Structural and Algebraic Obstructions for Sums of Two Primes

We study two types of restrictions on the set $U_1=\{p+q:p,q\text{ prime}\}$ of sums of two primes. First, for every finite nonempty set $C$ of positive integers whose elements have the same parity, we prove that there are arbitrarily large $n\in U_1$ such that $n-c\notin U_1$ for every $c\in C$. We derive related results for scaled sets and for subsets defined by the parity of the prime summands. Second, we establish impossibility results for several specified polynomial, fractional-linear, and affine-linear rules that might otherwise be used to generate Goldbach representations for all sufficiently large indices. The two parts concern structural and algebraic restrictions on Goldbach representations; neither implies a disproof of Goldbach's conjecture. A possible extension to arbitrary polynomial relations is formulated as an open question.