Step 1: Upper bound by squaring. For any real weights $\lambda_d$ with $\lambda_1 = 1$ and $\lambda_d = 0$ for $d > D$: $\left(\sum_{d|n, d|P(z)} \lambda_d\right)^2 \geq 1$ when $(n, P(z)) = 1$ and $\lambda_1 = 1$. Therefore: $S(\mathcal{A}, z) \leq \sum_{a \in \mathcal{A}} \left(\sum_{\substack{d | (a, P(z)) \\ d \leq D}} \lambda_d\right)^2 = \sum_{d_1, d_2 \leq D} \lambda_{d_1}\lambda_{d_2} |\mathcal{A}_{[d_1, d_2]}|$

Step 2: Separate main term and error. Substituting $|\mathcal{A}_e| = \frac{\omega(e)}{e}X + r_e$ with $e = [d_1, d_2]$: $S \leq X \sum_{d_1, d_2} \lambda_{d_1}\lambda_{d_2} \frac{\omega([d_1,d_2])}{[d_1,d_2]} + \sum_{d_1, d_2} |\lambda_{d_1}\lambda_{d_2}| |r_{[d_1,d_2]}|$

Step 3: Optimize the quadratic form. The main term is a quadratic form $Q(\lambda) = \sum_{d_1, d_2} \lambda_{d_1}\lambda_{d_2} h(d_1, d_2)$ where $h(d_1, d_2) = \omega([d_1,d_2])/[d_1,d_2]$. Using the multiplicativity of $\omega$ and Mobius factoring, this diagonalizes: $Q = \sum_{e | P(z)} \frac{1}{g(e)} \left(\sum_{e | d} \lambda_d\right)^2$ where $g(d) = \prod_{p|d} \frac{p - \omega(p)}{\omega(p)}$.

Subject to $\lambda_1 = 1$, the minimum of $Q$ (by Lagrange multipliers) is $1/G(z)$ where $G(z) = \sum_{d \leq D, d|P(z)} \mu^2(d)/g(d)$.

Step 4: Evaluate $G(z)$. By multiplicativity and the Mertens-type estimate: $G(z) = \prod_{p < z}\left(1 + \frac{\omega(p)}{p - \omega(p)}\right) \cdot (1 + O(D^{-c})) \asymp (\log z)^\kappa$ where $\kappa$ is the sieve dimension. The error $O(e^{-s/2})$ comes from the truncation $d \leq D = z^s$.

Step 5: Conclusion. $S \leq X/G(z)(1 + O(e^{-s/2})) + R$ where $R$ is the remainder depending on the individual error terms $r_d$. $\square$