Typically the burden of proof is on the one making the claim.
They have some of the best publicly available analysis on these topics. The full details and numbers are hidden behind the institutional accounts which are priced for investors (not something you sign up for personally) but they're generous with what they send out in their newsletter.
If you're not familiar with resources like this I could understand how you'd assume that the providers are hemorrhaging money on inference costs, because that is that story that gets parroted around spaces like Hacker News.
You could ignore all of that, though, and go check OpenRouter to see how much providers are selling high parameter count models. They're not entirely at the level of the SOTA models, but the biggest open weight models are not that far behind in complexity either. They're being sold an order of magnitude cheaper than what you pay for the APIs from the major players. We don't know exactly how big the major models are, but it's unlikely that they're more than 10X more compute intensive from the leaks we do have.
The cost of AI inference has been a heavily analyzed topic. I trust the professional analysts much more than the casual Hacker News commenter claiming they’re losing money per token because they’re repeating what they heard some other Hacker News commenter say
nobody needs to prove their suspicions
Analysts like Semi-Analysis have done a lot of modeling and estimates on the topic.
But two can play this game: There is absolutely no evidence to support that API prices do not have profitable unit economics.