Lots of drama this week on math proofs! I’m sure most have seen the back and forth between folks at OpenAI and Anthropic about a Navier Stokes proof.
The point of this post is not to dig in on all the details (too much commentary on that online already, and too much speculation…) Instead - here’s a very high level of one part of the saga (and then something that I found interesting). Producing a proof of the Navier Stokes existence and smoothness problem is one of seven Millennium Prize Problems that each have a $1m prize attached to it (ie solve it, get a $1m prize). And OpenAI said they did it. 10k agents working in parallel for 3-4 days, >100 pages of manuscript and a formalized proof. The drama around this are allegations that OpenAI piggybacked off research two professors had done (but not published) on the same topic (one of whom works at Anthropic).
Who gets credit, how it got discovered, etc, are not what I want to discuss today. What was interesting to me was how much of the commentary online centered around the cost / benefit of producing this proof. The “benefit” was the $1m prize. But at what cost? OpenAI said the agents sent ~2.7m messages and used ~130b output tokens. Many posts then attempted to do the math on the “cost” of solving this problem. At Astra’s list price of ~$50m / million output tokens, the 130b output tokens cost is ~$6.5m. That’s just output though. The input tokens for 2.7m messages passing context back and forth is most likely a much larger number. All in the total costs estimated from $10 - $40m. OpenAI has said that it was in the ballpark of “millions of dollars” of compute.
The lazy takes then become - “wait, they spent ~$10m+ to win a prize worth $1m?! How is that worth it!” Sam Altman even poked fun at people making these claims:
The broader point I want to make - for the most important breakthroughs, the “cost” of the first wave is never really that relevant. What’s most relevant is the binary - could we do it, and what’s the slope of the curve.
Noam Brown described this well. He talked about when OpenAI announced o3 in December ‘24 it cost $500k of compute to score 87.5% on ARC-AGI 1. Today, Astra scores higher for $20. In ‘25, OpenAI and DeepMind spent a ton of compute to get to IMO gold. For the 2026 IMP, anyone with a $20 / month ChatGPT subscription could do it. This is a pattern that will continue to repeat itself. Sure, maybe it cost $10m to solve Navier- Stokes today for a $1m prize. The focus should be on “look what a computer just did!” Not on “but look how much it cost.” Because the trend seems pretty clear - now that we’ve shown computers can solve problems of this difficulty, it should be obvious they will continue to solve more problems of this difficulty but at exponentially lower costs as time progresses. Epoch AI has the price to reach a given benchmark falling from 9x to 900x per year (depending on the milestone). GPT-4 came out in early 2023 at $60m / 1m output tokens. Now you can get that level of performance for <$1.
So let’s extrapolate this. This year it costs ~$10m to solve a Millennium Prize Problem. Next year it’s $1m. Year after that $100k. Year after that $10k. This specific trajectory of course won’t be exactly right, but the shape of it probably will be. Something that was once very out of reach for the vast majority of the population ($10m dollars, 10k agents, days of running time, etc) will (before long) cost less than a new foldable iphone and be accessible to everyone. That’s the real takeaway for me! And everyone just focusing on the mismatch in $1m price to cost spent is just completely missing the forest for the trees (or intentionally being dogmatic).
Tying this back to another (hotly) debated topic in AI - gross margins. Today, many folks will look at AI businesses with low gross margins and see a “broken business.” But this is making the same mistake as the cost / prize of the math proof I described above… Gross margins are point in time on a curve, and the cost of that curve is falling dramatically.
The pace of innovation is staggering, and only accelerating. Too often analysis of the industry looks at point in time metrics, and not at where things are heading. And the “where things are heading” can be tricky when the slope of the exponential is so steep. Last thing I’ll end with (maybe most important) - Noam also mentioned the model used at OpenAI for the proof was an unreleased model more powerful than astra (you can see his chart below). There are models that exist today that are already meaningfully more performant than what’s out in the public. The pace of the exponential will only steepen from here!
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