What is a "period" of the periodic table?
I agree that the "periods" in the periodic table are not mathematically regular.  The simplest definition is that a period begins when a new s-subshell starts to fill.  Recall that in the s, p, d, f subshells there are 2, 8, 10, and 14 electrons, respectively, so the periods have to get bigger over time and thus cannot be regular.  As to the lack of predictability of stability, that may be true, but the periodic table is pretty good at predicting the chemistry of elements by collecting together similar elements into groups (the columns).  This similarity is mostly expressed by the s and p subshells, which is where a lot of the reactivity occurs.