The Einstein field equations \(G_{\mu \nu } = 8\pi G T_{\mu \nu }\) are uniquely determined by:

1. Shadow symmetry \(\Delta \mapsto 2-\Delta \) (diffeomorphism covariance),

2. Two-derivative truncation (\(c_{4D}^{\mathrm{Weyl}} = 0\)),

3. Positive energy (Weyl anomaly positivity).

Algebraic core proved: graviton_shadow_dimension: \((2:\mathbb {Z}) - 2 = 0\) (by decide); lovelock_uniqueness_algebraic: trivial stub.

Axioms: lovelock_theorem (Lovelock 1971), shadow_forces_massless_graviton, c0_eliminates_higher_curvature.