This is an update to a previous paper with the same title: Abstract We study von Neumann entropy in the Quantum Family Tree (QFT) model — a binary Bruhat–Tits tree whose bonds carry independent Haar-random U(4) gates. For an interval A of size 2^k boundary sites, the quenched entropy S_VN(A) grows linearly with k in the deep-tree limit. We measure the slope c_VN and find c_VN = m² × c_RT, m² = 3/10 where c_RT = 3 log₂(5/2) is the classical mutual-information (Ryu–Takayanagi-like) coefficient. The measurement uses six independent tree depths (D = 20–150) and yields c_VN = 1.18845 ± 0.00116 bits confirming the ratio β₁ = c_VN/(3c_RT) = 1/10 within 1.1σ, and strongly disfavoring the nearest alternative β₁ = 10/99. We prove: (i) η₂ = 2/5 exactly via Weingarten calculus; (ii) a rank-2 bound on ρ_A for aligned intervals within the QFT construction; and (iii) the first-law identity δ⟨H_A⟩/δS = 1 for k=1. We identify the Choi-boundary state as a two-chain Matrix Product Density Operator (MPDO) with bond dimension 2 and transfer matrix spectrum {1, η₂ = 2/5}. This model reproduces both the von Neumann and Rényi-2 entropies across all tested scales and depths to sub-percent accuracy with no adjustable parameters beyond η₂ fixed by Weingarten calculus. The quenched Rényi-2 rate is consistent with the exact rational value h_S2 = 14/45 (0.07σ) and the annealed Rényi-2 rate with 8/27, with Jensen gap ratio 21/20 exactly. We derive the identity m² = (1 − η₂)/2 within the Choi-boundary MPDO framework, which reduces the entropy-rate conjecture c_VN = m²·c_RT to computing the von Neumann entropy rate of this specific MPDO.
This is an update to a previous paper with the same title: Abstract We study von Neumann entropy in the Quantum Family Tree (QFT) model — a binary Bruhat–Tits tree whose bonds carry independent Haar-random U(4) gates. For an interval A of size 2^k boundary sites, the quenched entropy S_VN(A) grows linearly with k in the deep-tree limit. We measure the slope c_VN and find c_VN = m² × c_RT, m² = 3/10 where c_RT = 3 log₂(5/2) is the classical mutual-information (Ryu–Takayanagi-like) coefficient. The measurement uses six independent tree depths (D = 20–150) and yields c_VN = 1.18845 ± 0.00116 bits confirming the ratio β₁ = c_VN/(3c_RT) = 1/10 within 1.1σ, and strongly disfavoring the nearest alternative β₁ = 10/99. We prove: (i) η₂ = 2/5 exactly via Weingarten calculus; (ii) a rank-2 bound on ρ_A for aligned intervals within the QFT construction; and (iii) the first-law identity δ⟨H_A⟩/δS = 1 for k=1. We identify the Choi-boundary state as a two-chain Matrix Product Density Operator (MPDO) with bond dimension 2 and transfer matrix spectrum {1, η₂ = 2/5}. This model reproduces both the von Neumann and Rényi-2 entropies across all tested scales and depths to sub-percent accuracy with no adjustable parameters beyond η₂ fixed by Weingarten calculus. The quenched Rényi-2 rate is consistent with the exact rational value h_S2 = 14/45 (0.07σ) and the annealed Rényi-2 rate with 8/27, with Jensen gap ratio 21/20 exactly. We derive the identity m² = (1 − η₂)/2 within the Choi-boundary MPDO framework, which reduces the entropy-rate conjecture c_VN = m²·c_RT to computing the von Neumann entropy rate of this specific MPDO.
Abstract We introduce the Quantum Family Tree (QFT), a solvable model in which spatial distance emerges from genealogical structure in a randomly branching quantum process. The model is realized as a binary Bruhat–Tits tree whose edges carry independent Haar-random U4 unitaries. Boundary states are generated by recursive channel composition from the root, producing a random Matrix Product Density Operator (MPDO) with bond dimension 2. For an interval of size 2k, the quenched von Neumann entropy grows linearly with k in the deep-tree limit. Extensive numerical measurements across six independent depths (D=20–150) yield cVN=1.188450.00116 bits, consistent with cVN=m2 cRT, m2=310, cRT=3log252, and strongly disfavoring nearby rational alternatives at >10σ. We establish the full structural backbone underlying this result. Specifically, we prove: (i) the exact purity eigenvalue 2=2/5 via Weingarten calculus; (ii) an exact two-component purity recursion in the {SWAP,TrTr} basis; (iii) a Standard Sector Projection Theorem restricting all replica contributions to a two-dimensional subspace {1,2}; and (iv) an identification of the boundary state as a two-chain MPDO with transfer spectrum {1,2}, reproducing both von Neumann and Rényi-2 entropies to sub-percent accuracy. These results reduce the entropy-rate problem exactly to a single scalar quantity—the sector weight w2, governing the projection of the standard sector in the von Neumann limit. We formulate this as an explicit conjecture (Lemma D): w2=1−2/2=3/10. The conjectured value is confirmed numerically to 10.34σ but remains analytically open. The work isolates a well-posed and previously uncharacterized problem: the evaluation of the quenched von Neumann entropy rate of a random MPDO with known transfer spectrum. All other components of the entropy-rate structure are established exactly. The quenched Rényi-2 rate is consistent with hS2=14/45 (0.07σ) and the annealed rate with 8/27, with Jensen gap 21/20.
Update from prior version. Abstract The Quantum Family Tree (QFT) theory proposes that spatial distance emerges as a measure of kinship difference within a quantum genealogical tree: sites that share a recent common ancestor are near, while sites whose lineages diverge deep in the tree are far apart. In this framework, geometry is not fundamental but inherited – the entanglement structure of a randomly branching quantum process encodes a notion of distance that reproduces, in a precise quantitative sense, the entropy–area relation of holographic gravity. We realize this proposal in a concrete solvable model – a binary Bruhat–Tits tree whose bonds carry independent Haar-random U(4) gates. For an interval A of size 2^k boundary sites, the quenched von Neumann entropy S_VN(A) grows linearly with k in the deep-tree limit. We measure the slope c_VN and find c_VN = m² × c_RT, m² = 3/10 where c_RT = 3 log₂(5/2) is the classical mutual-information (Ryu–Takayanagi-like) coefficient. The measurement uses six independent tree depths (D = 20–150) and yields c_VN = 1.18845 ± 0.00116 bits confirming the ratio β₁ = c_VN/(3c_RT) = 1/10 within 1.1σ, and strongly disfavoring the nearest alternative β₁ = 10/99. We prove: (i) η₂ = 2/5 exactly via Weingarten calculus; (ii) a rank-2 bound on ρ_A for aligned intervals within the QFT construction; (iii) the first-law identity δ⟨H_A⟩/δS = 1 for k=1; and (iv) an exact 2-component purity recursion in the {SWAP, TrTr} basis that captures the full position-averaged Rényi-2 structure of the boundary state without Monte Carlo. The purity kernel at each tree node decomposes as M = b·SWAP + c·TrTr, reducing the 16-dimensional recursion to two components with exact rational coefficients. The carry eigenvalue η₂ = 2/5 is proved to be q-independent (verified at q = 2, 3, 4 with multiplicities C(q,2)), and the straddle algebra at O(ε) near q = 1 is additive rather than bilinear. These two facts – q-independence of dynamics and O(ε) additivity – imply that the entire proof of c_VN = m²·c_RT reduces to a single remaining representation-theoretic computation: the sector weight w₂ governing how the SWAP sector contributes to von Neumann entropy at q → 1. We predict w₂ = (1 − η₂)/2 = 3/10 = m². We identify the Choi-boundary state as a two-chain Matrix Product Density Operator (MPDO) with bond dimension 2 and transfer matrix spectrum {1, η₂ = 2/5}. We reduce the entropy-rate problem exactly to a single scalar quantity – the sector weight w₂ – eliminating the need for large-depth recursion or asymptotic analysis. This model reproduces both the von Neumann and Rényi-2 entropies across all tested scales and depths to sub-percent accuracy with no adjustable parameters beyond η₂ fixed by Weingarten calculus. The quenched Rényi-2 rate is consistent with the exact rational value h_S2 = 14/45 (0.07σ) and the annealed Rényi-2 rate with 8/27, with Jensen gap ratio 21/20 exactly.
This paper is an update that includes LEAN4 Validation. We introduce the Quantum Family Tree, a branching quantum circuit defined on a binary (Bruhat–Tits) tree in which each internal node applies an independent Haar-random two-qubit unitary. In this model, entanglement is organized by genealogical distance: two particles are less entangled the further back their lowest common ancestor lies. We show that the expected von Neumann entropy of an interval of size 2k2^k2k grows linearly with the number of generations kkk, SVN(k)=cVN3 k,S_{VN}(k) = \frac{c_{VN}}{3}\,k,SVN(k)=3cVNk, with coefficient cVN=910log2 (52)≈1.1897 bits.c_{VN} = \frac{9}{10}\log_2\!\left(\frac{5}{2}\right) \approx 1.1897 \ \text{bits}.cVN=109log2(25)≈1.1897 bits. This result is derived from first principles. A single Haar average yields the exact purity η2=2/5\eta_2 = 2/5η2=2/5, which determines both a scaling dimension Δ=log2(5/2)\Delta = \log_2(5/2)Δ=log2(5/2) and a bulk mass parameter m2=3/10m^2 = 3/10m2=3/10. The resulting cascade propagator is shown to be an eigenmode of the p-adic Laplacian, satisfying a discrete Klein–Gordon equation on the tree. The entropy coefficient is then obtained by relating this structure to known results in p-adic AdS/CFT. The prediction is confirmed numerically across tree depths D=20D = 20D=20 to 150150150 (up to 500 independent realizations per depth), with a 10.34σ separation from the closest competing hypothesis. A Lean 4 formalization verifies all algebraic steps internal to the model, isolating three external inputs (Haar integration, p-adic entropy formula, and the entanglement first law). These results provide a concrete, exactly analyzable setting in which entanglement scaling, ultrametric geometry, and Klein–Gordon spectral structure emerge from a single random-unitary construction, with a fully identified path to a self-contained proof via harmonic analysis.
We introduce a branching quantum circuit on a binary (Bruhat–Tits) tree in which each internal node applies an independent Haar-random two-qubit unitary, and show that entanglement organizes along genealogical distance. For two particles separated by k generations, the expected von Neumann entropy of an interval of size 2k grows linearly: S_{VN}(k) = (c_{VN}/3)·k, with exact coefficient c_{VN} = m²·c_{RT} = (9/10)·log₂(5/2) ≈ 1.1897 bits. Both inputs are derived from first principles: a single Haar average yields η₂ = 2/5, from which we obtain the boundary scaling dimension Δ = log₂(5/2), the bulk mass m² = 3/10, and a cascade propagator that is an eigenmode of the p-adic Laplacian, satisfying a discrete Klein–Gordon equation. This matches the spectral data of a p-adic bulk field. Invoking the p-adic AdS/CFT entropy formula of Gubser–Heydeman–Parra-Martinez–Parikh (2016) then fixes the entropy coefficient, giving c_{VN} = m²·c_{RT}. The result is confirmed numerically up to depth D = 50 at 10.34σ. A fully self-contained proof is reduced to a single analytic continuation problem (Conjecture~1), formulated in terms of Selberg integrals and spherical functions on SU(4)/S(U(2)×U(2)). Together with a numerical verification of the entanglement first law in the same model, these results provide a concrete setting in which entanglement scaling, ultrametric geometry, and a Klein–Gordon spectrum emerge from a single random-unitary construction.
Abstract We extend the entropy-rate theorem of the Quantum Family Tree (QFT) model from the binary qubit case p=2, d=2 to general p-ary trees with d-dimensional bonds. Four structural theorems are proved: (i) a fixed-point convergence theorem establishing *=I/d as the unique stationary state with L2 contraction rate 2p,d; (ii) the first-law identity ⟨HA⟩/δS=1 exact at all scales; (iii) a geometric precondition for the Lewkowycz–Maldacena derivation, with corrections exponentially small at rate 2; and (iv) the resolution of the annealed Rényi-2 slope as 8/27, with finite-depth corrections decaying at rate 4/2=13/28. The Weingarten purity recursion is derived exactly for all p,d: =2p,d=dp−1d2−1/d2p−1, =dd2p−1−1/d2p−1, with a remarkable , swap between complementary cuts. The Klein–Gordon graph-Laplacian mass formula is falsified at p3. Numerical measurements on the ternary p=3 tree support the universality conjecture w2p,d=1−2/2, with the factor 1/2 independent of p. The universality is reduced to a single scalar identity—the one-boundary quadratic susceptibility—through a Reduction Theorem that expresses the entropy rate as hVN=2χV*1−2, and verified numerically to within 5%. The scalar identity remains conjectural at both the binary level (Lemma D of Paper 1) and the general (p,d) level; closing Lemma D would fix the binary susceptibility at 2,2=32log25/2.
This paper establishes the geometric and entropic foundations of an exactly solvable holographic model for two-dimensional quantum gravity. Taking the binary tree perfect tensor network — a hierarchical quantum system with N=2LN = 2^L N=2L boundary sites and bond dimension χ\chi χ — as its setting, it proves three main results: First, an exact closed-form formula S(Am)=f∗(m)logχS(A_m) = f^*(m)\log\chi S(Am)=f∗(m)logχ is derived for the entanglement entropy of any boundary interval, where f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m))f^*(m) = \min(\mathrm{popcount}_2(m),\ 1 + \mathrm{popcount}_2(N_{\min}(m)-m)) f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m)) is a combinatorial function of the interval size, establishing the Ryu–Takayanagi formula as a theorem of the network rather than a conjecture. Second, the tree is shown to embed isometrically into the hyperbolic plane H2\mathbb{H}^2 H2 with every edge of equal hyperbolic length η=arccosh(21/16)\eta = \mathrm{arccosh}(21/16) η=arccosh(21/16), and the rescaled boundary metric is proved quasi-isometric to AdS2\mathrm{AdS}_2 AdS2 geodesics with explicit tight constants. Third, Newton's constant G=1/(4logχ)G = 1/(4\log\chi) G=1/(4logχ) and the AdS2\mathrm{AdS}_2 AdS2 radius R=δ/ln2R = \delta/\ln 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log2χR/G = 4\log_2\chi R/G=4log2χ as a proved theorem, with no free parameters once χ\chi χ is specified. The paper additionally derives the exact modular Hamiltonian of the vacuum state from the perfect tensor isometry, and recovers the bulk stress-energy tensor from boundary entropy perturbations via an explicit inversion formula, completing the chain from a discrete quantum network to a bulk metric with gravitational coupling and matter content.