Large range of the K = 27 Schnirelmann density bound: at least n/25 even numbers up to n are sums of two odd primes for log n ≥ 300

For every integer $n$ with $\log n \ge 300$, at least $n/25$ of the integers $0 \le s \le n$ are a sum $s = p + q$ of two odd primes:

where $r(s)$ (Schnir.r) counts ordered pairs of odd primes $(p, q)$ with $p + q = s$.

This is the large-range part of the explicit Schnirelmann-density argument for $K = 27$ (every integer $> 1$ is a sum of at most 27 primes). It follows Riesel and Vaughan (1983), §§7–8: split $(h, 1002h]$, $h = \lfloor n/2004 \rfloor$, into $1000$ overlapping intervals $(kh, kh + 2h]$, weight each pair of primes from one interval by the truncated singular-series weight $w(m) = \prod_{p \mid m,\ 3 \le p \le E}(p-2)/(p-1)$, bound the weighted pair count from below with the Montgomery–Vaughan large sieve for primitive characters, and from above by $\#\{s \le n : r(s) > 0\}$ times the Siebert-type bound for one sum. Chebyshev's elementary bound $\psi(x) \ge 0.9212x - 5\log x$ replaces the Rosser–Schoenfeld prime number estimates, so no information on zeta zeros is used.