I used to work on DNA dyes. Typically when you see a 6 carbon ring with a chain of carbons attached, there is high probability of that molecule interfering with DNA replication.
Basically the mechanism works because the hexagon ring slides between the base pairs and this leads to a lowest energy state due to a phenomenon call pi orbital stacking resulting in the molecule getting stuck there. The carbon chain is mostly valuable in the sense that it distances the rest of molecule from interfering with the stacking process.
Take a look at ethidium bromide or pretty much any other intercalating dna stain and you’ll see similar characteristics. It’s also extremely carcinogenic.
https://en.m.wikipedia.org/wiki/Pi-Stacking_(chemistry)
It’s analogous to getting some cloth stuck in your zipper. Sometimes you can zip and unzip easily enough but sometimes it’ll get stuck. My understanding is that really DNA replication issues tend to be the root cause of some, possibly many cancers but really that’s outside my expertise.
So I would say that it is internally consistent with my limited knowledge of biochemistry that aspartame is carcinogenic.
I would strongly caveat this with saying that these structures occur in pretty high frequency across many forms of plant and animal life. Chemists in my lab used to joke about how potatoes contain 17 or so know carcinogenic compounds so why buy organic. My point is, if you go looking for correlations with cancer in many forms of food, you will find them.
I think for most people, aspartame is not likely to be major risk factor unless you are consuming it in extreme quantities and otherwise live a very healthy life.