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.
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=2χV1−2, 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 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.
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.
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.
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.
This is the thrid paper of the series of three: Spacetime and Gravity as Emergent Quantum Entanglement — A Proof of Concept The paper closes the three principal open questions of the series by identifying the physical boundary Hamiltonian, fixing all free thermodynamic parameters, and establishing that the boundary theory is maximally chaotic. Four results are proved: Theorem A identifies the boundary Hamiltonian as the Sachdev–Ye–Kitaev model: an explicit Collins–Śniady Weingarten calculation applied to the binary tree encoder's unitary 2-design property shows that the tree ensemble average and the SYK disorder average produce identical Gaussian coupling variance at leading order in 1/χ, yielding the same Schwinger–Dyson equations and the same conformal Green function G(τ) = −b·J⁻¹/²·sgn(τ)/|τ|¹/², establishing H_R = H_SYK as an equality of disorder-averaged effective field theories. Theorem B derives the previously free Schwarzian coefficient and topological dilaton — φ_r = (πb²N/16J)·log(J/Δ) and φ_0 = N/(2log₂χ) — making the near-extremal black hole entropy a parameter-free prediction. Theorem C proves that the quantum Lyapunov exponent saturates the Maldacena–Shenker–Stanford chaos bound at λ_L = 2πT, with scrambling time t* ~ β log N, placing the boundary theory in the fast-scrambling and GUE random-matrix universality class. Theorem D confirms the boundary stress-tensor two-point function ⟨T(x)T(0)⟩ = 6log₂χ/(2x²) as an unconditional theorem, consistent with an effective central charge c_eff = 6log₂χ. A corollary shows that the Bisognano–Wichmann assumption of Paper II is unnecessary once the explicit boundary Hamiltonian is in hand. Together with Papers I and II, this paper completes the derivation of the full holographic dictionary for the binary tree model — from a discrete quantum network to emergent geometry, physical time, gravitational field equations, and quantum chaos — entirely from first principles.