As well, that Basically I've managed to convince myself that there isn't a sensible mathematical definition of "real GDP" or the "price level" (also known as "GDP deflator")
is not a great intellectual achievment. GDP isn't well defined in infinitesimal small time intervals, or at least it will wiggle extremely around - e.g. on the day scale, GDP at night is much lower than in the morning. Therefore usually the smalles time unit in which GDP is published are months. This doesn't mean that you can't reasonably parametrise real GDP growth with analytical functions to make an economic theory.

If you have two collections of prices and volumes that are exactly the same 10 years apart, you have the same (nominal gdp change) but if the evolution of the economy took two sufficiently disparate paths over those 10 years you could have two very different (real gdp change) and (inflation).

Most economists teach a theoretical framework that has been shown to be fundamentally useless. -- James K. Galbraith