The Goldbach Survivor-Gap Problem for Two-Cloud Primorial Masks - A Finite Jacobsthal-Type Reduction of Binary Goldbach

This paper specializes the finite survivor-envelope framework to the binary Goldbach setting. For an even integer N, the Goldbach condition is encoded by a two-cloud survivor mask: a residue class x survives the primes up to z when neither x nor N minus x is divisible by any prime up to z. At the complete-sieve level z = floor (sqrt (N) ), any surviving value x in the interior interval between sqrt (N) and N - sqrt (N) gives a Goldbach representation by two primes. The paper develops the resulting finite survivor-gap problem, including the exact CRT product formula, the complete-sieve prime-interior criterion, localized survivor-count identities, the two-cloud recurrence on primorial tori, and autocorrelation-based routes toward windowed distribution estimates. It also compares the resulting obstruction problem with classical Jacobsthal-type gap questions. Internal version v04 adds computational validation from Goldbach Lab v03 up to 10¹0. The complete-sieve interior condition is verified for every even 4 < N <= 10¹0, while N = 4 is the expected boundary-only case corresponding to 2 + 2. The computation is included as validation of the finite framework and of the complete-sieve interior phenomenon over the tested range. The paper does not prove the Goldbach conjecture. Its purpose is to isolate the exact finite obstruction: proving the required survivor-window nonemptiness, or an equivalent two-cloud gap estimate, would imply the corresponding complete-sieve interior form of Goldbach.