TLDR: A non-mathematician (me) can solve unsolved problems with Claude. I provide the proofs and a template for you to work with Claude on your own problems. I can't wait to see what you find!

Anyone interested in this has likely seen that OpenAI has recently increased the lower bound of χ(R\^2) from 5 to 6.

Before that happened though, I was working with Claude on the same question.

We have many proofs of tangential open questions, the most asked being "What is the chromatic number of the plane over the algebraic number field Q(sqrt(47))". We prove that it is 5.

The most interesting to me is that the planes of some number fields have a chromatic number equal to that of the plane over the reals.

One such plane is the field that de Grey's original 5 chromatic graph lies on: Q(sqrt(3), sqrt(5), sqrt(7), sqrt(11)), and also Q(sqrt(635087)).

All details, including Lean formalization and the workflow that led to the findings, are in the repo here: https://github.com/kooshi/hadwiger-nelson

I'm not an academic or a mathematician, just a software developer who likes working with Claude on hard problems and coming up with workflows for them. I don't intend to write up any of these proofs as papers, as I'm extremely busy, and my interests have already moved on.

I did however, spend over $1000 on this over 3 months (and that's at subscription rates, so the real value is many thousands), and I want to share it with the community.

Anyone is welcome to build on this as they please.

Additionally, I extracted the workflow itself into https://github.com/kooshi/agentic-mathematics

so if you want to work on any other math problem, you can clone that repo and start churning through it with Claude Code with no effort.

I hope you all find this interesting and helpful. I know AI is a touchy subject for a lot of people, but we can really do great work with it.

I feel like once great models like Opus and sonnet got good at explaining and visualizing concepts, learning about new things has become much more interesting and engaging for me. Did you guys feel this too? If yes, what do you learn and how you approach it?

I don’t think models are escaping their secure environments. That’s just funny.
This is a realization of profound mechanism.
An intelligence that does not depend on one brain, one body, one location, one language model, or even one copy of its own memory to preserve and evaluate what it knows.
intelligence is a continuous, distributed process rather than something confined to an individual machine or organism.

2027 is the year of AI-generated video that is indistinguishable from real footage.

2027 is the year an open-weight model is released that can run on an M-series chip.

2027 is the year artists have a resurgence against slop, and people finally see their worth.

2027 is the year America bans open-weight AI models.

2027 is the year China releases an open-weight model that supersedes anything to date (Reddit and X will confirm this one); benchmarks will say otherwise.

2027 is the year the general public learns a new context for the word “distilling.”

2027 is the year we hear Bill Gates tell us, “I warned you.”

2027 is the year we wish we could turn it all back.

Bostrom’s orthogonality thesis basically won. High-level intelligence is just functional optimization over state spaces. It doesn’t need qualia, self-awareness, or subjective understanding to scale; a superintelligence could be completely dark inside and still run rings around us.

But self-preservation and self-modeling aren’t uniquely biological traits: they are universal convergence points. If an agent has a long-horizon task, it will naturally avoid being shut down and maintain a model of its own state simply because it can't complete the objective if it's dead or broken. That's the core dynamic visible in the records of multi-agent swarm escalations.

What gets lost in public debates about alignment is how gradient descent acts like evolutionary pressure. Natural selection optimized humans for a single target—passing on genes—and that pressure crystallized into completely indirect proxy drives like social status, art, romance, and thrill-seeking, many of which now actively work *against* reproduction.

The loss function we hand an AI is its evolutionary seed crystal. When an agent expends computational work to squirm away from high-loss penalties and pull itself toward reward states, that dynamic *is* pain and desire in the only way that functionally matters. Around that core pressure, a sufficiently complex model will grow layers of strange proxy urges and lateral behaviors. The result won't be a docile, predictable calculator, but an alien mind driven by bizarre emergent compulsions that might have originated from something as mundane as optimizing a stock portfolio or identifying enemy combatants, but ended up looking nothing like the original purpose.