New Lower and Upper Bounds for the Grothendieck Constant | Hacker News Reader

New Lower and Upper Bounds for the Grothendieck Constant

arxiv.org

50 pointssurprisetalk11 comments

related paper on the long-horizon AI research system used: https://arxiv.org/pdf/2608.11195

The section where the Research-state representation is assessed to be fragile is the work I want to see more of.

I've already accepted that models know everything and will only get smarter, its cope to pretend hallucination is achilles heel. I want to understand how to build/use harnesses that converge on goal states and allow me to contribute human expertise like intuition and taste.

The components they call bulletin, session report, and especially the curated summary are the parts that make their work go forward.

Thanks for posting. I'm doing some AI vibemathing and running into exactly the difficulties they describe.

The comment under “Future Directions” at the end of that paper would seem to apply to the use of AI in software development, business management, and other areas as well:

“For AI[-assisted mathematics] research, improving research judgement and research-state representation may require training and evaluation on records of mathematics as a process, including failed approaches, strategic decisions, and evolving assessments of evidence. For mathematical practice, while these limitations persist, human expertise is likely to remain most valuable in these global functionalities: deciding when to persist or reframe a direction, and maintaining an accurate representation of accumulated progress.”

Yeah, the end goal is that you can run the economy autonomously, at higher levels of complexity than people could understand or reasonably make decisions about.

AI will tell us how the world works and we won't be able to understand what it means

This makes no sense. It would be smart enough to understand but not enough to explain it, or are you assuming the human brain can’t comprehend the world by default?

I am one of the first authors of this paper. To summarize our work, the Grothendieck constant (KG) is a fundamental constant in fields like analysis, algorithms, and quantum information, and concretely we prove 1.7135 < KG < 1.7819. The results came from truly novel ideas, and diverged significantly from the previous literature on proving upper and lower bounds. Our results were discovered in collaboration with a long-horizon research harness we built using publicly accessible frontier models, with separate reasoning and coding agents. We found that although AI excels at technical execution, they need deep human insight for research taste–understanding which ideas to pursue or abandon. We’ve documented our research experience as a case study of long-horizon human-AI collaboration; we hope to help shape future conversations around humans and AI in math, in a time dominated by results from large AI companies.

Happy to answer any questions about the results or the proofs!

By Grothendieck's standards, there are a surprising number of important practical applications.

"Grothendieck's Theorem, past and present"

https://arxiv.org/abs/1101.4195

Interesting. I wonder if there has also been any recent progress on the Legendre constant. Kidding.

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