Question about interest calculation (mortgage)

I don't seem to understand something regarding how interest is paid on a mortgage. Say the loan is for $100,000 at a 5% rate for 10 years, paid monthly.

I would think that on the first month, the interest I have to pay $100,000 × (0.05 ÷ 12) = $416.67. However the mortgage calculator says that the first payment is actually $412.39. While it's not a huge difference, it's a difference nonetheless and I can't really figure out where it comes from.

My intuition is that it's somehow related to the fact that interest is compounded daily, but when I take r = 0.05 ÷ 365 and N = 365 × 10 payments (keeping leap years in mind for later), and calculate the first 30 days, I get $409.70, and the first 31 days give $423.32. I guess that the "actual" number is some kind of weighted average since the calculator doesn't ask at which month your loan starts.

So where is this $412.39 coming from? In reality when paying a mortgage, do you see the interest fluctuating as it decreases, depending on the number of days every month?