Why Hijri Date Conversion Breaks Naive Math Formulas

If you search "convert Gregorian to Hijri" you'll find dozens of one-line formulas floating around StackOverflow, most of them some variant of hijriYear = (gregorianYear - 622) * 33 / 32. They compile, they run, and they're wrong more often than they're right. Here's why, and what a correct implemen