Remark: Which parametrix is used

For the boundary current calculation, the singular reference distribution is explicitly $C_0$, and (eq:subtraction) proves the required local subtraction. For a local bulk observable, the appropriate object would instead be the four-dimensional Hadamard parametrix associated with $\mathrm{g}$. The imported Hadamard theorem guarantees its applicability to the selected bulk state; this paper does not identify the one-dimensional boundary subtraction with a bulk stress-tensor renormalization prescription. These distinctions are standard in curved-spacetime QFT HollandsWald2015.