Why does the "q + r + s" sum give the year's doomsday drift?

Each common year moves the doomsday +1 and each leap year +2, so the drift is YY + (leap years) = YY + YY÷4 — and the 12-split is just an easy way to compute that mod 7.

A normal year is 365 = 52·7 + 1 days, so the calendar shifts one weekday; a leap year shifts two. Over YY years that's YY + ⌊YY/4⌋ (mod 7). Writing YY = 12q + r, this equals 15q + r + ⌊r/4⌋, and since 15 ≡ 1 (mod 7), it reduces to q + r + s. So the trick is exact number theory, just arranged for small mental steps.