Let $A\subset \mathbb{Z}$ be such that $\exists\ c\in\mathbb{Z}$ such that $\forall\ n\geq c: \exists\ a,a'\in A$ with $a+a' = n.$ In other words, $A$ satisfies sort-of Goldbach conjecture, but for all integers $\geq$ some integer $c,$ rather than all even integers.
Proposition: Not every $n\geq c$ can be written as the sum of two members of $A$ in a unique way, that is;
$$\exists\ d\geq c,\quad \exists\ a_1,\ {a_1}',\ a_2,\ {a_2}'\in A\ \text{ such that } a_1 + {a_1}' = d = a_2 + {a_2}',\ \text{ where } \{ a_1, {a_1}' \} \neq \{ a_2, {a_2}' \}. $$
I tried to find a sequence that makes the conjecture fail, but I failed in doing this, so I suspect the conjecture is true but I don't see why it is true. Any ideas for how to tackle this sort of problem?
Maybe there is a counter-example with the sequence defined as: $a_1=1$ and $a_{n+1}$ is the least positive integer so that $ a_{n+1} - a_j \neq a_k - a_i$ for all $j,k,i\leq n.$ This sequence is: $1,2,4,8,13,21,31,45,60,76,97,119,144,\ldots.$ I'm not sure what the growth rate of this sequence is, or exactly how knowing the growth rate of this sequence helps answer the question, but I think this sequence is related to the question, because if we can show that a $c$ as in the question exists for this sequence, then since this sequence certainly satisfies the desired property of uniqueness of the pair $(a_1, {a_1}'),$ this sequence would be a counter-example.