Using multiplicity one for imaginary roots gives the wrong affine denominator

By F2 the factor for $\delta$ in F1 is $(1-e^{-\delta }{)}^2=1-2e^{-\delta }+e^{-2\delta }$; the proposed replacement is $1-e^{-\delta }$. A positive root below $\delta$ that is imaginary must equal $\delta$: in F2's computed loop list the other positive roots have nonzero finite-root part and squared length two, hence are real, while positive imaginary roots are $n\delta$. Terms from $n\ge 2$ cannot contribute at degree $\delta$ because their simple coordinates exceed those of $\delta$.

Let $R$ be the common product of all factors relevant at or below $\delta$ other than the $\delta$ factor. Its constant coefficient is one by F1. If $r$ is its coefficient at $e^{-\delta }$, the true product has coefficient $r-2$ and the modified product has coefficient $r-1$. Cross products with the nonconstant part of the $\delta$ factor require the zero coefficient of $R$, already one; no other terms can reach this degree. Thus the modified coefficient exceeds the true one by exactly one, refuting the claim. The zero-degree coefficients agree, so that agreement cannot detect the error. Rank-one imaginary multiplicity one would give no such witness; the rank-two diagonal space in F2 is essential. The calculation is finite and choice-free.