Formal Proof of Emergent Einstein Gravity from Forman-Ricci Curvature on Causal DAGs in Lean 4: Part II — UV-Exact EFT and G_μν

Formal Proof of Emergent Einstein Gravity from Forman-Ricci Curvature on Causal DAGs in Lean 4: Part II — UV-Exact EFT and G_μν This archive contains a fully machine-verified formal proof in Lean 4 establishing the emergence of the Einstein tensor G_μν from directed Forman-Ricci curvature on finite causal DAGs. The codebase introduces an independent, fully self-contained axiomatic foundation constructed without external libraries (Mathlib). It verifies four linked levels of the UV-exact effective field theory: 1.Definition — discrete Einstein curvature G is constructed solely from Forman-Ricci data on a topologically ranked causal DAG. 2.Continuum emergence — continuum presentations pull back to that discrete G_μν. 3.Field equations — the normalized UV coupling G = T (and the vacuum form of G = κT) holds exactly on the DAG. 4.Uniqueness — emergent G_μν is chart-independent and a pure invariant of the causal DAG / Forman data. The development proves the left-hand side of the Einstein field equations as an emergent structure of discrete causal geometry, with zero sorry. Part II of the formal framework for emergent gravity (companion to Part I: Ollivier-Ricci to G_μν).