= Solution
The <mass> statement needs a nonzero <parallel spinor> and an isolated <asymptotically flat spacetime>; the zero <spinor> solves the equation on every geometry and cannot imply anything about <mass>. Assume the usual complete regular spin initial data and <dominant energy condition> used by the <positive energy theorem>, with the <spinor> tending to a nonzero constant $\epsilon_\infty$ at infinity. Let $V^a=\bar\epsilon\gamma^a\epsilon$ be its future causal <Dirac current>.
The <Nester two-form> can be written, up to normalization, as
$$
B^{ab}=\bar\epsilon\gamma^{abc}\nabla_c\epsilon
-\overline{\nabla_c\epsilon}\,\gamma^{abc}\epsilon.
$$
It is bilinear in $\epsilon$ and $\nabla\epsilon$. For a spacetime-parallel <spinor> it vanishes identically, and hence so does its asymptotic flux. The <ADM boundary term of the Nester two-form> identifies that flux, up to a fixed positive normalization, as
$$
E V_\infty^0-P_iV_\infty^i=0.
$$
The <positive energy theorem> gives $E\geq|P|$. Since $V_\infty$ is nonzero future causal, a future timelike ADM momentum would have strictly positive contraction with it. Consequently the ADM momentum is zero or null:
$$
\boxed{M_{\rm ADM}=\sqrt{E^2-|P|^2}=0.}
$$
In a rest frame, when one exists, this immediately says $E=0$. The usual regular asymptotically flat rigidity conclusion also excludes a nontrivial null-momentum configuration and gives flat initial data. This is the spinorial boundary-charge proof, not an inference that every Lorentzian manifold with a <parallel spinor> is flat. Without the asymptotic and global hypotheses the local <spinor> equation alone does not define, let alone determine, an ADM <mass>.
For the modified connection, it is important to fix the <Clifford algebra> normalization. Write
$$
\{\gamma_a,\gamma_b\}=2s\,g_{ab}I,\qquad
\gamma_{ab}=\frac12[\gamma_a,\gamma_b],\qquad s>0.
$$
The compatible <spinor curvature identity> is
$$
[\nabla_a,\nabla_b]=\frac1{4s}R_{abcd}\gamma^{cd}.
$$
Indeed, rescaling conventional <gamma matrices> by $\sqrt s$ rescales $\gamma_{ab}$ by $s$, while leaving the geometric spin connection unchanged. Since the connection is torsion-free and $\nabla\gamma=0$, the cross terms cancel in the modified commutator:
$$
[D_a,D_b]\epsilon=
\left(\frac1{4s}R_{abcd}\gamma^{cd}+c^2[\gamma_a,\gamma_b]\right)\epsilon.
$$
Thus the <Killing-spinor integrability with rescaled gamma matrices> equation is
$$
\boxed{(R_{abcd}\gamma^{cd}+8s c^2\gamma_{ab})\epsilon=0.}
$$
A factor of two in the convention for antisymmetrization multiplies the whole zero equation and cannot change this relative coefficient.
The two printed coefficients are consistent with $s=2$, namely $\{\gamma_a,\gamma_b\}=4g_{ab}I$. In that convention the displayed integrability equation becomes $R_{abcd}\gamma^{cd}+16c^2\gamma_{ab}=0$ on each solution. With the customary convention $s=1$, the coefficient is instead $8c^2$, and the final Ricci coefficient below is $-12c^2$. The PDF does not state its Clifford normalization, so these alternatives must be distinguished rather than mixing them.
In four spacetime dimensions the complex Dirac <spinor> fibre has dimension four. Four independent solutions of $D\epsilon=0$ span that fibre at every point: a solution vanishing at one point vanishes everywhere by <parallel transport>. Therefore the integrability matrix annihilates every <spinor> and is the zero matrix. The six bivector matrices $\gamma^{cd}$ are linearly independent, as is seen by taking traces against them; their trace pairing is a nondegenerate multiple of the metric on two-forms. Expressing $\gamma_{ab}=g_{ac}g_{bd}\gamma^{cd}$ therefore gives
$$
R_{abcd}=-4s c^2(g_{ac}g_{bd}-g_{ad}g_{bc}).
$$
This proves that <maximal Killing spinors force constant negative curvature>, not merely an Einstein <Ricci tensor>. Contracting in four dimensions gives
$$
\boxed{R_{ab}=-12s c^2g_{ab}
=\begin{cases}-24c^2g_{ab},&s=2,\\-12c^2g_{ab},&s=1.\end{cases}}
$$
For $s=2$ this is the requested Einstein equation. Since $c>0$, its <Ricci tensor> automatically has rank four; under the maximal-spinor hypothesis the additional rank assumption is redundant. If the stated four-dimensional solution space is interpreted directly as the pointwise kernel of the algebraic integrability equation, the same spanning and trace argument applies.
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