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 = 2L N=2L boundary sites and bond dimension χ χ — as its setting, it proves three main results: First, an exact closed-form formula S (Am) =f∗ (m) log⁡χS (Aₘ) = f^* (m) 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) = (popcount₂ (m), \ 1 + popcount₂ (N_ (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 H2H² H2 with every edge of equal hyperbolic length η=arccosh (21/16) = arccosh (21/16) η=arccosh (21/16), and the rescaled boundary metric is proved quasi-isometric to AdS2AdS₂ AdS2 geodesics with explicit tight constants. Third, Newton's constant G=1/ (4log⁡χ) G = 1/ (4) G=1/ (4logχ) and the AdS2AdS₂ AdS2 radius R=δ/ln⁡2R = / 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log⁡2χR/G = 4₂ R/G=4log2χ as a proved theorem, with no free parameters once χ 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.