Spacetime and Gravity as Emergent Quantum Entanglement - A Proof of Concept: Paper I — The Binary Tree Network, Exact Entropy Formula and AdS₂ Metric Convergence

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 χχ χ — as its setting, it proves three main results: First, an exact closed-form formula S(Am)=f∗(m)log⁡χS(A_m) = f^*(m)logχ 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(popcount_2(m),\ 1 + popcount_2(Nₘᵢₙ(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 H2H^2 H2 with every edge of equal hyperbolic length η=arccosh(21/16)η = arccosh(21/16) η=arccosh(21/16), and the rescaled boundary metric is proved quasi-isometric to AdS2AdS_2 AdS2 geodesics with explicit tight constants. Third, Newton's constant G=1/(4log⁡χ)G = 1/(4logχ) G=1/(4logχ) and the AdS2AdS_2 AdS2 radius R=δ/ln⁡2R = δ/ln 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log⁡2χR/G = 4log_2χ R/G=4log2χ as a proved theorem, with no free parameters once χχ 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.