I think we can safely assume save file. Especially if it is nearly as large as he is making us think.
Actually, if the plot is entirely linear it could be very long and still represented as one single integer.
From what we've seen, it's not entirely linear, but is perhaps a collection of several distinct "stage beaten/not beaten" flags, or perhaps a few linear plots. Still, he could optimize the save files for really, really long games of this type to obscene extremes, like a few bytes per file.
Therefore, unless they're doing something crazy that I'm just not seeing, he should really have files. There's not much information to save and it would be very convenient. At the very least, each profile should have a subspace emissary file attached.