Frame Comparison Removes the Scalar Obstruction to Entanglement Equilibrium

Jacobson derived the Einstein equation from the requirement that the entanglement of small balls be stationary at fixed volume. For massive or otherwise non-conformal matter the derivation rests on a conjecture about the form of the entanglement, and Casini, Galante and Myers showed that the actual small-ball entanglement contains logarithmic and R^{2Δ} terms that dominate the stress-tensor term as R → 0. We show that the difference between the first-order entanglement of the same ball cut in two Lorentz frames contains, to order R^d, only the stress-tensor term with the Casini–Huerta–Myers coefficient and second derivatives of light scalars: every term that obstructed the derivation is a Lorentz scalar and cancels. Requiring this difference to vanish is equivalent to entanglement equilibrium with a scalar local curvature term, and it gives G_ab + Λ g_ab = 8πG [T_ab + (∂_a∂_b − g_ab ∂²)X] with Λ constant, for every unitary, Lorentz- and CPT-invariant quantum field theory with a conformal short-distance limit; light scalars can enter only as constant curvature couplings, and at first order in conformal perturbation theory they do not enter at all. Pre-registered lattice computations at 48–60 digits confirm it. For the free massive Dirac fermion in d = 2, the case of Casini, Galante and Myers, the logarithm is reproduced, b/(a_R m²) = 0.033 − 0.941 ln(mR), and the frame difference returns the stress-tensor coefficient to 2.5 × 10⁻⁴ while the single-state ratio is 1.3–4.2. For massive scalars in d = 3 and d = 4 the single-state ratio reaches 2.3 × 10³ and 5.1 × 10³; the frame difference returns 1 to 0.2% and, for the median state, 0.13%, converging with the ball radius in d = 4 from 0.92% to 0.20% to 0.13%.