= Solution
Fix the current sign convention by defining the localized variation as $\delta_\epsilon S=-\int d^dx\,(\partial_\mu\epsilon^a)j_a^\mu$. For a first-derivative Lagrangian invariant without a boundary term under constant parameters, this means $j_a^\mu=-\partial\mathcal L/\partial(\partial_\mu\phi)\,t_a\phi$; include the usual improvement term if the constant variation is a total derivative. On solutions, arbitrary compactly supported parameters imply $\partial_\mu j_a^\mu=0$. The <Noether charge> is $Q_a=\int d^{d-1}x\,j_a^0$ and is conserved if the spatial flux vanishes. This sign convention matches the printed <Ward identity>; reversing the current also reverses the corresponding generator convention.
Assume an invariant regulated <functional measure>, invariant vacuum boundary conditions and no <quantum anomaly>. Changing variables in the normalized <path integral> gives $0=\langle\delta_\epsilon X\rangle+i\langle X\delta_\epsilon S\rangle$. Integration by parts then yields
$$
-i\int d^dx\,\epsilon^a(x)\partial_\mu\langle j_a^\mu(x)X\rangle=\langle\delta_\epsilon X\rangle.
$$
For a product of <scalar fields>, the local <Ward identity contact terms> are
$$
\boxed{-i\partial_\mu\langle j_a^\mu(x)\phi(x_1)\cdots\phi(x_n)\rangle
=\sum_{r=1}^n\delta^d(x-x_r)\langle\phi(x_1)\cdots(t_a\phi)(x_r)\cdots\phi(x_n)\rangle.}
$$
Away from the insertions this is current conservation. Time-ordering or the distributional functional identity supplies the contact terms.
For the gauge theory write $[X,Y]^a=f^{abc}X^bY^c$ and use the <left-acting BRST differential> $s$, with $\delta_\epsilon=\epsilon s$. It obeys the <graded Leibniz rule> and
$$
sA_\mu=D_\mu c,\quad sc=-\tfrac12[c,c],\quad s\bar c=b,\quad sb=0.
$$
Assume the structure constants satisfy the <Jacobi identity> and the dot product is invariant; antisymmetry alone would not be enough. Since $c$ and $D_\mu c$ are odd, their <Lie brackets> are symmetric in these two arguments. Consequently
$$
s(D_\mu c)=D_\mu(sc)+[sA_\mu,c]
=-\tfrac12D_\mu[c,c]+[D_\mu c,c]=0.
$$
Also $sF_{\mu\nu}=D_\mu D_\nu c-D_\nu D_\mu c=[F_{\mu\nu},c]$. The <graded Jacobi identity> gives $s[c,c]=0$, so $s^2c=0$; the other three fields have zero second variation immediately. For independent odd parameters, $\delta_{\epsilon'}\delta_\epsilon=-\epsilon'\epsilon s^2=0$. These are off-shell <BRST nilpotence> identities because the <Nakanishi-Lautrup field> $b$ is retained.
The <Yang-Mills theory> variation is proportional to $F^{\mu\nu}\cdot[F_{\mu\nu},c]=0$. The gauge-fixing variation is $(\partial^\mu b)\cdot D_\mu c$, while the ghost variation is its negative; the $b^2$ term does not vary. Equivalently these terms are $s\Psi$ for the <gauge-fixing fermion> $\Psi=(\partial^\mu\bar c)\cdot A_\mu+\xi\bar c\cdot b/2$. <BRST nilpotence> makes this expression invariant. Ghost-number scaling also leaves every term invariant: the ghost and antighost factors carry opposite weights.
Here are the two explicit currents in the same sign convention as the Ward identity. Localizing the even ghost parameter gives coefficient $(\partial_\mu\theta)[\bar c\cdot D^\mu c-(\partial^\mu\bar c)\cdot c]$. Localizing the odd parameter, keeping it on the left, gives coefficient
$$
(\partial_\mu\epsilon)\left[-F^{\mu\nu}\cdot D_\nu c-b\cdot D^\mu c-\frac12(\partial^\mu\bar c)\cdot[c,c]\right].
$$
The last sign comes from moving the odd parameter through the odd antighost derivative. Since the current was defined as minus this coefficient,
$$
\boxed{j_G^\mu=(\partial^\mu\bar c)\cdot c-\bar c\cdot D^\mu c,\qquad
j_B^\mu=F^{\mu\nu}\cdot D_\nu c+b\cdot D^\mu c+\frac12(\partial^\mu\bar c)\cdot[c,c].}
$$
These are the <ghost-number Noether current> and the <BRST current in derivative-b gauge fixing>. If the opposite Noether sign is used, both displayed currents acquire an overall minus sign. Integrating the gauge-fixing term by parts changes the Noether representative by the associated boundary improvement; mixing the two Lagrangian conventions without that improvement gives incorrect signs.
With no BRST anomaly, the conserved odd <BRST charge> has $Q_B^2=0$. The <BRST cohomology> identifies closed states $Q_B|\psi\rangle=0$ modulo exact states $Q_B|\chi\rangle$. Nilpotence puts every exact state in the closed space. In the usual indefinite gauge-fixed state space, a Hermitian BRST charge makes exact states orthogonal to closed states; the standard no-ghost/positivity assumptions then give a physical <inner product> on the quotient. The physical sector is its ghost-number-zero component,
$$
\boxed{\mathcal H_{\mathrm{phys}}=H^0(Q_B)=\frac{\ker Q_B\cap\mathcal H^0}{Q_B\mathcal H^{-1}}.}
$$
The <ghost number> assigns $+1$ to $c$, $-1$ to $\bar c$ and zero to gauge and auxiliary fields. Gauge-invariant observables and the chosen vacuum have <ghost number> zero; unphysical ghost excitations are removed in BRST pairs. Thus physical representatives are expected to satisfy $Q_G|\psi\rangle=0$. This zero-grading selection is part of the physical-state prescription, not a consequence of nilpotence alone.
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