9 Removing the genericity hypothesis
The main theorem above assumes a generic parameter \(s\) (the spectator product is nonzero for every selection with invertible coefficient matrix). We remove this hypothesis by continuation in the parameters \(s\). The flag-residue sum has, as a function of \(s\), the shape (holomorphic numerator)/(polynomial denominator), so it is meromorphic in \(s\) and the generic-to-all step is the analytic identity principle in \(s\) — carried out after clearing denominators, so that no removable singularities arise.
The domain \(\Omega = \{ s \in \mathbb {C}^R : \mathrm{Re}(s_j) {\gt} 0 \text{ for all } j \} \). It is convex, hence preconnected and open.
The integral \(F(s) = \int _{\mathbb {R}^n} h(x) / \prod _j (f_j(x) - s_j i)\, dx\), viewed as a function of the parameter \(s\).
The subset \(U \subseteq \Omega \) where, for every selection whose coefficient matrix is invertible, the spectator product is nonzero. It is open, and (when nonempty) dense in \(\Omega \).
\(D(s) = \prod _{\sigma \in \Pi \text{-stable}} \mathrm{nonSelProd}_\sigma (s)\), the product of the spectator denominators. It is polynomial in \(s\).
\(N(s) = (2\pi i)^n \sum _\sigma \bigl(\prod _{\sigma ' \neq \sigma } \mathrm{nonSelProd}_{\sigma '}(s)\bigr)\, h(z_\sigma (s))\, \det _\sigma ^{-1}\), obtained by clearing the common denominator from the flag-residue sum. Since the pole point \(z_\sigma (s)\) is affine in \(s\) and \(h\) is holomorphic, \(N\) is holomorphic in \(s\).
If \(f\) and \(g\) are analytic on a preconnected set \(\Omega \) and agree on a nonempty open subset \(U \subseteq \Omega \), then they agree on all of \(\Omega \).
Agreement on the open \(U\) gives eventual equality at a point of \(U\); the analytic identity principle then propagates the equality across the preconnected \(\Omega \).
Assume \(h\) holomorphic, the convergence hypothesis (for all \(s\) with positive real part), and that \(F\) is analytic in \(s\) on \(\Omega \). Then, with no genericity hypothesis,
Where \(D(s) \neq 0\) this recovers the closed-form stable-residue formula; where \(D(s) = 0\) it expresses that \(N/D\) has a removable singularity equal to the integral.
On the generic locus \(U\) the main theorem and the closed form give \(D \cdot F = N\). Both sides are analytic in \(s\) on \(\Omega \) (\(D\) and \(N\) are built from analytic pieces; \(F\) by assumption), and \(U\) is a nonempty open subset of the preconnected \(\Omega \); analytic continuation extends the identity to all of \(\Omega \).