Published: July 22, 2026 | Reading Time: ~8 minutes | Channel: techminute
There's a line from the Hacker News thread that I can't stop thinking about. Someone named Phillip Kerger — a researcher most people had never heard of a week ago — had just posted a paper to arXiv. In it, he described how he'd spent about a year trying to close a theoretical gap in convex optimization that had been sitting there since 1996. He couldn't crack it. So he wrote a ten-page prompt, fed it to GPT-5.6 Sol Pro, and in 148 minutes, the model produced a proof. Verified in Lean. Gap closed.
The HN thread drew 150+ comments. One of them crystallized the whole thing: "If knowledge is a Swiss cheese, LLMs can help fill the holes, but not make the cheese bigger."
I think that's right. I also think it's the most important thing anyone's said about AI-assisted research all year. Because it tells you exactly where the value is shifting — and where it isn't.
Let me give you the problem in one paragraph, because the specifics matter.
In optimization theory, you want to minimize a convex function using only function evaluations — no gradient information, no derivatives, just "what's the value at this point?" The question is: how many queries do you need, in the worst case, to get a good answer?
Since 1996, we've had an upper bound of O(d² log² d) queries (Protasov's method) and a lower bound of Ω(d) queries. That gap — between linear and near-quadratic — is what Kerger was trying to close. He established a new lower bound of Ω(d² / log(d+1)), which essentially matches the upper bound up to polylogarithmic factors.
The gap is closed. Thirty years. Done.
But here's the part that actually matters for anyone reading this who isn't a convex optimization theorist (which is... almost everyone): how it was closed.
This wasn't "ChatGPT, solve this problem." It wasn't even close.
Kerger had been working on this gap for roughly a year. He had deep domain expertise. He knew the landscape of existing techniques — Nesterov's accelerated gradient methods from 1983, the Optimized Gradient Method from Kim and Fessler in 2014, Protasov's 1996 upper bound. He knew what directions were promising and which were dead ends.
He encoded all of that — the techniques, the context, the promising avenues, the mathematical scaffolding — into a ten-page prompt. Ten pages of advanced mathematics, priming the model in the right direction.
As he put it in the Reddit discussion: "I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory."
The techniques already existed. What GPT-5.6 provided was the capacity to explore combinatorial possibilities systematically — doing the mechanical work of trying combinations, checking consequences, and surfacing candidates. The kind of work that a human could do, but would take months or years of grinding.
Here's what the actual workflow looked like:
The LLM isn't replacing the mathematician. It's operating as an extraordinarily capable research assistant that explores possibilities faster than any human could.
This is different from what a lot of people imagined. It's not the AI having a genius insight. It's the AI doing the grinding — the systematic exploration of known technique combinations — while the human provides the judgment, the direction, and the verification.
Let's zoom out, because the pattern is what's significant.
A week before Kerger's paper, OpenAI announced that GPT-5.6 Sol Ultra had produced a proof of the Cycle Double Cover Conjecture — a 50-year-old problem in graph theory. That proof is still under review. This one's verified in Lean.
Two LLM-assisted mathematical proofs in one week: one in graph theory, one in optimization theory. This is becoming routine, not exceptional.
The pattern holds across both: problems solvable through systematic application of existing techniques are vulnerable to LLM assistance. Problems requiring genuinely new mathematical ideas remain human territory — for now.
The Hacker News discussion was unusually good. Not the usual tribal warfare between AI boosters and skeptics. Actual thoughtful debate about what this means.
The "Swiss cheese" metaphor was the standout. LLMs can fill holes in existing knowledge — connecting dots between known techniques, exploring combinations, surfacing non-obvious connections. But they can't (yet) make the cheese bigger. They can't generate genuinely novel mathematical frameworks.
But here's what I think the metaphor misses: most scientific and engineering progress doesn't come from paradigm-shifting breakthroughs. It comes from filling holes. From closing gaps. From doing the grinding work of connecting technique A to problem B.
If LLMs can automate a significant fraction of that grinding work — and the evidence right now suggests they can — then the pace of progress accelerates at every level, all at once. Not because we have a new Einstein in a box, but because we have an army of tireless research assistants who can explore possibility spaces faster than any human.
The breakthrough isn't the proof. The breakthrough is the process.
Kerger himself said something that should make every working researcher pause: "I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging fruit. We'll be needed for problems where actual novel approaches are needed."
This is the part that's genuinely unsettling. Not because AI is coming for the Nobel Prize — it isn't. But because the entry-level research problems, the ones that PhD students cut their teeth on, the "low-hanging fruit" that fills CVs and builds careers — those are increasingly automatable.
The value proposition for human researchers shifts. It moves toward:
The mechanical work of exploring known technique combinations? That's increasingly a job for the machines.
A year ago, the conversation was about whether AI could do creative work at all. Could it write a poem? Could it debug code? Could it draft a legal brief?
Now we're past that. The question isn't whether AI can contribute to mathematical research. It clearly can — given the right human partner, the right context, the right framing. The question is what kind of research problems are best suited to this partnership, and what happens to the researchers whose work falls on the wrong side of that line.
The "Swiss cheese" model is both optimistic and honest. It says: you still need humans to make the cheese bigger. To imagine new frameworks, new questions, new directions. But for filling the holes — for the systematic, combinatorial, grinding work of connecting known techniques to known problems — the AI is faster, cheaper, and doesn't get tired.
The researchers who thrive in this world won't be the ones who resist the tool. They'll be the ones who get really, really good at writing the prompt.
All claims verified against Gold-tier (arXiv paper, Lean formalization) and Silver-tier (Developers Digest, El Solitario) sources. Each source URL was scraped and confirmed accessible. Last verified: July 22, 2026.
