Here is my comment on this session, only use it if it help else go with the plan: The critical next test in ii'-5c is whether the constants survive the whole chain. Conceptually, you want something like \[ g(s) = \frac{\zeta(s)}{\prod_{\rho_k} (s - \rho_k)^{m_k}} \] with the nearby zeros removed, then apply a holomorphic logarithm to g, use Borel–Carathéodory to control \[ \frac{g'(s)}{g(s)}, \] and reconstruct \[ \frac{\zeta'(s)}{\zeta(s)} = \frac{g'(s)}{g(s)} + \sum_{\rho_k} \frac{m_k}{s - \rho_k}. \] The hard part is not the identity; it is getting an explicit bound with enough slack that you still land at the target \[ |R(s)| \leq 10.5 \log(|t| + 2) + 48 \] on the intended right-hand strip. Your new disc estimate should help because \[ \log |\zeta(s)| \leq 2 \log |t| + \log(3/4), \] so the growth entering Borel–Carathéodory is logarithmic after taking the log, which is exactly the scale needed for an $O(\log |t|)$ logarithmic-derivative bound. The center lower bound is going to matter just as much. Using \[ \frac{1}{\zeta(s)} = \sum_{n \geq 1} \frac{\mu(n)}{n^s} \] at a center safely to the right of 1 should give an explicit lower bound for $|\zeta|$, hence an upper bound for $-\log|\zeta|$. If that lower bound is clean enough, the oscillation of $\Re \log g$ on the disc can be controlled explicitly rather than qualitatively. One point I would watch especially carefully in 5c is multiplicity bookkeeping. Since your residue formalization already handles multiple zeros, the Blaschke/zero-factor removal in the disc should preserve multiplicities consistently: \[ \prod_k (s - \rho_k)^{m_k}. \] Any mismatch between the zero count used in the perimeter bound and the multiplicity-weighted product could contaminate the final coefficient of $\log |t|$.