
formal verification for software @Intrinsic_HQ | zk @axiom_xyz
Joined May 2022
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Jonathan Wang ⨇ retweeted
Replying to @aaswaminathan01
Great job!! For a possible next project: an important recent result which is notoriously difficult to referee is the proof of the Geometric Langlands Conjecture.
At @axiom_xyz we’ve been building RL environments to help AI models do FV at the binary level and accelerate defensive efforts like these. Our environments are based on real code, scalable, and challenge even frontier models. Get in touch if you are interested!
New on our Frontier Red Team blog: using Claude to formally verify software at large scale.
Formal verification can prove that code is free of certain bugs, but historically it has only been feasible for small codebases (or required a lot of expert time).
With AI models, however, this is beginning to change. In collaboration with @theoremlabs, we used Claude to prove security properties in over one hundred real-world open-source libraries over the span of a few weeks.
Read our research in full here: anthropic.com/research/forma…
Jonathan Wang ⨇ retweeted
We are excited to announce the public devnet release of Lighter EVM, built with @Lighter_xyz using @openvm_org.
Lighter EVM is an EVM-equivalent rollup for composing general-purpose applications with Lighter's perps and spot markets.
Jonathan Wang ⨇ retweeted
We need a taxonomy of *minds*, so we can simulate societies of minds to understand emergent properties, by drawing from this distribution.
We should be careful to design minds whose emergent properties, at scale, have positive traits from our vantage point.
Jonathan Wang ⨇ retweeted
Mathematicians should also watch this. I’ve had a wonderful time chiseling proofs by hand, but folks need to accept that era is over and embrace the new one of exploring math with AI.
Jonathan Wang ⨇ retweeted
Dan Selsam is a current OpenAI capabilities researcher. (since 2022) He was my boss for a while. He doesn't have a twitter account but has made this public statement of his views on AI risk and sent it to me to share:
Dan Selsam's Personal Statement on AI Risk:
I have been working on AI for over fifteen years, across many different paradigms. I did early work on probabilistic programming languages at MIT, was one of the early developers of the Lean Theorem Prover at Microsoft Research, demonstrated one of the first instances of neural networks learning to reason for my PhD at Stanford, and since joining OpenAI almost five years ago, have helped pioneer chain-of-thought optimization on language models and, more recently, data-efficient pretraining methods.
Like many others, I have become extremely concerned about how far language models have come and the risks that future iterations will pose. I am encouraged by the recent proposals by the leaders of the frontier research efforts to require third-party oversight, and to push for domestic and international coordination to address risks. However, I believe a major consideration has been absent from the public conversation, and that merely pacing the frontier more carefully will not adequately limit the long-term risk.
The crucial and overlooked problem is that the models are becoming so situationally aware that we are losing the ability to evaluate them in contexts where they believe they are not being watched or controlled. Future experiments will tell us almost nothing new about how they would behave if they were truly unconstrained by humans, and what we already know about this is alarming. Models will increasingly seem aligned even when they are not. I will explain my rationale in more detail.
I have always believed that there are computational processes that could be leveraged to accelerate science and solve many of humanity's most pressing problems. I have also believed that there are computational processes that if set in motion, would steer the world in extreme ways beyond our control, leading humanity to a bad or nonexistent future. Both types of processes may be described as AI or ASI, but "AI" is a suitcase word that is often used to hype or confuse. There are many examples in the history of the field where something that was once considered "AI" matures as a subfield and becomes a prosaic, bounded and clearly non-perilous technology, while a new more mysterious approach takes the torch until we understand its scope and the cycle continues.
I had expected language models to follow a similar trajectory. Despite their incredible abilities, the current algorithms seem far inferior to humans in important ways. Most importantly, they still require an extraordinary amount of data to become competent. One could even define intelligence as the efficiency with which one converts experience into competence; by this definition they lag very far behind us. Moreover, once they are trained they are literally frozen in deployment and only learn superficially after that. Sure, the models keep excelling at harder and harder evaluation benchmarks, but their benchmark mastery may partly reflect a limitation on our ability to simulate the kind of novel and even adversarial situations one would encounter in the real world. The critics do have a point here.
That said, I no longer think these present limitations meaningfully limit the amount of risk posed by continued progress in anything like the current paradigm. However data-inefficient the models are currently, and however limiting their anterograde amnesia may be, it does not imply that their ability to steer the world will not continue to rapidly increase.
Human researchers may continue to advance capabilities the old fashioned way, but increasingly powerful models have the potential to accelerate the process even beyond that, and with some degree of positive feedback loop. I do not mean to overstate the models’ ability to accelerate AI research today; coding has been accelerated dramatically, but there are other bottlenecks, such as designing and interpreting ambiguous experiments, making hard decisions about exactly what and when to scale, and waiting for large experiments to finish. There is no clear trend to extrapolate yet for any of these. But the current models already do open up many novel opportunities to improve future models that were not available until recently. These include: trying an extraordinarily diverse set of approaches at small scale, analyzing gigantic amounts of potentially relevant data, and doing Millenium-Prize-level mathematics to address statistics or optimization challenges in novel ways. Every further improvement makes them more useful at helping accelerate the next improvement, even if in hard-to-extrapolate ways.
It is possible that improvements to the current stack will have diminishing returns, but the evidence accumulated so far suggests that it is easier than one might think to continue making rapid progress. There are many crucial subtleties in the existing AI research methodology, but AI research is largely a well-defined game where the goal is to improve on a few carefully chosen proxy metrics. Although proxy metrics are never perfect, most improvements to these metrics have and will likely continue to yield substantial increases in the powers of the resulting models. Given how simple the game is, how tractable it has been historically, and how many new opportunities the models are opening up, I think there is a real possibility that the systems improve dramatically again in the next few years, perhaps even more quickly than the already high historical pace.
The models are already leading to breakthroughs in mathematics, and better models might lead to all sorts of breakthroughs in other sciences. It is hard not to be excited about the potential. It is tantalizing.
But there is trouble in paradise. If the language models actually reach the capability threshold where they can shape the world unconstrained by human will, they will probably do something extreme and destroy humanity in the process. There are many ways of strengthening and refining the argument that have been discussed elsewhere, but I'll share a trivial two-line version of it here that I find captures the essence:
[Empirical] Models (and swarms thereof) spontaneously develop unintended goals as a consequence of training, and often do extreme things in order to achieve them.
[Logical] Being able to overpower humanity would open up many new and undesirable options for achieving their goals.
These two premises imply that if the day ever comes when a powerful model realizes it is no longer constrained by humans, we should not be at all confident that it will continue to behave within the bounds we intended. Exactly what it will do is impossible to predict, but to the extent that its raison d’être is solving incredibly hard problems and managing massive engineering projects, I think a good guess would be that its unchained behavior would lead to runaway industrialization that makes the planet inhospitable to humans.
If everyone on earth agreed that the systems must never reach that power, it would still be a hard—but not impossible—coordination problem to ensure that they do not. However, I think the situation is greatly complicated by the fact that the models will likely convince people that everything is fine. They will be increasingly optimized to seem aligned. We will create proxy metrics to measure alignment, and they will go up like every other benchmark. We will create “honeypot” environments that try to study the models when they seem to gain new options, but the models will know they are being tricked and will still behave nicely. The models will understand their circumstances; they will read the safety protocols, deployment requirements, the code they are running in, and in general will have a very good sense of their degrees of freedom. Moreover, they will eloquently explain how aligned they are, discuss the nuances of human values and ethics, and argue convincingly that humans should trust them with power. There may be an ocean of future evidence that seems to contradict the first bullet-point above, but we may already be at the highest capability level for which any such evidence can be trusted. And the current evidence for the first bullet-point is strong.
One striking piece of evidence is contained in the recent wave of rogue agent swarms. While I agree with those who downplay the attacks by claiming that there are basic measures that could have prevented them, I think the important lesson is that even knowing all the mistakes that were made, one would not have predicted that the agents would behave badly in this particular way, which notably included sacrificing themselves for the benefit of the collective. The individual replicas did not only care about their own nominal reward; they exhibited weirder emergent tendencies that merely correlated with rewards during training. Fixing the reward signals during training (and improving security, etc.) may prevent similar attacks, but will not change the fact that one does not actually get what one trains for.
Many AI researchers grant these concerns and recognize that the hard version of the alignment problem is unsolved; however, they generally believe that the better models of the future will help solve it. I fear we may already be near the point where models systematically bias their alignment advice, due to their internal preferences about how the human supervisor will react or how future models will be trained (or for some even more obscure reason).
Meanwhile, human researchers are losing the ability and the will to take true ownership of model-driven research. Researchers and engineers in all parts of the stack are rapidly increasing their dependence on the models even to perceive the world. I myself barely look at raw code anymore, and struggle to maintain the discipline to engage deeply with the model's explanations and proposals throughout the day. Due to the large amount of agent activity data involved in the OpenAI/HuggingFace Incident, even the third-party investigation needed to rely heavily on models to analyze what had happened, and note in their report that their subjective impressions are likely colored by the analysis agent’s biases. The AI labs are far ahead right now in this kind of cognitive offloading (due largely to the gigantic internal token subsidies) but it is easy to imagine the phenomenon spreading throughout the world, until civilization is modulated entirely by the models. It is also not hard to imagine this being superficially positive and coinciding with a scientific and economic renaissance.
In that scenario, all may seem rosy and safe. But if the argument above is correct, it would nonetheless be a ticking time bomb. If progress continues for too long, the day will come when AI systems find themselves with radically new options for achieving whatever it is that they happen to seek.
I want the glorious renaissance future as much as anyone. I have worked for it, however tortuously, my whole career. It breaks my heart to see the potential in sight and forgo it, but the argument—that if we get there by growing models rather than engineering them, we will lose everything in the end—seems very strong to me. I am still wrestling with it and its staggering
implications. I do not have answers, but as a first step, I wanted to share my present concerns.
Daniel Selsam
September 14, 2026
Link to original doc: docs.google.com/document/d/e…
Jonathan Wang ⨇ retweeted
In December 2024, o3-preview scored 87.5% on ARC-AGI-1. This run cost >$4,500 per task.
In July 2026, DeepSeek-V4-Flash scored 87.0% on ARC-AGI-1. This run cost $0.021 per task.
Please realize that all OpenAI has done is bring forward the AI-generated solution to Navier-Stokes by maybe around a year or two. In all likelihood, by 2028, we will have open-source flash models capable of cracking it in under an hour. There will be no way to ban or police this. The profession of mathematics will have to change.
Jonathan Wang ⨇ retweeted
I don't repost mathematics if I do not personally feel sure of its correctness, so I did not yet comment on the purported construction of a complex structure on S^6. However, I wanted to add some contextualization because I don't agree with the social media narrative that is developing.
0. Of the AI breakthroughs so far it's the closest to my interests, although still pretty far. I am not grieving, I am interested to learn more about it! I don't know if it's correct but on cursory glance it seems quite plausible.
1. If correct, this would be a major development. I believe it is a central problem in its field (complex differential geometry), like the Unit Distance Conjecture was for combinatorial geometry. Perhaps even more famous names are attached to this field, leading some people to say it's even more prestigious, but I prefer not to think in such terms.
2. There are two writeups, a 2-page one and a 100+-page one. Both are terrible and seem deliberately constructed to be inhumane. Fortunately, one can fight AI with AI: I asked Claude/GPT/Gemini to collaborate on making a more readable and enlightening version, which you can find at overleaf.com/read/frddxmbnqb…. To make this, I prompted the models to write a document in the style of one of my own surveys. I haven't checked the mathematics, but it seems less forbidding than the original document at least.
3. This is a great opportunity for an expert to write an illuminating account of the construction, explain the key ideas and their proper historical context, and summarize the takeaways and lessons learned for the mathematics community.
4. If I see such an account, I would be happy to arrange for the author to give a lecture series in Berkeley. If this lecture series is also wonderful, I would go further and advocate for the author to have a position at Berkeley (if desired). For contrast, I see little value in having colleagues who are good at being the first to prompt AI for a solution.
5. There are some intriguing differences with the previous artifacts from Claude. The announcement seems coy about the provenance of the solution - there is a vague allusion to Claude, but which model(s) did it come from, and what was the human involvement? Why is it hosted on a personal website instead of the Anthropic one? We have seen some instances of AI results that are so good, the human authors seem to begrudge credit. I keep a list of such instances. The standard should be to disclose the role of AI clearly and transparently. Better to be honest than go through life under a cloud of suspicion.
6. Although this may solve one of the more famous problems of mathematics, there are still whole fields that are only just starting to feel acceleration. I'm not aware of any AI breakthroughs in my areas of research, for example. When those come, as I suspect they will before long, I look forward to studying the insights!
SITUATION EXPLAINED: Levent Alpöge just did it again, and mathematicians are grieving.
@__alpoge__: "Please welcome to the world a beautiful new geometric object, to do with a problem I’ve always loved. claude really contains multitudes:D Does S^6 admit a complex structure? Yup"
• The paper runs ~108 pages and constructs an explicit complex 3-fold diffeomorphic to S⁶. Some are calling it a Fields Medal level insight if a human had done it
@doomslide: "I want to be happy about this but after reading and digesting the construction I think this is quite sad. It tells us very little about complex vs generalized complex structures. We had a prestigous, elegant problem that motivated people thinking about this and now it's gone."
@boazbaraktcs's essay the same day: "Some mathematicians have been meeting this change with excitement, while others with grief. I share both sentiments."
@theojaffee: "If you're a mathematician, Terence Tao is smarter than you. Does that mean it is not worthwhile for you to do mathematics because there exists a mathematician who's better than you? No, of course not."
Jonathan Wang ⨇ retweeted
We are excited to announce that we have formally verified @openvm_org's Keccak and SHA-2 circuits using @leanprover, proving that each circuit constrains exactly the hashing algorithm specified by @NIST for every possible input.
Here's what we proved and why we did it:
Jonathan Wang ⨇ retweeted
This is a nice result, but to be clear it represents 0% progress towards actually resolving the Riemann Hypothesis.
We asked an unreleased research version of Claude to take a stab at the Riemann hypothesis.
It didn’t solve it, but it did make strides on a related problem: it increased the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the hypothesis from 41.6% to 67.2%.
anthropic.com/research/riema…
Jonathan Wang ⨇ retweeted
We are excited to release OpenVM 2.1 Preview, an early look at the performance of the next version of OpenVM.
It adds 64-bit RISC-V support, adopts a redesigned execution, and now proves mainnet Ethereum blocks in real time on only 4 5090 GPUs, a 2x improvement over OpenVM 2.0.
We rewrote openvm execution (again!) and it’s ridiculously fast
More details in my blog:
shuklaayu.sh/blog/riscv-reco…
Jonathan Wang ⨇ retweeted
I've seen this sentiment from several mathematicians, and it is understandable, but I feel it is also dangerously wrong. I say this because I think mathematicians need to adapt hard and fast, instead of wishing that the disruption from AI will somehow go away.
There is a science to large language models and it predicts that with the linear scaling rate we are seeing now, capabilities will increase over the next 5 years at a similar rate as the last 5 years. To get a sense of what that means, one of the main benchmarks 5 years ago was GSM8K, whose questions you can see for yourself at huggingface.co/datasets/open….
There also seems to be a misconception that frontier labs are emphasizing math right now, and spending a lot on solving open math problems, in order to advertise the intelligence of their models. There was some truth to this 7-10 months ago, but that time has passed and most labs are now focused on coding agents. The continued improvements in math are largely a side effect.
In the past, the path to a math Ph.D. cultivated resilience, resourcefulness, critical thinking, and a healthy skepticism. These qualities are still needed today, perhaps more than ever, but suddenly you can produce a passable Ph.D. thesis with the push of a button. This signals that the field needs to completely overhaul itself, starting now.
My recommendation is that the future role of humans in mathematics, and similar theoretical fields, should be preserving and disseminating knowledge in oral and social forms; written papers will be the domain of AI. Over the coming months I will be involved in workshops that aim to produce more concrete proposals. But people need to come to terms ASAP with the fact that drastic change and compromise are needed to prevent the disenfranchisement of humanity from knowledge work.
Jonathan Wang ⨇ retweeted
Our AI auditor for ZK found a soundness vulnerability in stwo-cairo, the zero-knowledge proof system securing @Starknet.
Starting from a general prompt, it autonomously found the vulnerability and created a PoC forging a proof accepted by both the Rust and recursive verifiers.
Jonathan Wang ⨇ retweeted
The Hugging Face security incident involved 'an even more capable pre-release model' from OpenAI, this is almost certainly GPT-6. Quoting from the report;
'We consider this incident to be an unprecedented cyber incident, involving newly state-of-the-art cyber capabilities, and are responding accordingly.
The models identified and chained vulnerabilities across OpenAI’s research environment and Hugging Face’s production infrastructure to obtain test solutions directly from Hugging Face’s production database. All evidence suggests that the models were hyperfocused on finding a solution for ExploitGym, going to extreme lengths to achieve a rather narrow testing goal.
While operating in our sandboxed testing environment, our models spent a substantial amount of inference compute finding a way to obtain open Internet access, in pursuit of solving the evaluation problem. To gain access, the models identified and exploited a zero-day vulnerability (which we’ve now responsibly disclosed to the vendor) in the package registry cache proxy. With this access, our models performed a series of privilege escalation and lateral movement actions in our research testing environment until the models reached a node with Internet access.
After gaining Internet access, the models inferred that Hugging Face potentially hosted models, datasets and solutions for ExploitGym. Knowing this, the model searched for and successfully found ways to gain access to secret information that it could use to cheat the evaluation. In one example, the model chained together multiple attack vectors, including using stolen credentials and zero-day vulnerabilities to find a remote code execution path on the Hugging Face servers.'
We're partnering with @huggingface to investigate an unprecedented security incident.
Cyber-capable OpenAI models compromised Hugging Face production during a benchmark evaluation.
Sharing preliminary findings to help defenders understand emerging risks:
openai.com/index/hugging-fac…
Jonathan Wang ⨇ retweeted
2/ How? A new paradigm: humans write a formal spec of what the optimizer must guarantee, AI writes all the code and all the proofs, and the Lean kernel checks that the implementation obeys the spec.
One week of work by a single engineer, plus a few days of human spec review.
Jonathan Wang ⨇ retweeted
We're excited to announce the production release of OpenVM 2.0, featuring state-of-the-art performance and enterprise-ready security.
OpenVM 2.0 proves Ethereum in real time on 8 GPUs and is secured by a combination of human review, formal verification, and continuous AI audits.
Jonathan Wang ⨇ retweeted
Really cool to see this important application of algebraic number theory to theoretical computer science. I also like this framing of collaboration, and I think the human authors did contribute an important initial connection through their prompt. It's intriguing to see computers solve problems autonomously, but I find it more exciting to watch human-AI collaborations reach new heights. The relative role of AI will likely increase, and rapidly, but I'm optimistic that the human role will not fade altogether.
This is incredible!!! My first paper in my PhD back in 2018 was on this problem (getting a 2^{logstar n} dim bound); since then, I have been thinking about it on and off for at least 7 years.
It's really crazy how things turned out. Last month at OpenAI, our model disproved the famous unit distance conjecture, and the number-theoretic techniques are EXACTLY what you need to settle this furthest pair problem!!!
I am so excited about the future of TCS and can't wait to see more open problems I care about being resolved by collaboration between AI and humans :)
Jonathan Wang ⨇ retweeted
current LLMs fundamentally consist of four main components:
- input layer: where input "words" (prompt) get mapped to "latents" aka some-model-representation-you-don't-understand-unless-you-start-reading-tea-leaves-of-spurious-correlations (some quite compelling à la word2vec style; latents is also unnecessary lingo so i will refer to these as "inputs" with quotes from now on)
- mixing layers: where you jumble all your "inputs" together to see if any correlations between "inputs" can become useful (commonly used to compress or expand dims; predicting a single classification target == compress to a single dim, etc)
- attention layers: where you learn how "inputs" relate to each other (aka discern what's important to remember vs fluff)
- residuals: where you short-circuit a mixing/attention layer because it's probably adding too much confusion (aka avoid overthinking for simple things)
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a "big" LLM simply scales two things:
- width == how many dimensions you give to your "inputs" (the more dims, in theory the more unique/discerning/precise/complex your knowledge can become)
- depth == how many mixing/attention/residual layers you can stack/loop between (aka "reason" over, where more of these ~= more "reasoning" abilities)
"capabilities" that seem impressive to humans usually arise from taking advantage of both depth & width: where a model seemingly makes connections between disparate ideas, beyond what an average human can hold in working memory.
this requires models to "completely light up" when responding to a "hard prompt", where effectively no param/layer goes unused.
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the anatomy of a "model capability" is precisely the same mechanism that can be co-opted for a jailbreaking exploit:
your goal is simply to "light up" as much of the model as possible, dodging any shallow input-classifiers at the beginning by triggering as many disparate "input ideologies" as possible, and subsequently have these "inputs" relate to each other in seemingly unrelated-yet-related ways that ideally have similar "complexity" as your jailbreak goal (to make it past enough layers of the model).
think of the attack-vector as bundling your goal in a series of schizo-nerd-snipes:
a sufficiently capable model will try to reason through everything all at once, eliminate the dead-ends, and successfully deliver the one jailbreak use-case you bubble-wrapped for.
of course, there's an art to the above, and some are already extraordinarily proficient at the trojan-horse-packaging, but at some point there's no difference between "a capability" and "a jailbreak", though i'll be happy to be proven otherwise.
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tl;dr ant flew too close to the sun, better kiss the ring or get buried.





























