Building Mathematical Superintelligence
Joined January 2024
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Harmonic retweeted
@HarmonicMath is hiring a Formal Verification Engineer to develop Aristotle and apply it to formally verify production hardware and software!
The work spans formal methods research, product development and customer deployments.
Join our team in Palo Alto or London!
🔗 jobs.ashbyhq.com/Harmonic/5e…
Harmonic retweeted
Today we're launching Miles v0.1, an open-source RL framework for LLMs and multimodal models.
RL training is easy to start and hard to debug. Miles helps you ensure your run is correct, use hardware efficiently, and keep RL running at scale.
Over the past 9 months, 72 contributors have landed 1,326 commits, 85 GPU E2E CI tests, battle-testing Miles on frontier open models like Kimi K3, DeepSeek V4, Qwen 3.8, GLM 5.2, Inkling, MiniMax H3, etc.
Miles powers frontier-model development and production RL workloads at @humansand, @periodiclabs, @modal, @DecagonAI, @Eigent_AI, @nebiusai, @IBM and more, on both @NVIDIAAI and @AIatAMD hardware.
Here is what we built, and why teams picked Miles🧵
Harmonic retweeted
Luckily, I had recently been contacted by @PietroMonticone from harmonic.fun, who asked whether I had any formalization projects I would be interested in undertaking using their Aristotle system. I'm happy to say that it has just successfully formalized our paper.
Harmonic retweeted
A pleasure to help @wtgowers and his colleagues get their preprint formalised with Aristotle (@HarmonicMath).
The long case analysis that a referee would understandably prefer not to check by hand is now formally verified in @LeanProver.
The code should be available on @GitHub before long. After some further cleanup, parts of the supporting API may also be suitable for upstreaming to Mathlib.
Harmonic retweeted
Happy to share that our paper
“Mapping Uncharted Symmetries: Machine Discovery in Combinatorics”
has been accepted to the ICML 2026 AI4Math Workshop.
We study AI for discovery in algebraic combinatorics, with verification in @leanprover using @HarmonicMath’s Aristotle.
1/11
Harmonic retweeted
Today at the New York Number Theory Seminar, Wouter and Pietro were discussing their new paper. Really cool use of the AI-human feedback loop, with Aristotle as the main AI ingredient.
I explained how I think formalization feels like doing the low-tech steps of algebraic geometry proofs with commutative algebra. You can have high-brow intuitions, but eventually one has to prove all the details.
I wonder how this will evolve, but we are definitely at the level of assembly language being written by models like Aristotle. Tactics feel like the first glimpses of higher-level programming principles. The next step might be a more conversational style of working with the models.
I am now working on a tighter integration of Lean with computer algebra languages. Stay tuned!
Here's what András Sárközy, Erdős's most prolific collaborator, asked 25 years ago:
"How small can one make the maximal gap between the consecutive elements of a multiplicative Sidon set selected from {1, 2, ..., n}?"
In particular: does there exist a multiplicative Sidon set A ⊆ {1, 2, …, n} such that every sub-interval of [1, n] of length at least √n contains at least one element of A?
The answer is yes. The solution was autonomously discovered and formally verified in #Lean by Aristotle. We then improved the bound below √n and Aristotle formalised our proof too.
Harmonic retweeted
Here's what András Sárközy, Erdős's most prolific collaborator, asked 25 years ago:
"How small can one make the maximal gap between the consecutive elements of a multiplicative Sidon set selected from {1, 2, ..., n}?"
In particular: does there exist a multiplicative Sidon set A ⊆ {1, 2, …, n} such that every sub-interval of [1, n] of length at least √n contains at least one element of A?
The answer is yes. The solution was autonomously discovered and formally verified in #Lean by Aristotle. We then improved the bound below √n and Aristotle formalised our proof too.
Harmonic retweeted
I used @HarmonicMath's Aristotle to formalize Erdős problem #426 in Lean 4…
and ended up fully verifying a stronger bound that the original paper only suggested 🧵
Erdős offered $25 for a disproof, and $100 if the conjecture was true 👇
Harmonic retweeted
Mel Nathanson — How Harmonic's Aristotle solved some of my problems. gist.github.com/kim-em/20212… #LeanProver #ITP #Aristotle #AI4Math
Harmonic retweeted
Nathanson has just published the recording of his talk about Aristotle’s solutions and it is very interesting to watch!
“I tried to figure out what it did that I didn’t do to solve the problems.”
“The incredibly clever idea that Aristotle had was…”
youtu.be/VBIxv-6m7sk
Harmonic retweeted
Great to see Lean at the centre of this closed loop: from autonomous discovery, to formal verification, to mathematical exposition that humans can read, rewrite, and improve.
To my mind, what seems most important here is not so much the results themselves, nor even the particular methods used in the proof.
What really matters is the workflow: the closed loop from autonomous discovery of the right constructions, to formal verification of the proof in Lean, to informalised exposition back into a manuscript that mathematicians can read, understand, rewrite, and improve.
It suggests a framework in which formal proof is not merely a static final certificate, but an active part of mathematical research: a medium through which ideas are found, organised, tested, explained and made available for human judgment.
That dynamic loop is the real story for me. The natural next question is how far it can be pushed.
Harmonic retweeted
Very nice result! I autoformalized it with Aristotle by @HarmonicMath.
Here is the interactive code:
live.lean-lang.org/#project=…
Harmonic retweeted
Finally tried it out on a real research project, with amazing results!! Aristotle was key to making this paper happen.
nitter.cf/aaswaminathan01/status…
A decade ago, I abandoned the first math problem my PhD advisor ever gave me.
This week, I finally solved it—and formally verified it—using @AnthropicAI's Claude Code, @OpenAI's Codex, and @HarmonicMath's Aristotle.
Here’s how AI turned my 10-year-old notes into a 15,000-line Lean 4 proof. 🧵
RT @SwishMoe: Been using Aristotle Agent by @HarmonicMath internally for math-heavy LLM evals, and it’s honestly one of the best math tool…
Harmonic retweeted
This seems to be working well. I tried it on a few lengthy but simple proofs that I got from GPT-5.4 Pro. It seems to have worked flawlessly.
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Harmonic retweeted
Really impressive work! It took just a few hours to formalize two classic algorithms in game theory. One of them took me three weeks, and I wrote nearly 2K lines Lean code, while Aristotle agent only spent three hours and used 400 lines. I am so surprised
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Harmonic retweeted
It correctly formalized a graph theoretic proof from our recent STOC paper in one shot. It’s only 1 page on paper, but I’m still impressed; 450 correct lines of Lean is no small feat—that used to take me days!
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