, 156 qubits).

Key results: Response equation, RG flow, causality, Conjoined Theorem, hardware validation.

PDF preprint

The central open problem

Lemma D. Prove that the sector weight w2 = (1 − η2)/2 = 3/10. This single identity, if proved, would complete the entropy-rate theorem across all three papers and provide a rare example of an exact entropy formula in a nontrivial random quantum system. The conjecture is confirmed numerically to 10.34σ but remains analytically open.

Proved constants

η2 = 2/5  •  m2 = 3/10  •  β1 = 1/10  •  pstat = 2/3  •  hS2qu = 14/45  •  hS2ann = 8/27  •  Jensen gap = 21/20

Citation

K. Donahue, "Emergence of Distance from a Quantum Family Tree,"
submitted to Commun. Math. Phys., April 2026. nothingmatters.life

Code

Simulation code: github.com/kevin-nothing-matters/QFT