Hadamard state $\omega$ $$\int_{\mathbb{R}}{\mathrm{d}}\tau \, \langle \rho^\mathrm{ren} \rangle_\omega (\gamma(\tau)) f^2(\tau) \geq
- \mathcal{B} \, ,$$ modulo local curvature terms, where $\mathcal{B}$ is of the form $$\begin{aligned}
\mathcal{B} &=& \int_{{\mathbb{R}}^+}\frac{{\mathrm{d}}\xi}{2\pi}\xi \bigg[ f\otimes f \, \vartheta^*{\,}^\psi{\mathrm{H}}^{(+)}_4 
\bigg]^\wedge(-\xi,\xi) {\nonumber \\}&\,& \quad - \int_{{\mathbb{R}}^-}\frac{{\mathrm{d}}\xi}{2\pi} \xi \bigg[ f\otimes f
\,\vartheta^*\big(i\mathrm{S}_{\mathrm{sp}}-{\,}^\psi{\mathrm{H}}^{(+)}_4 \big) \bigg]^\wedge(-\xi,\xi) \, .\end{aligned}$$ Here $\vartheta=\gamma\otimes\gamma$, $S_{\mathrm{sp}}$ is the fundamental solution to the Dirac equation, ${\,}^\psi{\mathrm{H}}^{(\pm)}_k$ are scalar distributions created from ${\,}^\psi H^{(\pm)}_k$ and $\hat{\,}$ denotes the Fourier transform which in our conventions is given by $$\hat{f}(\xi) = \int {\mathrm{d}}x \, f(x) \, e^{i\xi\cdot x} \, .$$

The structure of this paper is as follows: In section \[dirac background\] we present a review of the formulation of the classical (§\[dirac background 1\]-\[dirac background 2\]) and quantum (§\[quantum dirac hadamard\]) Dirac fields and their Hadamard states. We then direct our attention to a microlocal description of the Hadamard series for the Dirac field in section \[microlocal review\]; in particular we review the Sobolev wave-front set and its properties (§\[microlocal review 1\]) before applying the theory to the matter in hand and obtaining estimates on the singularities of the Hadamard series (§\[microlocal results\]). Finally, a point-splitting lemma is presented in section \[splitting section\] before our main result 