Lemma: Local Hadamard form from the closed expression

On a sufficiently small developing neighbourhood of each regular point, in a local parallel frame,

with $H_{\beta }$ smooth; in fact the local regular remainder is real-analytic in the corresponding real variables.

Proof. Take a lift with $\delta$ near zero and $Q$ near $1$. The quotient in (eq:reducedF) is meromorphic in $(Q,\delta )$, because both its numerator divided by $\sinh \eta$ and its denominator are even analytic functions of $\eta$ near $\eta =0$. Its denominator factors as

For $\delta \ne 0$ sufficiently small, either term in (eq:Sgeom) computes the pole residue. It gives

analytic across the pole. Analytic division extends the same statement through $(Q,\delta )=(1,0)$. Multiplication by the prefactor in (eq:Wclosed) gives the direct kernel in (eq:localHadamard), because

The remaining analytic function has a smooth Lorentzian boundary value. The direct term is the ordinary Minkowski positive-frequency distribution in the developing chart.