\section{Introduction}

In 1948 R\'enyi \cite{Renyi} proved that there is a constant $K$ such that every even integer $N \ge 4$ is the sum of a prime and a number
with at most $K$ prime factors. For large $N$ Chen \cite{Chen} showed that one may take $K = 2$, but Chen's and R\'enyi's arguments are
ineffective. An explicit version of Chen's theorem, valid for $N > \exp(\exp(32.7))$, was obtained by Bordignon, Johnston and Starichkova
\cite{BJS}, and Johnston and Starichkova \cite[Theorem~1.3]{JS} then stated that every even $N \ge 4$ is $p + \eta$ with
$\Omega(\eta) \le 395$, where $\Omega$ counts prime factors with multiplicity; arXiv versions v1 and v2 of \cite{JS} had stated $369$,
the value quoted in \cite{HJ}. Under the generalised Riemann hypothesis they obtained $31$, and Johnston's thesis
\cite{Johnston} improved this to $16$.

\begin{theorem}\label{thm:A}
Every even integer $N \ge 4$ can be written as $N = p + \eta$, where $p$ is prime and $\eta \ge 2$ satisfies $\Omega(\eta) \le 18$.
\end{theorem}

\begin{theorem}\label{thm:B}
Assume that no Dirichlet $L$-function has a zero in $\operatorname{Re} s > 7/8$. Then every even integer $N \ge 4$ can be written as
$N = p + \eta$ with $p$ prime, $\eta \ge 2$ and $\Omega(\eta) \le 14$.
\end{theorem}

\begin{theorem}\label{cor:GRH}
Assume the generalised Riemann hypothesis for Dirichlet $L$-functions: every nontrivial zero of every Dirichlet $L$-function,
$\zeta(s)$ included, has real part $1/2$. Then every even $N \ge 4$ is $p + \eta$ with $p$ prime, $\eta \ge 2$ and $\Omega(\eta) \le 9$.
\end{theorem}

\subsection*{Where the old bound came from}
In arXiv v3 of \cite{JS}, the proof of Theorem 1.3 combines a large-range sieve producing at most $39$ prime factors with the
small-range result of Hathi and Johnston \cite{HJ} ($N = p + \eta$ with $\eta$ squarefree and coprime to $2 \cdot 3 \cdots 13$). At its
stated threshold $X_2 = \exp(e^{7.9})$, the latter argument gives $395$ by counting consecutive primes starting at $17$. Here we apply
\cite[Theorem 4.2]{JS} only for $N \ge X_2' = \exp(2736.86)$, where Appendix~\ref{app:js} verifies all its hypotheses. Our counting
bounds improve both ranges.

\subsection*{The method}
For $N$ below the sieve threshold we count primes $p \le N - 2$ with $N - p$ coprime to $1155$, using a single modulus of the explicit prime
number theorem for progressions of Bennett, Martin, O'Bryant and Rechnitzer \cite{BMOR18}. Against this we bound the number of integers
$n \le N$ coprime to $2310$ with $\Omega(n) > K$, by a Hardy--Ramanujan-type count \cite{HR}: Chebyshev's identity
$\log n = \sum_{d \mid n} \Lambda(d) = \sum_{p^a \parallel n} a \log p$, combined with a bound for $\theta$ as in Hardy and Wright
\cite[\S22.18]{HW}, followed by the exact coefficients of $\exp(zS'(X) + C_{11}(z))$, which majorises the finite Euler product
$\prod_{13 \le p \le N} (1 - z/p)^{-1}$ coefficientwise for $N \le X$, in place of Rankin's trick \cite{Rankin} (Lemmas~\ref{lem:Rstar}
and~\ref{lem:C3}). For $N \ge e^{1800}$ we apply the weighting to the squarefree part of $n$ only. An integer $n = N - P$ that is to be
excluded inherits the residue class of $P$ modulo $1155$, so in $n = p n'$ the prime $p$ lies in as many reduced classes as the primes we
count, and the class factor cancels (Lemma~\ref{lem:C4}).

Above the threshold we count inside the set sifted by Johnston and Starichkova: their lower bound for the number of $p$ with $N - p$ free of
primes below $N^{1/40}$, with the coefficient re-evaluated in Appendix~\ref{app:js}, is compared with the number of $N^{1/40}$-rough
integers with many prime factors. The same weighting reduces this to reciprocal sums over rough cofactors. In logarithmic units these are
bounded by products of the measure $\eps\,\delta_1 + \mathbf 1_{[1,\infty)}(u)\,du/u$, which keeps the Mertens density of the rough primes
(Lemmas~\ref{lem:Fp} and~\ref{lem:R3}), and the pre-sieve modulo $1155$ lowers the count again (Lemma~\ref{lem:P}).
Theorem~\ref{thm:B} replaces the sieve above the threshold by the explicit linear sieve of Bordignon, Johnston and Starichkova \cite[Theorem~6]{BJS}, in the form of
\cite[Theorem~2.1]{Johnston}, with the remainder controlled
by an explicit form of the prime number theorem for characters under the $7/8$ hypothesis, and uses the same counting inside the sieved set.
Theorem~\ref{cor:GRH} pre-sieves to larger moduli below $e^{73}$, using a prime number theorem for progressions under GRH
\cite{BSGRH}, counts inside Johnston's GRH sieve \cite[Proposition~4.18]{Johnston} up to $\exp(7.7 \cdot 10^6)$, and uses the explicit
Chen theorem under GRH of Bordignon and Starichkova \cite{BSGRH} beyond.

\subsection*{Credits and prior art}
None of the devices is new. Johnston's thesis \cite[\S6.2, after Question 6.2]{Johnston} already suggests that, for $N \le \exp(73)$ under
GRH, the few $\eta = N - p$ with exactly $16$ prime factors could be ``carefully excluded'' from the representation function to reach $15$;
we implement this idea systematically and unconditionally.

The good-count template, a log-weighted representation function fed by \cite{BMOR18}, is that of Hathi and Johnston \cite{HJ}; its
restriction $(N - p, 1155) = 1$ is their coprimality condition $(n - p, k) = 1$ with $k = 1155$, a pre-sieve. Recent explicit work on sums of
two almost primes pre-sieves by $3$ (an arXiv preprint \cite{Campbell}) and by $3$ and $5$ (a deposit, not peer reviewed
\cite{deMorais}); Fan's explicit estimates for rough numbers pre-sieve by $2$, $3$ and $5$ \cite[\S2]{FanRough}. Lemmas~\ref{lem:C4}
and~\ref{lem:P} pre-sieve the count of the integers to be excluded as well; the analogy is with Chen's switching, but the step here is
elementary.

The weighting $\log n = \sum_{d \mid n} \Lambda(d)$ is Chebyshev's identity. Landau's count of integers by their number of prime factors
\cite[vol.~I, \S56]{Landau} brings in $\theta$ through a lemma on sums $\sum_{p \le x} F(p, x)$ and weights each product by its number of
prime factors: it counts the solutions of $pQ \le x$, with $Q$ squarefree with a fixed number of prime factors, in two ways
\cite[vol.~I, p.~208]{Landau}. The logarithmic weight, with $\theta$ as the first case of an induction on the number of prime factors, is
the device of the proof of Landau's theorem in Hardy and Wright \cite[\S22.18]{HW}, which their notes attribute to Wright (1954); the
lemmas of Hardy and Ramanujan \cite{HR} do not use it: they count each integer with $k \ge 2$ distinct prime factors $k - 1$ times, once
for each prime factor other than the largest \cite[(2.23), (3.14)]{HR}, and similarly with multiplicity \cite[(4.23)]{HR}. Pollack \cite[eq.~(3)]{Pollack} states a Hardy--Ramanujan inequality for integers composed entirely of primes from a given
set, rough integers included; it counts $\omega$, and its constants are absolute but not numerical. The power-series identity behind
Lemma~\ref{lem:Rstar}, its recurrence, and the coefficientwise bound for the exponential of a nonnegative series are in Mathar
\cite{Mathar} and Lichtman \cite[Proposition~3.1, (3.3) and Lemma~5.4]{Lichtman}; the closest explicit use in sieve work is Fan's
evaluation of elementary symmetric sums by Newton's identities \cite[\S2]{FanRough}.
The one-dimensional step of Lemma~\ref{lem:Fp} is the explicit partial-summation lemma quoted from Greaves \cite{Greaves} in
\cite[Lemma~20]{BJS} and used in \cite[Lemma~52]{BJS}; the split at $N^{1-\delta}$ is the usual Chebyshev split.
Splitting off the largest prime factor and counting rough cofactors with explicit integrals
appears in the explicit Chen theorem of \cite[Lemma~52]{BJS} (after Nathanson \cite{Nathanson}); explicit Hardy--Ramanujan inequalities that
split off the largest prime are in Bayless, Kinlaw and Klyve \cite{BKK}; Ford \cite{FordHR} combines the largest-prime split with Rankin's
trick for sifted sets; explicit counts of rough products with a fixed number of prime factors are in Loebenberger and N\"usken \cite{LN}.
Fan \cite[Theorem~1.1]{Fan} gives a weighted Hardy--Ramanujan inequality for sifted sets, including factor counts with multiplicity; its
constants are not supplied numerically. The idea of counting the distribution of $\omega$ over shifted primes goes back to Erd\H{o}s
\cite{Erdos35}. What is new here is the specialised explicit form of these bounds and the numerical results. Platt and Trudgian
\cite[\S5]{PT} note that their results on zeros of quadratic Dirichlet $L$-functions would improve explicit results of this type, including
those of \cite{JS}.

\subsection*{Related large-range results}
Recent public drafts give Chen-type results for very large $N$: under the hypothesis of Theorem~\ref{thm:B}, every even
$N \ge \exp(51046)$ is $p + P_2$, and under GRH every even $N \ge \exp(24274)$ \cite{c249}; under GRH and published inputs of Bordignon and
Starichkova \cite{BSGRH}, every even $N \ge \exp(\exp(13.375))$ \cite{Hubbard}. They do not bridge to the range of numerical verification, and our
proofs do not use them. Separately, a recent manuscript \cite{QRH} claims that no Dirichlet $L$-function vanishes in
$\operatorname{Re} s > 7/8$. We have not verified that claim and do not use it: Theorem~\ref{thm:B} is stated under this hypothesis. If the
claim holds, the conclusion of Theorem~\ref{thm:B} holds unconditionally.

\subsection*{Certification}
Every inequality whose truth depends on a numerical evaluation was checked in interval or ball arithmetic. The certificates of record
make one exception: the discrete convolutions that bound the integrals $J_m$ of Lemma~\ref{lem:Fp}, whose terms are nonnegative, are
evaluated in binary64 under an explicit bound on the rounding error; an independent audit recomputed them in ball arithmetic.
\S\ref{sec:numerics} says which implementation certifies which step. The scripts, their parameter files and outputs form the supplement.