\section{Notation and inputs}\label{sec:inputs}

We use $p$ for primes and $q$ for a modulus, except where $q$ is explicitly a prime divisor. All unqualified logarithms are natural,
and $\log_2 x = (\log x)/(\log 2)$. Further, $\Omega(n)$ and $\omega(n)$ denote the number of prime factors of $n$ with and without multiplicity, $P^+(n)$
and $P^-(n)$ the largest and smallest prime factor of $n > 1$, $\Lambda$ is von Mangoldt's function, and $L = \log N$. We write
$\theta(x) = \sum_{p \le x} \log p$, $\theta(x; q, a)$ for the same sum restricted to $p \equiv a \pmod q$,
$\pi(x; q, a)$ for the number of primes $p \le x$ with $p \equiv a \pmod q$, $\li(x) = \int_0^x dt/\log t$
(principal value), $B = 0.2614972128\ldots$ for Mertens' constant and $c_\theta := 1 + 1/36260$. For even $N$ let
\begin{gather*}
 \cA = \{ N - p : p \le N,\ p \nmid N \},\\
 U_N = 2 e^{-\gamma} \prod_{p > 2} \Bigl(1 - \frac{1}{(p-1)^2}\Bigr) \prod_{\substack{p > 2\\ p \mid N}} \frac{p - 1}{p - 2}
 \ \ge\ U_{\rm low} := 2 e^{-\gamma} \cdot 0.6601618158 .
\end{gather*}
For $z > 1$ let $S(\cA, z)$ be the number of elements of $\cA$ with no prime factor below $z$. A prime dividing $N$ cannot divide $N - p$ when
$p \nmid N$, so these elements are $z$-rough. We use the following published inputs and the derived bound (I4$'$).

\begin{itemize}
\item[(I1)] \emph{Goldbach's conjecture holds for all even $4 \le N \le 4 \cdot 10^{18}$} \cite{OSHP}.
\item[(I2)] \cite[Theorem 1.2 with (1.10)--(1.11)]{BMOR18}: for $3 \le q \le 10^4$, $(a, q) = 1$ and $x \ge 8 \cdot 10^9$,
 $\bigl| \theta(x; q, a) - x/\varphi(q) \bigr| < x / (840 \log x)$.
\item[(I3)] Explicit Mertens and Chebyshev bounds of Dusart \cite[Theorem 6.10 and Proposition 5.1]{Dusart}.
 With the function $d(x) := 1/(10 \log^2 x) + 4/(15 \log^3 x)$ one has
 \[
 -d(x) \le \sum_{p \le x} \frac{1}{p} - \log\log x - B \quad (x > 1), \qquad
 \sum_{p \le x} \frac{1}{p} - \log\log x - B \le d(x) \quad (x \ge 10372),
 \]
 and $\theta(x) < c_\theta x$ for $x > 0$.
 We write $D(\ell) := 1/(10\ell^2) + 4/(15\ell^3)$, so that $d(x) = D(\log x)$.
\item[(I4)] Johnston and Starichkova \cite[\S4, proof of Theorem 1.3]{JS}: with their parameters
 $Y_0 = 7.9$, $\alpha_1 = 7$, $\alpha_2 = 2$, $C = 3.98$, $u_0 = 400$, $\delta = 1.3$, $\alpha = 1/4$ and $M = 40$, their Theorem 4.2 gives
 $S(\cA, N^{1/40}) > 5\, U_N N/\log^2 N$ for the even $N$ to which it applies. That theorem assumes $\log\log x_1(X_2) \ge Y_0$ with
 $x_1(N) = N/\log^5 N$. Applied with $X_2 = N$, this holds exactly when $\log N - 5\log\log N \ge e^{7.9}$, that is from
 $\log N = 2736.8551\ldots$ on; the threshold stated with these parameters in the proof of \cite[Theorem 1.3]{JS} (arXiv version v3, 11 April 2025) is
 $X_2 = \exp(e^{7.9})$, which is smaller; we have not consulted the journal version, which may differ at this point. We apply \cite[Theorem 4.2]{JS} only for $N \ge X_2' := \exp(2736.86)$, where all its
 hypotheses hold (Appendix~\ref{app:js}), and only through (I4$'$); the coefficient $5$ above is not used. Commented-out lines of the
 arXiv v3 source record a different admissible tuple, $(Y_0, \alpha_1, \alpha_2, C) = (7.8, 7, 2, 431.57)$, whose sieve output is not
 evaluated here; no error in \cite[Theorem 1.3]{JS} is asserted.
\item[(I4$'$)] For every even $N \ge X_2' = \exp(2736.86)$, $S(\cA, N^{1/40}) > c(L)\, U_N N/\log^2 N$, where $c(L) = 9.8036$ for
 $2736.86 \le L \le 3000$ and $c(L) = 26.218$ for $L \ge 3000$. This is a derived consequence of Johnston--Starichkova,
 \cite[Theorem 4.2]{JS}, applied separately with $X_2 = N$, using the parameters of (I4) and the corrected definition of $a(X_2)$ with
 denominator $\log^\delta N$. This definition is the one supported by the proof of \cite{BJS} imported in case (b); the exponent
 $1 + \delta$ printed in \cite{JS} (arXiv v3), \S2.1, is a misprint. Appendix~\ref{app:js} proves the two coefficients, using interval
 arithmetic on $2736.86 \le \log N \le 3000$ and decreasing majorants for $\log N \ge 3000$.
\item[(I5)] \cite[Corollary 4]{BJS}: every even $N > \exp(\exp(32.7))$ is the sum of a prime and a squarefree number $\eta > 1$ with at
 most two prime factors (their proof counts representations with $\eta > 1$: \cite[proof of Corollary 4]{BJS}).
\end{itemize}

For Theorem~\ref{thm:B} we use in addition the explicit linear sieve of Bordignon, Johnston and Starichkova \cite[Theorem~6, Table~1]{BJS},
in the form of \cite[Theorem~2.1, Table~2.1]{Johnston}, the lemmas of \cite[Chapter 4 and Appendix A]{Johnston}, and an
explicit bound for $\psi(t, \chi)$ under the $7/8$ hypothesis (\S\ref{sec:qrh}). For Theorem~\ref{cor:GRH} we use, under GRH:
\begin{itemize}
\item[(I6)] \cite[Lemma~3]{BSGRH}, from \cite[Theorem 1]{EHP}: if $3 \le d \le \sqrt{X_2}$ and $X_2 \ge 4 \cdot 10^{18}$, then
 for all real $x \ge X_2$ and all integers $a$ with $(a, d) = 1$, $\bigl| \pi(x; d, a) - \li(x)/\varphi(d) \bigr| \le 0.640 \sqrt x \log x$.
\item[(I7)] \cite[Proposition 4.18]{Johnston}: for even $N \ge e^{73}$ and $z = N^{1/11}$, the set of elements of $\cA$ free of primes below $z$
 has more than $0.146\, U_N N/\log^2 N$ elements.\footnote{In \cite[(4.35)]{Johnston} the factor $M|\cA|U_N/(\log N)^2$ should read
 $M|\cA|U_N/\log N$, as (4.37) in its proof shows; with this factor and $|\cA| > N/\log N$, (4.35) has the order $U_N N/\log^2 N$ of
 \cite[(4.39)]{Johnston}, which is the form used here.}
\item[(I8)] \cite[Corollary~1]{BSGRH}: every even $N > \exp(\exp(15.85))$ is the sum of a prime and a squarefree number
 $\eta > 1$ with at most two prime factors.
\end{itemize}