Einstein Equation as Φ-Self-Consistency and Bianchi Identity from Diff(M⁴) Symmetry in ODTOE — Overview
Closing stage 3 of programme §XIV.3. Einstein equation G_μν+Λg_μν=(8πG/c⁴)T_μν derived as Φ-self-consistency condition on pairs (g,T). Bianchi identity ∇_μG^μν=
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Closing stage 3 of programme §XIV.3. Einstein equation G_μν+Λg_μν=(8πG/c⁴)T_μν derived as Φ-self-consistency condition on pairs (g,T). Bianchi identity ∇_μG^μν=0 established along two independent paths: kinematic (contraction of second Bianchi identity) and Noether (diffeomorphism invariance of observer action). Theorem C.T1: pair (g,T) solves Einstein equation iff it is fixed point of map Φ_C; existence via Banach fixed-point theorem. Theorem C.T2: dual-path Bianchi with 50-digit verification |∇_μG^μν|_{Path1}−|∇_μG^μν|_{Path2}<10⁻⁴⁵. Theorem C.T3: ODTOE singularity theorem as structural analog of Hawking–Penrose theorem.