Prime-improved denominator bound for each partial-fraction coefficient

Let $P$ be admissible, let $n>0$, and let $(c_{s,k})$ be any partial-fraction datum for the mission rational function $R_n$, including its factor $h_0+2t$. For every $1\le s\le S=q-r$ and every $k$ in the full pole interval,

The product is empty, hence $1$, for $s=S$. This is an individual coefficient estimate; it contains no zeta sum or harmonic constant. It packages the elementary-brick arithmetic used in Lemma 19: the rough bound $D_{m_0n}^{S-s}c_{s,k}\in\mathbb Z$, with $m_0=\max(\eta_r,\eta_0-2\eta_{r+1})$, together with the prime valuation improvement by $\phi(n/p)$. Since $m_j\ge m_0$, the tail lcm product clears the rough denominators; each prime selected in $\Phi_n$ occurs once in each tail lcm because $p\le m_Sn\le m_jn\le\eta_0n<p^2$. The prime cutoff here is exactly that of the mission's 2001 note. The normalized local coefficients must be computed after symbolic cancellation of the pole factors.