Sums of two odd primes have density at least 1/25 for $23 \le \log n \le 300$

Let $r(s)$ be the number of ordered pairs $(p,q)$ of odd primes with $p+q=s$ (platform definition Schnir.r). For every natural number $n$ with $23 \le \log n \le 300$,

where $s$ ranges over $\{0,1,\dots,n\}$.

This is the middle range of an explicit Schnirelmann argument for the bound $27$ on the number of primes needed to write an odd number. It follows Riesel and Vaughan's small-shift argument: with $Q$ the first $150$ odd primes and $R(s)=\#\{a\in Q : s-a \text{ an odd prime}\}$, Cauchy–Schwarz gives $\#\{s\le n: R(s)>0\}\ge (\sum R)^2/\sum R^2$. The first moment uses Chebyshev's bound $\psi(x)\ge 0.9212x-5\log x+5$; the second moment is bounded by Siebert's prime-pair bound $8C\,K(d)\,x/\log^2x$ with $K(d)=\prod_{p\mid d,\,p>2}\frac{p-1}{p-2}$ (platform theorem TaoFivePrimes.siebert_prime_pair_bound).

Formalization Note $\log$ is the natural logarithm Real.log of $n$ cast to $\mathbb{R}$; the count is the cardinality of the filter of Finset.range (n + 1).