I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which there are infinitely many prime numbers $p_k > p_n$ which have $g_k = g_n$? Or, is it something else? If I do have this correctly, is p_n equal to the first prime of gap $N$?

I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which there are infinitely many prime numbers $p_k > p_n$ which have $g_k = g_n$? Or, is it something else? If I do have this correctly, is p_n equal to the first prime of gap $N$? Does this mean that for any gap greater than g_n there is no "last one" only "last know use"?