= Virasoro–Shapiro amplitude pole residue
{c}
{title2=$s_n=4(n-1)/\alpha'$}
For $a=-1-\alpha's/4$, $b=-1-\alpha't/4$ and $a+b+c=1$, the amplitude has generic <simple poles> at $a=-n$. The <gamma function> <residue> and recurrence give
$$
\operatorname*{Res}_{s=s_n}A^{(4)}=-\frac{8\pi g_s^2}{\alpha'(n!)^2}\prod_{j=1}^n(b-j)^2.
$$
The <residue> polynomial has degree $2n$, consistent with closed-string exchange up to spin $2n$. When $a=-n+\delta_a$ and $b=-m+\delta_b$, the reciprocal factor $1/\Gamma(a+b)$ removes a putative double pole: the leading ratio is $\binom{n+m}{n}^2(1/\delta_a+1/\delta_b)$. Thus channel singularities must be counted using the kinematic constraint, not by multiplying independent numerator poles.
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