Spacetime and Gravity as Emergent Quantum Entanglement — A Proof of Concept: Paper III — Boundary Hamiltonian, Chaos Bound Saturation, and the SYK Duality

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.