\section{Computations}\label{sec:numerics}
{\sloppy

Apart from the discrete convolutions bounding the integrals $J_m$, described below, every numerical step was carried out in outward-rounded
interval arithmetic (mpmath \texttt{iv}) or ball arithmetic (Arb, via python-flint), with decimal inputs treated as exact rationals and
Euler's constant and Mertens' constant enclosed. Each positive comparison was certified by at least two separately written implementations,
one of which is, for some steps, an independent audit (separately written scripts, in the supplement's \texttt{audit/} directory, that recompute the same quantities in Arb); the details follow. The scripts named below, with their parameter files and outputs, form
the supplement to this paper, which is the \texttt{anc/} directory of the paper's record, and each is named by its path in it. Its
\texttt{run\_all.sh.txt} (a bash script) reruns every certificate; \texttt{compare.py}
compares each output with the stored one and checks every bound that this section quotes from the outputs; \texttt{SHA256SUMS.txt} and
\texttt{REPRODUCTION.md} record the files' hashes, the software versions and a complete run.

\subsection*{The middle range}
For Lemmas~\ref{lem:C3num}, \ref{lem:C4num} and~\ref{lem:C3GRHnum}, the coefficients $R^*_j$ and $R^{(w)}_j$ are computed from the recurrence
in the remark after Lemma~\ref{lem:Rstar}, all of whose terms are nonnegative. The prime sums $P_{13}(k)$, $P_w(k)$, $A_a$ and $A^{(w)}_a$
are summed exactly over the primes up to $10^5$ (or $1000$--$2000$ for large $k$), and every truncation is carried as an enclosure: the tails
$\sum_{n > P} n^{-k} \le P^{1-k}/(k-1)$, the integral tails of $A_a$ for $a \le 80$, a geometric bound for $a > 80$, and an enclosure of
Mertens' constant. So for \texttt{author/cert\_C3s.py} and \texttt{third/t2\_check/c3\_grh.py} the computed interval encloses the left side
itself; its upper end proves a condition and its lower end disproves it at the stated parameters. \texttt{second/cert\_C3.py} and
\texttt{second/cert\_C4.py} compute upper bounds.
\begin{itemize}
\item Lemma~\ref{lem:C3num}: at $(e^{1800}, 18)$, the enclosure is $[0.209698, 0.209801]$ in mpmath \texttt{iv} (\texttt{author/cert\_C3s.py})
 and the Arb upper bound is $0.2098006747\ldots$ (\texttt{second/cert\_C3.py}); the independent audit's bound (\texttt{audit/t2v6/audit\_c4.py}) is
 $< 0.2097564$. At $(e^{1800}, 17)$ the enclosure is $[0.564615, 0.564888]$. At $(e^{350}, 14)$ the Arb upper bound is
 $0.1497600658\ldots$, and \texttt{author/cert\_C3s\_v7.py}, which evaluates \texttt{author/cert\_C3s.py}, gives the enclosure
 $[0.149686, 0.149761]$. At $(e^{62}, 9)$ the mpmath enclosure is $[0.241321, 0.241386]$ (the case $w = 11$ of
 \texttt{third/t2\_check/c3\_grh.py}, which is Lemma~\ref{lem:C3}), and the independent audit's Arb bound
 (\texttt{audit/t2v6/audit\_grh9.py}) is $< 0.2413750$.
\item Lemma~\ref{lem:C4num}: the Arb upper bound is $0.4068036291\ldots$ (\texttt{second/cert\_C4.py}), of which the term with $(1 + \eps_Y)$
 contributes at most $0.360897\ldots$ and the rest at most $0.045906\ldots$; the independent audit's bound
 (\texttt{audit/t2v6/audit\_c4.py}) is $< 0.4067970$. The right side is enclosed as $0.98665909713\ldots$.
\item Lemma~\ref{lem:C3GRHnum}: \texttt{third/t2\_check/c3\_grh.py} (mpmath \texttt{iv}) gives $< 0.147535$ and $< 0.139061$ against right
 sides enclosed above $0.2281295$ and $0.2171299$; the independent audit's Arb bounds (\texttt{audit/t2v6/audit\_grh9.py}) are
 $< 0.1475264$ and $< 0.1390529$.
\end{itemize}

\subsection*{The integrals $J_m$}
For Lemmas~\ref{lem:S4num}, \ref{lem:GRHsievenum} and the grids of Theorem~\ref{thm:B}, $J_m(w)$ is bounded above by cell sums: the region
$\{u_i \ge 1, \sum u_i \le w\}$ is covered by the cubes $\prod [1 + n_i/H_c, 1 + (n_i+1)/H_c)$ whose lower corners lie in the region, and
each cube carries the weight $\prod 1/(H_c + n_i)$ or $\prod \log(1 + 1/(H_c + n_i))$, which bounds $\int du/u$ over the cell. So
$J_m(w) \le \sum_{n \le (w - m)H_c} D_m(n)$, where $D_m$ is the $m$-fold discrete convolution of the cell weights. The convolutions are
direct (not by FFT), in binary64 with all terms nonnegative; every weight is multiplied by $1 + 10^{-15}$ (the logarithmic weights by
$1 + 10^{-14}$) and every convolution and extracted sum by $1 + 10^{-10}$, which exceeds the accumulated rounding error
($< 3.38 \cdot 10^{-12}$ relative for the lengths used). The assembly of Lemma~\ref{lem:R3} and of the conditions is in interval or ball
arithmetic. In every application of Lemma~\ref{lem:Fp} the certificates bound $N_j(w)$ by $\sum_{i=0}^{j}\binom{j}{i}\eps^i J_{j-i}(w)$, with
$\eps$ exactly as stated there. This is at least the closed form of Lemma~\ref{lem:Fp}, because $J_{j-i}(w-i) \le J_{j-i}(w)$, and the same
holds for the cell sums. So the values quoted below are upper bounds for the left sides as printed. The independent audit checked the cell bounds,
every inflated weight and every extracted sum used against Arb enclosures (\texttt{audit/t2v6/check\_float\_cells.py}), and recomputed the
convolutions wholly in 256-bit Arb for Lemma~\ref{lem:S4num} (both cell choices), Lemma~\ref{lem:GRHsievenum} and all $950$ intervals of
Theorem~\ref{thm:B}; every condition holds there as well.

\subsection*{The Johnston--Starichkova range}
Lemma~\ref{lem:S4num}: at $H_c = 400$ and $\delta = 0.01$, the certificates also use the common remainder
$(L_0^2 e^{-\delta L_0} + 5 L_3^2 e^{-L_0})/(c\,U_{\rm low})$ on both pieces. It dominates the remainder in \eqref{eq:S4}, since
$t^2 e^{-\delta t}$ decreases for $t \ge 2/\delta$. In addition, \texttt{third/t2\_check/cert\_S3P\_K18.py} replaces $0.6601618158$ in
$U_{\rm low}$ by the smaller certified lower bound $0.66016175369\ldots$ of the twin-prime constant
(\texttt{third/js\_cert/tp\_bound.py}). Each of these changes can only increase the left side. With them,
\texttt{third/t2\_check/cert\_S3P\_K18.py} (logarithmic cells; the certificate of record) gives $1.339973\ldots \cdot 10^{-6}$ and
$0.448050660\ldots$ on the two pieces, and \texttt{second/cert\_S4.py} (rational cells, $U_{\rm low}$ as defined) gives
$1.339973\ldots \cdot 10^{-6}$ and $0.4540790\ldots$. The independent audit's bounds (\texttt{audit/t2v6/audit\_s4.py}), made in the same
way with the twin-prime bound $0.6601617536$, are $< 0.448050660231$ and $< 0.454079084257$. The coefficients $9.8036$ and $26.218$ are read
from the certificate of (I4$'$), Appendix~\ref{app:js}, which has two interval implementations: one in mpmath \texttt{iv}
(\texttt{third/js\_cert/js\_cert.py}, with its twin-prime constant \texttt{third/js\_cert/tp\_bound.py}) and one in
256-bit Arb (\texttt{audit/audit\_js.py}).

\subsection*{The interval grids for Theorem~\ref{thm:B}}
The range $350 \le \log N \le 51046$ is covered by $501$ intervals and $51046 \le \log N \le 159011874857757$ by $449$ intervals, generated
with successive endpoint ratios $1.01$ and $1.05$, the last interval of each grid shortened to reach its endpoint. On each interval the
parameters $(M, \lambda)$ are chosen by a floating-point search, and then every hypothesis of Lemma~\ref{lem:T} (integrality, $L_1 \ge 300$,
$\lambda \le 100$, $H \ge 25$, $z_1 \ge 10372$ as an interval comparison, $s_1 > 2$ and $s_1, s_2 \le 4$ ($s_1 \le s_2$ because $s$
increases with $L$), positivity of $F(s_1)$ and of the $V$ factors, and $z_1 \ge 8 \cdot 10^9$ on the second grid) and the final inequality
are checked in interval or ball arithmetic, with $\delta = 0.05$ and rational cells with $H_c = 200$; the selected $(M, \lambda)$ of every
interval are in the supplement's parameter files. The certificates are \texttt{second/cert\_qrhB3.py} and \texttt{second/cert\_qrhB3p.py}.
Both call the author's \texttt{margin\_iv} from \texttt{author/cert\_qrh\_range.py} with its own $K$ set to $M - 1$, so that it returns
$\mathrm{main} - \mathrm{rem} - e^{-L_1}$; \texttt{margin\_iv} applies the gates on $M$, $L_1$, $\lambda$, $H$, $s_1$, $s_2$, $F(s_1)$ and
the $V$ factors, but of the gates on $z_1$ it checks only $z_1 \ge 694$ and $z_1^{M-1} \ge 10372$. The gate $z_1 \ge 10372$ is applied by
the two scripts themselves, which search only over $\log z_1 \ge \log 10372 + 0.01$ and assert $z_1 > 10372$ in ball arithmetic on every
interval, and the gate $z_1 \ge 8 \cdot 10^9$ by \texttt{second/cert\_qrhB3p.py}, also in ball arithmetic. Over the two grids, $z_1 > 10493$
(with $M \le 48$) and $\log z_1 \ge 1595.1875$ (with $M \le 32$). No interval fails; the least relative margins
$(\mathrm{main} - \mathrm{rem} - e^{-L_1} - \mathrm{bad})/\mathrm{main}$ (with $\mathrm{bad}_P$ on the second grid) are $0.9030\ldots$ and
$0.4237\ldots$, and the independent audit's Arb calculation of the second (\texttt{audit/t2v6/audit\_v6.py}, all 950 intervals) is
$> 0.42371750647$. The constant $c_E$ (\texttt{cE\_i} in \texttt{author/cert\_qrh\_range.py}) is the term-by-term majorant of
Proposition~\ref{prop:psi} on the rectangle $t \in [e^{L_1/2}, e^{L_2}]$, $q \le H(L_2)$: the terms decreasing in $L$ are evaluated at $L_1$
and the increasing ones (the $Z_1$ term, with $T' \le 1 + e^{L_2/8}$ and $q = H(L_2)$) at $L_2$.

\subsection*{The sieve range under GRH}
Lemma~\ref{lem:GRHsievenum}: \texttt{third/t2\_check/cert\_grh\_sieve.py} covers $[73, 7.7 \cdot 10^6]$ by $238$ intervals with endpoint ratio
$1.05$, takes $H_c = 400$, chooses $\delta$ among $0.30$, $0.25$, $0.20$, $0.15$, $0.10$, $0.05$ and $0.01$ with $\delta L_1 \ge 2$ on each interval, and
bounds $S(z_1)$ for $z_1 < 10372$ by the values at primes of $\sum_{p \le x} 1/p - \log\log x - B$, which decreases between consecutive
primes, together with $D(\log x)$ above $10372$. The largest left side is $0.0002645896\ldots$, on the last interval. The independent
audit's Arb grid (\texttt{audit/t2v6/audit\_grh9.py}; exact rational endpoints, $H_c = 400$) gives every row below $0.000265$.

\subsection*{What limits the method}
For Theorem~\ref{thm:A} the binding steps at $K = 17$ are Lemma~\ref{lem:C3} at $e^{1800}$, Lemma~\ref{lem:C4} (upper bound about $1.07$)
and the second piece of Lemma~\ref{lem:S4} (about $14.3$). For Theorem~\ref{thm:B}, $K = 13$ would need the sieve range to start lower than
$e^{350}$, which needs Lemma~\ref{lem:T} with $z_1 < 10372$ (possible through the general form of Lemma~\ref{lem:Fp}, with
$\eps = S(z_1) + D(\log z_1) + 1/z_1$) and Lemma~\ref{lem:Q} below $L = 300$, and the top range, where $13$ is not established on $103$ of
$449$ intervals at the parameters found by the search. Under GRH the binding step for $K = 8$ is the middle range at $e^{73}$.\par}