\section{The range of the Johnston--Starichkova sieve, and the proof of Theorem~\ref{thm:A}}\label{sec:js}

Put $X_3 := \exp(\exp(32.7))$, $L_0 := 2736.86$, $L_3 := e^{32.7}$ and $z_0 := e^{L_0/40}$.

\begin{lemma}\label{lem:S4}
Let $K \ge 1$ be an integer, and let $\eps$, $\mu_j$ (in either form of Lemma~\ref{lem:R3}) and $A_a$ be evaluated at $z = z_0$, $M = 40$. Put
\[
 \Phi_P := \sum_{k = K+1}^{39} \Bigl[ G_P\Bigl(1 + \frac{480}{840 \log z_0}\Bigr) \mu_{k-1} + \sum_{a=2}^{k} A_a(z_0)\, \mu_{k-a} \Bigr], \qquad G_P = \frac{9}{32}.
\]
For each piece $(L_a, L_b, c) \in \{ (L_0, 3000, 9.8036),\ (3000, L_3, 26.218) \}$ choose $0 < \delta < 1$ with $\delta L_a \ge 2$, and suppose
\begin{equation}\label{eq:S4}
 \frac{L_b\, \Phi_P}{(1 - \delta)\, c\, U_{\rm low}} + \frac{L_a^2 e^{-\delta L_a} + 5 L_b^2 e^{-L_a}}{c\, U_{\rm low}} < 1 .
\end{equation}
Then every even $N$ with $X_2' \le N \le X_3$ is $p + \eta$, where $p$ is prime, $\eta \ge 2$ has no prime factor below $N^{1/40}$, and
$\Omega(\eta) \le K$.
\end{lemma}

\begin{proof}
Fix $N$ in a piece, $L := \log N \in [L_a, L_b]$ and $z := N^{1/40} \ge z_0 > 8 \cdot 10^9$. By (I4$'$), more than $c\, U_N N/L^2$ primes
$p \le N$ with $p \nmid N$ have $N - p$ $z$-rough; the elements $N - p$ are distinct, and at most one of them equals $1$. A $z$-rough $n < z^{40}$
has $\Omega(n) \le 39$. The bad elements, those with $\Omega(N - p) \ge K + 1$, are: at most $N^{1-\delta}$ elements $n \le N^{1-\delta}$; at most
four elements $N - q$ with $q \in \{3, 5, 7, 11\}$; and the elements of the sets $\mathcal B_k$ of Lemma~\ref{lem:P}, $K < k \le 39$. Since
$\eps$, $\mu_j$, $A_a(z)$ and $1/\log z$ decrease as $z$ grows, Lemma~\ref{lem:P} at $z$ is bounded by its value at $z_0$, so
$\sum_k \#\mathcal B_k \le \Pi(N)\, N\Phi_P/((1 - \delta)L)$, after multiplying the terms $a \ge 2$ by $\Pi(N) \ge 1$. As
$U_N \ge U_{\rm low}\Pi(N)$ (Remark~\ref{rem:P}), the number of bad elements plus one is less than the number of good ones as soon as
\[
 \frac{L\, \Phi_P}{(1 - \delta)\, c\, U_{\rm low}} + \frac{L^2 N^{-\delta} + 5 L^2/N}{c\, U_{\rm low}} < 1 .
\]
Here $L\Phi_P \le L_b \Phi_P$; $L^2 e^{-\delta L} \le L_a^2 e^{-\delta L_a}$, since $t^2 e^{-\delta t}$ decreases for $t \ge 2/\delta$; and
$L^2/N \le L_b^2 e^{-L_a}$. So \eqref{eq:S4} gives a good element $\eta = N - p \ne 1$ with $\Omega(\eta) \le K$, and $\eta \ge z \ge 2$.
\end{proof}

\begin{lemma}\label{lem:S4num}
With $K = 18$ and $\delta = 0.01$, the left side of \eqref{eq:S4} is $< 1.34 \cdot 10^{-6}$ on the first piece and $< 0.4481$ on the second.
\end{lemma}
\begin{proof}
Interval computation, \S\ref{sec:numerics}, using the crude form of $\mu_j$ ($1/(z_0 - 1) \le \eps$ holds at $z_0 = e^{68.4\ldots}$) and the conservative replacements described there.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:A}]
For $4 \le N \le 4 \cdot 10^{18}$ use (I1). For $4 \cdot 10^{18} \le N \le e^{1800}$ use Lemma~\ref{lem:C3} with $\delta = 1/5$ and $K = 18$
(Lemma~\ref{lem:C3num}). For $e^{1800} \le N \le X_2'$ use Lemma~\ref{lem:C4} with Lemma~\ref{lem:C4num} ($K = 18$). For $X_2' \le N \le X_3$ use
Lemma~\ref{lem:S4} with Lemma~\ref{lem:S4num} ($K = 18$); the two pieces meet at $\log N = 3000$. For $N > X_3$ use (I5). \qedhere
\end{proof}

\begin{remark}
The stated parameters do not give $K = 17$: at $e^{1800}$, the left side of \eqref{eq:C3} with $\delta = 1/5$ exceeds $0.5646$
(Lemma~\ref{lem:C3num}); at $K = 17$ the computed upper bound in Lemma~\ref{lem:C4} is about $1.07$, and on the second piece of
Lemma~\ref{lem:S4} about $14.3$, so neither is established. These calculations do not prove that every choice of parameters fails, and no
global optimisation is claimed. Lemma~\ref{lem:C3} alone cannot give $K = 18$ up to $X = X_2'$ for any $\delta$: there the sum over $k$ in
\eqref{eq:C3}, before the factor $(1 - \delta)^{-1}$, already exceeds $0.4961 > G_0$ (from the enclosure of the left side at $\delta = 1/5$,
which is $> 0.628287$); this is why Lemma~\ref{lem:C4} is needed.
\end{remark}