An Exact Identity for the Twin Prime Counting Function Derived from Minimal Goldbach Partitions

Assuming the Goldbach conjecture, we establish a quantitative connection between the twin prime counting function $\pi_2(x)$ and the distribution of minimal primes in Goldbach decompositions of even integers. For each even integer $k \geq 4$, define $p(k)$ as the smallest prime $p$ such that $k - p$ is also prime. We introduce a generating polynomial whose coefficients $c_m(n)$ count the number of even $k \leq n$ for which $p(k) = m$, where $m$ is prime. These coefficients are characterized in terms of avoided prime constellations. This framework yields an exact identity: for even $n \geq 8$,

$\pi_2(n-5) = \pi(n-3) + \pi(n-5) + H(n) - \frac{n}{2},$

where $H(n) = \sum_{\substack{m \geq 7 \\ m \text{ prime}}} c_m(n)$.