Goldbach problems for Polignac families: twin, cousin and sexy summands

For an even integer $d$, let $\PP_d$ be the set of primes $p$ such that $p-d$ or $p+d$ is also prime; thus $\PP_2$, $\PP_4$ and $\PP_6$ are the twin, cousin and sexy primes. We study the binary problem of representing an even integer $n$ as $p+q$ with $p\in\PP_a$ and $q\in\PP_b$, which for $a=b=2$ is the twin-prime Goldbach problem of Zwillinger and Dubner. By an exact, integer-only witness search we determine the exceptional sets for all six pairings of twin, cousin and sexy primes up to $10^8$, and for all $120$ pairs of gaps $a,b\le 30$ up to $10^7$. Every even $n\ge 6$ is the sum of a twin prime and a cousin prime in this range, whereas twin$+$twin and cousin$+$cousin have $34$ and $36$ exceptions, the largest being $4208$ and $2952$. We prove a structure theorem modulo~$3$ that explains this contrast: when $3\nmid a$, the representations of $s$, $s+a$, $s+2a$ are locked together in the exact ratio $1:2:1$, so equal-gap exceptions occur in arithmetic triples, while for $a\ne b$ the class $n\equiv 0\pmod 3$ has two independent routes. As a corollary, the twin-prime Goldbach problem for multiples of $6$ is equivalent to a Goldbach problem for the index set $\{k:6k\pm1\text{ both prime}\}$. We formulate a Hardy--Littlewood type conjecture for $r_{a,b}(n)$, prove that its singular series is bounded below by an explicit absolute constant $c_0\approx 4.1512$ (halved when $a=b$), and show that the prediction, refined by inclusion--exclusion, matches the data to within $1\%$ near $10^7$ and reproduces the number of observed exceptions. Any proof of the existence part for a single pair $(a,b)$ would establish Polignac's conjecture for gap $a$ or gap $b$.