Goldbach Midpoint Transitions and Twin–Cousin Rescue: A Computational Study to One Billion

We study Goldbach representations about a moving pivot, distinguishing the loss of a
direct continuation from the loss of all representations. For pivots congruent to 1 modulo
3, direct extinction has an exact primality criterion. Dirichlet’s theorem then supplies
infinitely many such pivots. We conjecture that each extinction is rescued by a twin pair
and a prime pair at distance four, and show that this conjecture implies binary Goldbach
and infinitely many twin primes. An equivalent four-prime convolution leads to a stronger
residue-specific conjecture. An odd-only Eratosthenes sieve and a complete fallback search
test every pivot through 109: among 284,220,751 residue-specific targets, no failure occurs,
and no extinction pivot occurs in the other residue classes. A stricter right-hand orientation
already fails at 277. We give exact algorithms, finite counts, code, and the logical limits of
these observations; neither Goldbach nor the twin-prime conjecture is proved.