Universal Grid Mechanics — The Gravity Hamiltonian: Recovering Newton, Structural MOND-like Dynamics, and the Mathematical Proof That General Relativity Cannot Be Recovered

Within Universal Grid Mechanics (UGM), a gravity-sector Hamiltonian is derived solely from the admissibility axioms and the primitive L6 operator, without importing any gravitational data. The Hamiltonian contains two coupled sectors: a gradient-energy sector linking the structural field S̄ to matter-content ρκ, and a memory sector linking M̄ to the write–release balance. Stationarity recovers the UGM scalar gravity closure ΓU∇²(S̄+M̄) = ρκ + M̄ with ΓU = π√3/3. In the low-memory limit M̄ ≪ S̄, the Newtonian sector is isolated: ∇²Φ = 4πGρκ with G = √3/4π² in natural units, recovering the inverse-square law as the Green's function. In the memory-dominated limit M̄ ≫ S̄, the closure becomes a screened Poisson equation exhibiting MOND-like galactic dynamics. The MOND transition acceleration g† is structurally identified as the crossover condition ℛeq = 1, giving g† = (ωWτM − 1)S̄0/τM; its numerical value awaits micro-derivation of ωW from L6 dynamics, characterised as an E1/E2 bifurcation problem in V02.10. The full derivation chain produces three observer branches from one Hamiltonian: Branch I (Newton, ℛ→0); Branch II (MOND-like, ℛ≫1, quasi-steady); and Branch III (history-dependent, ∂τℛ≉0, where MOND-like scaling explicitly fails). Branch III yields quantitative falsifiable predictions: the lag-fraction formula Δaeff/aN ≈ −αM exp(−tdyn/τM) produces a parameter-free ordering of the Bullet Cluster, El Gordo, and isolated disks keyed to τ0 ≈ 3 Gyr. General Relativity is structurally excluded by the memory-accumulation structure of Axiom 4 and the single-realized-branch update structure (Admissible Path Interpretation, API, V02.10). The global contraction theorem for the S̄–M̄ coupled system is closed by Theorem 7.3 of V02.10. This paper closes the scalar gravity sector only. Tensor closure, covariant conservation, lensing, and gravitational-wave correspondence remain subsequent derivation targets. Open gaps are precisely identified, including two parallel open tracks: Track A (2V branch universality criterion for ωW) and Track B (dimensional bridge B(ℓ, Amax) = 0).