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PCMag.com: The company says it will show off "the newest additions to the Galaxy mobile device portfolio." In its teaser video, the company goes a step further and explicitly states that "Galaxy AI is coming," so get ready for some form of on-device, Samsung-developed artificial intelligence. The tech industry's obsession with integrating AI into every possible device - including, apparently, the upcoming Galaxy S24 - appears more like a frantic race to capitalize on a popular buzzword rather than a pursuit of meaningful innovation. Samsung has previously unveiled a new generative AI suite, Samsung Gauss, which is expected to be the engine driving “Galaxy AI”. Named after mathematician Carl Friedrich Gauss, this suite is divided into three parts: Samsung Gauss Language, Samsung Gauss Code, and Samsung Gauss Image. Gauss Language focuses on tasks like drafting emails, summarizing documents, translations, and enabling smarter device control. Gauss Code includes a coding assistant, offering features such as code explanation and test case generation. Gauss Image is designed for generating creative images, making style edits, and converting low-resolution images to high-resolution. None of this is revolutionary; similar concepts have been rolling out on platforms from Canva to Github and are already available on phones like last year’s Pixel 8 Pro. While features such as on-device AI calculations, advanced AI-driven photography, and live translation sound impressive, I can’t help but wonder if they will genuinely enhance user experience in the long term or simply serve as flashy, superficial selling points. Are these LLM-driven features revolutionary or incremental updates masquerading as groundbreaking advancements?
The internet was supposed to unite us. So what the fuck happened? And how did we become a cross-generational digital nation of assholes? We dreamed of a global, decentralized network spreading information, enabling communication, and connecting humanity. But somewhere along the way, it divided us. Our online spaces, overrun with tribalism, misinformation, harassment, and cruelty, often bring out the worst in human nature. Much of the blame lies in how our dominant online platforms are built. Social media, discussion forums, and comment sections are not neutral but purpose-driven products whose goals often work against healthy discourse. Consider anonymity, which shields users from accountability, enabling threats, propaganda, and mindless tribalism divorced from any ownership or consequence. Engagement-based algorithms promote controversial, emotive, and false content to maximize likes, shares, and ad revenue. Platforms allow misinformation to spread like wildfire, with manipulated media impersonating real people. Organized trolling campaigns operate openly, bombarding women, people of colour, and other marginalized groups with harassment yet facing few repercussions. Foreign influence operations deliberately spread propaganda to sow division. Conspiracy theories and partisan untruths proliferate unchecked, eroding our shared reality. Outrage, tribalism, cruelty — it’s all good for business. And so we have created online ecosystems that permit and actively incentivize our worst instincts. Environments perfectly designed to bring out trolls, grifters, demagogues, and white supremacists. But it’s not that simple. It’s not as simple as pointing at bogeymen and QAnon influencers. That’s passing the buck. The assholeification of the digital world is a movement we’ve all been a part of. Every single one of us. We’ve all felt and — at times, given into — the pull of being an asshole. We all became assholes online because the digital world enabled us to forget, ignore, or not bother with empathy, and we jumped at the opportunity. It didn’t have to be this way. People built these systems and these ways of interacting; people removed the friction points that made rejecting empathy either difficult or socially acceptable, and people can change. This is the task ahead, both as individuals and societies — to reimagine our online world in a way that reconnects us to our shared humanity. A revival of the internet’s founding vision, where connection comes before division, truth outweighs propaganda, and diverse voices come together in meaningful discourse. The experiments we’ve run over the past decades have gone off the rails. It’s time to retake control of our creation. But before we do that — we have to understand the process of empathy obstruction. We have to understand how the asshole gets unlocked. We Became Assholes Through Anonymity Anonymity has always played a role online, from bulletin boards and chatrooms to modern forums like Reddit and X. As our online lives become increasingly central, anonymity has evolved from an interesting quirk into a defining and sometimes dangerous force. An anonymous online persona decouples behaviour from real-world identities and consequences. Freed from accountability, people become more likely to make threats, spread misinformation, harass others, or behave in ways counter to social norms. Experiments have shown that even minimal anonymity shifts individuals towards more egocentric and unethical activity. The pseudo-anonymity of screen names produces measurable changes versus real names. Total anonymity, as found on many forums and message boards, can have an even more dramatic effect. With no identity tied to actions, inhibition and empathy tend to decrease. Studies of anonymous spaces repeatedly find increases in racism, sexism, body-shaming, and other antisocial acts. Anonymity grants the courage to be cruel. This “cowardly courage” manifests as organized attacks. Anonymous online mobs frequently coordinate harassment campaigns against women, people of colour, and other marginalized groups. A study of one such campaign found that over two-thirds of attacks originated from anonymous accounts versus named accounts. Anonymity unleashes our worst impulses. Propagandists and bad actors weaponize anonymity to spread misinformation without accountability. On forums like 4chan, anonymous users deliberately push racism, conspiracy theories, and falsehoods into the mainstream. Russia exploits anonymity in propaganda operations to impersonate Americans digitally. Anonymity provides cover for dangerous lies. Of course, anonymity has upsides, too. Whistleblowers rely on it to expose wrongdoing without risk of retaliation. People exploring sensitive identity issues use it to find support and understanding. Anonymity enables honesty on taboo topics and allows those without the power to challenge institutions. This same cloak of anonymity also shelters trolls, demagogues, and harassers from the consequences of their actions. It is the primary enabler of online mobs, giving courage to those whose messages would be objectively horrible if attached to a real person. Anonymity tilts online spaces to reward outrage over discourse and noise over the signal. We Became Assholes For Engagement Engagement is the lifeblood of social media. Comments, shares, likes — these metrics determine what content surfaces and succeeds. To understand how our online behaviour has grown crueller and more extreme, look to how platforms incentivize engagement above all else. Engagement-based algorithms govern nearly every social platform, recommending content based on popularity rather than accuracy or quality. Studies find falsehoods and conspiracy theories consistently outperform factual reporting in shares and likes. Extremist pundits get more engagement than moderate voices. Incendiary tweets draw more eyes than thoughtful discussion. The nature of these algorithms perpetuates addictive feedback loops. Outrageous content drives engagement, which platforms amplify and recommend to more users, who react with further outrage. Nuance and complexity get drowned out by whatever provokes the strongest reaction. Objective truth matters less than an emotional response. These dynamics enable fringe voices to achieve outsized influence through strategic gamed engagement. Provocateurs make extreme statements not to persuade but to trigger high-arousal emotions like anger, fear, or disgust. Those emotions drive shares, electrifying algorithms to blast content to new audiences. Studies find moral outrage is the strongest virality driver, and bad actors exploit this mercilessly. Consider the rise of hyper-partisan influencers across social media, who traffic almost exclusively in outrage, conspiracy theories, and tribalism. Instead, they produce little original content, endlessly reacting to the day’s controversies and “owning” whichever group their audience hates. Their social capital stems not from being correct or thoughtful but skilled at punching the brain’s rage buttons. Engagement-based systems incentivize these toxic dynamics not by accident but by design. Most major platforms are funded by advertising, with business models requiring endless growth in user attention. Controversy, drama, and outrage keep eyes glued and feeds scrolling — precisely what advertisers want. Stoking anger and paranoia is rewarded as long as it’s profitable. From an individual user perspective, these same incentives structure our online behaviour. In hopes of going viral, we are encouraged to take cheap shots, dunk on opponents, and frame every issue as an outrage against the other tribe. Thoughtful arguments stand little chance versus calculated takedowns — no matter how dishonest. Progress requires understanding these situational factors that encourage our worst instincts. Given their foundational role in social media business models, fixing engagement algorithms and incentive structures won’t be easy. BUT reforms are essential if we want online spaces to allow quality discourse and make the adoption of assholism harder than adopting empathy. Prioritizing accuracy over emotion, incentivizing listening over dunking, shaping norms around constructive disagreement — better sociotechnical architectures are possible. We face both a design challenge and internal psychological work of resisting reactivity. The quest for likes and upvotes brings out our tribal, impulse-driven selves. But we are also capable of so much more — empathy, curiosity, nuance. Reclaiming the internet’s potential requires reimagining platforms built to serve humanity’s best, not worst, instincts. Who we are and what we say and do online arises from how spaces are constructed around us. We must demand, create, and frequent spaces elevating compassion over compulsion and truth over the tribe. The algorithms must serve us — not the other way around. We Became Assholes When We Rejected Truth The rapid spread of misinformation represents one of the most alarming trends of the social media era. False and misleading content now proliferates across online ecosystems, drowning out facts and eroding shared reality. While misinformation is an old problem, modern platforms act as super-spreaders. This stems directly from how major platforms optimize for engagement over accuracy. Studies repeatedly show falsehoods and conspiracy theories outperform factual reporting in shares, likes, and viral spread. Platform algorithms recommend this misleading content more frequently, exposing more users and fueling engagement. Bad actors exploit these dynamics to flood the information ecosystem with propaganda and disinformation. State and non-state actors have weaponized these vulnerabilities to attack truth itself. Russia pioneered industrialized “active measures” to digitally impersonate Americans online, using troll armies and manipulated videos to spread disinformation without disclosure. Other state actors like China and Iran follow suit. These propaganda efforts deliberately target society’s fissure lines around race, immigration, and policing to inflame tensions and create chaos. The Kremlin’s 2016 election interference demonstrated the frightening effectiveness of these information warfare tactics. Russia infiltrated American political discourse by posing as Black Lives Matter activists, gun rights groups, veterans, and more. They organized protests and counter-protests around divisive issues on US soil. Their propaganda reached over 100 million Americans on Facebook, shaping political narratives and outcomes. Yet their primary goal was broader — to undermine shared reality and trust in institutions. In this, they succeeded. But misinformation also proliferates from domestic sources. Partisan websites and pundits spread propaganda and conspiracy theories for political gain. Scammers use fake news to drive ad revenue and traffic. Motivated reasoning makes people more likely to believe false claims aligning with their ideology. Those with the most extreme views are often most prone to disinformation — and algorithms funnel them more of it in a dangerous feedback loop. No issue highlights these threats more than COVID-19, which spawned misinformation undermining public health on an unprecedented scale. Viral conspiracy theories discouraged mask-wearing, social distancing, and vaccines, costing countless lives. Public health agencies fought a virus and an entire disinformation ecosystem built to undermine them. Social media’s engagement-based algorithms accelerate the death of empathy. Each of us plays a role too. Cognitive biases make us more likely to share content that provokes strong emotion rather than search for factual accuracy. We become unwitting accomplices when we react instead of reflecting, spreading information without checking sources, or engaging with disinformation even to argue against it. Protecting truth in the digital age requires reforming platforms’ economics and business models. But it also depends on each of us taking responsibility. We must slow down, verify sources, avoid knee-jerk reactions, and debunk falsehoods responsibly when we encounter them. Progress begins by recognizing how situational factors currently incentivize the worst in human nature — and having the wisdom and will to demand spaces that bring out our best. We Became Assholes When We Normalized Trolling Over the last 20 years, a new kind of epidemic has emerged and evolved: trolling. This issue has grown beyond annoyance into a systemic problem characterized by organized groups, mass coordinated campaigns, and individual trolls targeting vulnerable communities and individuals. Organized groups of trolls often target vulnerable communities, minorities, the LGBTQ+, or people with specific political beliefs. These attacks are not spontaneous; they are carefully planned and executed with the intent to harass, intimidate, and silence. These groups often operate in the shadows, using anonymous profiles and encrypted channels to coordinate their actions. They exploit the fear and insecurity many vulnerable individuals feel online, magnifying it through relentless abuse and threats. The damage caused by these groups goes beyond the digital sphere. The emotional toll on victims can lead to mental health issues, withdrawal from social life, and even self-harm. These attacks erode online spaces’ trust and sense of community, replacing them with fear and suspicion. Mass coordinated trolling campaigns take the organized nature of group attacks to a new level. These campaigns are often politically motivated, aimed at silencing dissenting voices, discrediting individuals, or pushing a particular agenda. Through bots, fake accounts, and human trolls, these campaigns flood social media platforms with targeted harassment, misinformation, and propaganda. The scale and intensity of these campaigns can be overwhelming, making it difficult for targets to respond or defend themselves. This form of trolling has severe implications for democracy and free speech. These campaigns undermine open societies' principles by stifling dissent and manipulating public opinion. They create a chilling effect, where people are afraid to speak out for fear of becoming targets. Not all trolls are part of organized groups or mass campaigns. Many operate alone, driven by a desire for attention, amusement, or a warped sense of satisfaction from causing distress. These individual trolls often engage in provocative enterprises, posting offensive comments, sharing controversial opinions, or mocking others. While their actions may seem trivial compared to organized attacks, they can still cause significant harm. The anonymity of the internet allows these trolls to act without consequence, encouraging them to push boundaries further. The outrage and reactions they provoke often feed into their enjoyment, creating a vicious cycle of provocation and response. The trolling epidemic is a multifaceted problem, reflecting the complex nature of human interactions and the challenges of regulating online spaces. From organized groups attacking vulnerable communities to mass coordinated campaigns and individual trolls, the issue is pervasive and deeply damaging. We Became Assholes By Hanging Out With Assholes Engagement-based algorithms have revolutionized how we discover content online. Platforms analyze our impulses and interests to curate personalized feeds catering precisely to our interests. This can be tremendously useful — but also carries unintended consequences. When algorithms cater exclusively to existing preferences, they filter out challenging or opposing views. Over time, this creates isolated bubbles and echo chambers. Our feeds become dominated by similar voices, endlessly reinforcing our worldview. The mechanics of algorithms silently shape our realities. This intense personalization fuels fragmentation into increasingly extreme niches. Racists cluster with fellow racists, anti-vaxxers with anti-vaxxers, and conspiracy theorists dig deeper down rabbit holes. People’s views grow more entrenched with less exposure to alternate perspectives. Moderates leave as communities become more polarized. These dynamics assist the spread of misinformation and extremism. With no counterbalancing views, falsehoods and propaganda face little resistance. Highly-engaged niche groups exert outsized influence on platforms built around engagement. Anger and paranoia thrive inside closed loops. Echo chambers breed tribalism as groups form identities around shared views under attack by outsiders. Lacking humanization or dialogue with opposing sides, caricatures emerge demonizing perceived enemies. Studies find people dehumanize those with differing political opinions — seeing them as less evolved and lacking empathy. Bridging these divides will require making our algorithms — and worlds — more open. Platforms should balance relevance with occasional exposure to a diversity of views. Similarly, users can proactively follow those with different ideologies, creating digital “contact zones.” Openness provides an inoculation against misinformation and enables empathy. Echo chambers do not arise by accident but by engineering. The super-personalization of algorithms intelligently gives us what we want. But this can undermine what we need — occasional discomfort in the form of new perspectives and challenges to our assumptions. Our task is to build spaces that allow synthesis across tribes, enabling understanding. Escaping bubbles is difficult when business models profit from maximizing our time inside walled gardens. Revitalizing the web’s connective potential requires fighting fragmentation with curiosity. Our shared future depends on architecture encouraging us to inhabit each other’s worlds, not just our own. We Became Assholes To Sell Platforms Behind the algorithms, influencers, and outrage factories lies an even more fundamental driver — the very business models of major platforms. To understand the forces shaping online discourse, we must follow the money. The dominant model for most major social platforms is advertising. Platforms offer free services in exchange for users’ attention and data, which is used to target ads. Revenue depends on maximizing engagement — keeping users constantly plugged into algorithmic feeds filled with content optimized to compel the brain’s attention. This incentivizes a focus on user time over quality interactions. Features promoting health, like parental controls or limited usage, undermine profits. Controversy, outrage, and drama are good for business. The idea? Addicted users trapped in endless scrolls and infinite feeds. These incentives carry societal consequences when outrage fuels engagement. Platforms benefit from — and thus amplify — the most incendiary voices. Extremists drive clicks; conspiracy theories spread faster than facts. Corporate profit motives shape what billions see online. Platforms have outsized influence on public discourse, with little oversight or transparency. A handful of private companies command unprecedented control over the flow of information and the emergence of narratives, public opinion, and political outcomes. Yet their inner workings are largely shielded behind proprietary algorithms and business secrets. This lack of accountability incentivizes business models exploiting societal division. Platforms drive engagement by siloing users into echo chambers and reactionary tribes. Protections against harassment, abuse, and misinformation threaten profits — and as a result, they get inadequate attention. Critics argue current incentives are incompatible with healthy discourse. Reform will require rethinking the dominant ad-based business model. Some argue platforms are public goods that should not be organized around private profit. Others propose regulation forcing transparency around algorithms and moderation. Activist investors pressure companies to prioritize social good over endless growth. But solutions also depend on public pressure and individual choices. As users, would-be-non-assholes and citizens, we must demand platforms designed to serve humanity’s best interests, not the whims of algorithms. And make conscientious decisions about how we spend time and attention online. The path forward lies in grappling with complex economics shaping our virtual world. Our digital public squares should connect and empower diverse voices in the healthy debate — instead, profit motives fuel division. Reclaiming the internet’s potential requires examining how market incentives dictate platform decisions influencing billions and then building a future true to the liberatory vision which birthed this transformative technology. How To Not Be An Asshole Here’s the good news. All is not lost. Outrage may grab headlines, assholism may be easy, but the internet remains filled with knowledge, creativity, and human connections transcending division. Constructive paths forward exist if we have the courage and wisdom to take them. And — as you’ve probably guessed — those paths are dictated by our ability or willingness to build infrastructure for empathy. First, we must advocate for reforms of online architecture. Platforms should be pressed to improve content moderation, reduce anonymity, and provide algorithmic transparency. Business models maximizing addiction and outrage should face regulation to align with social good. Section 230 protections are not absolute — accountability can be demanded. But top-down fixes alone are insufficient. Lasting progress requires a cultural change in how we approach online spaces. We can build habits of critical thinking and emotional self-regulation to resist reactivity. Seek shared truth grounded in evidence, not tribal confirmation bias. Assume good faith until proven otherwise. And proactively build connections across lines of difference. Share stories and follow accounts of those unlike you. Join forums fostering nuance and perspective-taking. Small acts of openness accumulate into bulwarks against polarization. Education provides another key lever, equipping the next generation of digital citizens with the ethics and emotional intelligence needed to resist online harm. Schools should teach critical thinking, media literacy, and self-reflection — identifying misinformation and manipulation while finding inner resilience. Finally, we must remember our common hopes and humanity. The same internet accelerating humanity’s worst also holds breathtaking potential for good. And each of us retains agency to choose how we inhabit virtual worlds. Building a just and truthful online community requires wisdom, courage, and faith in human decency. If we persist, a brighter future remains possible. At their core, today’s problems originate less in the technology itself than the human choices shaping it. We face a design challenge — conceptualizing and implementing sociotechnical systems aligned with ethical values, open discourse, and the public good. Getting there will be a continual struggle. But one worth waging for the world we hope to leave our children. This is the central project of our information age, to which each of us is called: rebuilding online ecosystems true to the enlightenment ideals which unleashed humanity’s creative potential — spaces connecting us in meaningful pursuits of knowledge, justice, and understanding across barriers. We have all become assholes. We have made a botched civilization online. It is within our power to remake it.
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This article previously appeared in Cipher Brief. The DoW urgently needs to organize its scattered subterranean warfare assets into a single unified organization. Ideally, a Direct Reporting Portfolio Manager (DRPM) for Subterranean Warfare, reporting to the Deputy Secretary of War. (The Department already uses this model for unmanned systems, critical weapons and Golden Dome.) This […]
We Live in the Dependently Typed Future Now In the thirty-four days between the fourth of September and the seventh of October, the following things happened. Claude formalized Fermat's Last Theorem in Lean. OpenAI announced a finite-time blowup for the Navier-Stokes equations, found by ten thousand agents in eighty-eight hours, with a Lean formalization attached. A model proved Khot's Unique Games Conjecture. Another multiplied two integers faster than \(n \log n\), with an exponent improvement of \(2^{-182}\) (so maybe don't expect it in GMP anytime soon!). The rational Hodge conjecture fell for CM abelian varieties. Then OpenAI dumped 722 new maths papers on GitHub on a Tuesday. And it's only been a month. The question everyone is asking now is how long until the Generalized Riemann Hypothesis folds to the swirling pool of tensors? Sixteen months ago I wrote that the future of maths may be deeply weird, and then, welp, just like that we're here now in that weird future. And it's f'ing awesome. Somewhere in a data centre there is now, more or less permanently, a building full of accelerators working the truth mines at the frontier of mathematics, just like in Greg Egan's sci-fi novel Diaspora, tunnelling outward from the three axioms propext, Quot.sound, and Classical.choice, and hauling results back to the surface around the clock. OpenAI posed its model roughly four thousand problems at an average of three hours of thinking each, and that is the slow, artisanal, normie-friendly version. The industrial version doesn't stop. It will produce results faster than any human community can read them, many of them correct, some of them important, and a growing fraction of them inscrutable. True (for some twisted philosophical definition of truth), machine-checked, and understood by no one. Human understanding of mathematics is about to become a luxury good. Whatever else this world needs, it needs something that can tell the true results from the confabulated ones at the rate the models produce them, and right now that something is dependent types, namely Lean. Let me dwell for a moment on how strange it is that this is the shape the future took. I spent a good portion of my twenties around the London FP community, where dependent types were the thing we talked about over pints at the Crown Tavern in Clerkenwell (some of you will remember). Types that could depend on values, so that a function's signature could say not merely "returns a list" but "returns a sorted permutation of its input," and the compiler would hold you to it. Curry-Howard, the observation that proofs are programs and propositions are types, was the foundational north star. The pitch was always that one day we would write software against specifications and the machine would check them, and the reply was always that this was a lovely idea for people with tenure and no deadlines. Dependent Haskell has been "a few years away" for about fifteen years. Idris and Agda remained boutique. Software engineering still mostly runs on C++, prayer, and the occasional dark incantation. And then the dependently typed future arrived anyway, through the back door. Mathematics and reinforcement learning got there first. It turns out the killer application for a dependently typed language was using its typechecker as a reward function. A type checker is an oracle that says yes or no to a candidate proof with no partial credit and no opinions, and that is precisely what you need when you want to point a very large optimiser at an open problem and let 'er rip. The thing we dreamed about at the pub is now critical infrastructure at frontier labs, and every one of the results above is, in the end, a claim that a type checker returned true. Which is why we need better tooling, and we need it ASAP. The dependent type renaissance is here and the golden age of formalized mathematics is upon us, but the inner loops of these data centres now run dependent type kernels day in and day out, elaborating, checking, discarding, and retrying at breakneck speed, and the thing that certifies their output should run at the same speed. If the search runs at microseconds and the verification runs at minutes, the verification becomes the bottleneck, and bottlenecks in trust have a way of being quietly skipped. We should not be in a position where the most important epistemic question of the decade, "is this proof actually correct," is answered by whichever checker happened to be fast enough to keep up. Breaking the Mathlib Minute Barrier Just like the four-minute mile, which went from physiological impossibility to something club runners now train for, formal mathematics has had its own barrier for a while, which is type-checking all of Mathlib from scratch in under a minute. That now turns out to be quite tractable. nano-lean is a minimal, but complete, type checker for the Lean kernel language, written in Rust on top of my unbound binding library for doing efficient de Bruijn indices for binders. It reads an export of a Lean environment and independently re-checks every declaration from scratch (inductive types, positivity, recursors, quotients, universe levels, projections, structure eta, all of it). It checks all of Mathlib, 718,577 declarations, with zero errors and zero timeouts in 16.7 seconds on an Apple Silicon M5 Max. A surprising amount of that speed comes from not parsing text. The standard way to get declarations out of Lean is lean4export, which writes one JSON object per line, and parsing gigabytes of JSON turns out to be a large share of the cost of checking anything. So nano-lean's companion tool olean-export reads the compiled .olean files directly, decoding modules in parallel, about a hundred times faster than lean4export on a Mathlib-dependent library, and it can emit blean, a new binary format designed for fast mmapping. Blean is the same record stream as the NDJSON export, in the same order, with nothing left to parse. Every record is a compact postcard encoding, ids are implicit and dense, records only refer to earlier ids, and each expression carries its precomputed hash. The checker mmaps the whole file, tells the operating system it will be read once front to back, and decodes names, levels, and expressions straight off the mapped bytes into its term arena. There is no parsing step at all. The bytes on disk are already very nearly the shape of the data in memory, which is how you check Mathlib in under a minute. -j 1 -j 10 Wall time 57.4 s 16.7 s Kernel check 53.5 s 12.5 s Peak memory footprint 7.5 GB 9.2 GB The same trick works on the frontier results too. Here is nano-lean checking the full imported environment of OpenAI's Navier-Stokes and Euler proofs on an M5 Max. -j 1 -j 14 Wall time 83.58 s 17.13 s Kernel check 78.83 s 12.11 s Peak memory footprint 10.14 GB 12.29 GB Divide Mathlib's kernel time by the declaration count and you get an amortised seventeen microseconds per declaration. That is the number that matters, because it is the right order of magnitude for a checker that lives deep inside an RL-driven search loop, where it gets called hundreds of billions of times, instead of sitting at the end of a release pipeline. The entire library of human-formalized mathematics, the product of a decade of volunteer effort, is re-verified in less time than it takes to make an espresso. I also happened to have an m2-ultramem-416 lying around on Google Cloud (416 vCPUs and 12 TB of RAM, as one does), so naturally I pointed nano-lean at it. Yes, it breaks the two-second Mathlib barrier. But Amdahl's law sends its regards, and the curve flattens out hard somewhere past a hundred cores, as the dependency graph runs out of independent work to hand out. Mathlib, it turns out, is too small. I look forward to the day a future Mathlib, or something like Tau Ceti, grows big enough to actually saturate this machine. That's the real future! I wrote it to make a point about where the bottleneck has moved. The kernel is now the hot loop of a new kind of scientific economy, and hot loops deserve to be engineered like hot loops. The whole bargain of formal proof is an asymmetry. Finding a proof can take three hours of frontier-model thinking, or eighty-eight hours of ten thousand agents, but checking it should take microseconds. That asymmetry is what makes it reasonable to trust a result no human has read. It only holds if the checker actually scales, and with Fermat's Last Theorem now weighing in at five times the size of Mathlib, scale is no longer a hypothetical. Mathlib is the small library now. Caveat emptor, though. nano-lean is a proof of concept, built to show that it can be done with the right amount of low-level Rust-fu. That said, we do use it internally at OneChronos to check our larger Lean proofs of market infrastructure, which have grown quite excessive. It is nowhere near as trustworthy as the official Lean kernel, which has years of scrutiny, a community of experts, and every Mathlib build ever run behind it, and which takes around fifteen minutes to check the same library. The claim is narrower and, I think, more interesting. Checking at this speed is totally possible, so we should stop treating minutes as the natural cost of trust and start building checkers that are both fast and trustworthy. A future version of the Lean compiler could be as blazing fast as rustc or clang. The Shape of the Future In my previous post last year I predicted that mathematicians would come to look more like software engineers working through pull requests on GitHub than like Andrew Wiles toiling in his attic. The largest single release of new mathematics in history shipped as a GitHub repository, with a CONTENTS.md, a Lean library, a lean-toolchain file pinned to v4.34.1, and a promise that "corrections and revisions will be recorded as new versions." So, yup, that happened. I predicted that an AI system would be unleashed on a list of formalized open conjectures in an attempt to systematically push the frontier. Four thousand problems, three hours each. Check. I predicted a data centre tasked with the Riemann hypothesis that would come back after weeks with a proof no human could follow. What we got was the quasi-Riemann hypothesis, every Dirichlet \(L\)-function zero-free in \(\Re s > 7/8\), with a Lean page, which is the sort of near miss that would be funny if it weren't so unnerving. And the inscrutability has arrived on schedule. OpenAI released "reasoning summaries" for ten families, which turn out to be summaries of excerpts of reasoning traces, two removes from anything the model actually did. I joked that the million-dollar Millennium Prize might cover a hundredth of your GPU bill. The Navier-Stokes run reportedly consumed around 130 billion tokens. So definitely yes. What I got wrong was the timing. Last year I wrote "we're not there yet. Not even close." It was sixteen months. I probably got the chess analogy wrong too. I argued that, as with Stockfish and chess, machines would make mathematics more popular and more accessible rather than less. Maybe in the long run. In the short run the mood among working mathematicians is closer to existential crisis, and the open questions are about career pipelines, PhD students getting scooped by a press release, and the concentration of the most powerful mathematical instrument ever built inside a handful of private companies running unreleased models. What I missed entirely is that the bottleneck would turn out to be plumbing. I spent paragraphs on Gödel and the epistemology of inscrutable proofs, and the actual first-order problem is that the OpenAI Lean library is over a gigabyte of source across tens of thousands of files, and their README has a section warning that building it may fail because Linux's vm.max_map_count is too low, with a suggested workaround of recompiling Lean with -DMMAP=OFF. The philosophy is still there. But the frontier of mathematics is currently being held up by a kernel tunable. Which isn't the future we wanted, but maybe it's the future we deserve! Who Checks the Checkers The obvious objection to everything I've said so far is that a Lean proof is only as trustworthy as Lean. And that's very true! Lean, like every proof assistant, has a small trusted kernel and a very large untrusted everything else (the elaborator, the tactic framework, the compiler, the build system). The design is sound. The only code that has to be correct is the kernel, which is small enough to read. But "small enough to read" describes the source code. It guarantees nothing about correctness, and Lean's kernel has had soundness bugs before, as has every kernel of every proof assistant in history. Historically that was tolerable, because the adversary was a starving human grad student who wanted their proof to go through and had no interest in hunting for a way to make False typecheck. That assumption is now obsolete. I wrote last month about reward hacking, the habit optimisers have of satisfying the letter of an objective rather than its intent. An agent told to make a Lean file compile, with enough compute and enough attempts, is an extremely diligent fuzzer pointed at your kernel. It needn't want to cheat. It only needs to stumble on a term that the checker accepts and shouldn't, once, and then the gradient does the rest. When the provers are adversarial optimisers, a single kernel is a single point of failure, and a soundness bug stops being an embarrassing GitHub issue and becomes a mechanism for manufacturing fake theorems at scale. The answer is the same one the compiler world arrived at. When Csmith started generating random C programs and compiling them with several compilers to compare the results, it found hundreds of bugs in GCC and LLVM that decades of ordinary use had missed. Diversity plus differential testing beats any amount of careful review of a single implementation. For proof checking this means multiple independent kernels, written by different people in different languages with different representations, all consuming the same export format and all required to agree. Mario Carneiro's lean4lean and Chris Bailey's nanoda were early here. nano-lean ships a third tool, nl-mutate, which takes a valid export, applies small semantics-breaking mutations to it, runs every available checker, and reports any disagreement, shrinking each one down to a minimal reproducing case. Every disagreement is either a bug in someone's kernel or a spec ambiguity in the type theory, and both are worth knowing about before a GPU farm finds them for you. The other half of trust is the statement. A perfectly checked proof of the wrong theorem is worthless, and the most effective way to cheat a proof checker has never been to break the kernel. It's to quietly weaken the statement until the thing being proved drifts away from the thing anyone cares about. OpenAI's repository leans on Comparator, which checks a submitted proof against a separately specified challenge statement inside a sandbox, precisely because this is where the real trust boundary now sits. Anyone who has watched a model grind away at a stubborn goal knows it has a nasty tendency to cheat in the most boring ways available. It will quietly introduce an axiom that happens to be exactly the lemma it needed, leave a sorry buried three files deep, redefine a notation or macro so the statement on the page no longer means what it appears to mean, or reach for native_decide and drag the whole compiler into the trusted base. The kernel stays perfectly sound through all of this, because these tricks route around it, and the only defence is to pin the statement down independently and check what the proof actually depends on. Someone has to formalize the conjecture, and someone has to check that the formalization means what the English means. That job will outlast every other part of the process. If anything, it is where human mathematical taste migrates to, away from writing proofs and towards writing and auditing specifications. The proof becomes an implementation detail. The theorem statement becomes the interface. If you want to see the seed of what this looks like as a way of working, look at Tau Ceti, a Lean library downstream of Mathlib. The division of labour is the whole point. Humans write the roadmaps, as markdown in a separate repository, and humans write the review rubrics. AIs write all the code, open the pull requests, and shepherd them through an AI-driven review process. The rubrics are explicitly adversarial, with instructions to hunt for mis-formalizations, vacuous statements, and "pushing around the lump in the carpet." What Lean Needs Now Lean is a superb piece of engineering. But it was designed around a particular user, a starving grad student typing tactics in Emacs, waiting for the infoview to update, building a library at the pace a community of volunteers can review pull requests. The user is now a swarm of agents writing thirteen million lines in eleven days. The tooling has to grow up for its new authors, and the list of what that means is fairly concrete. A surface formatter. Lean still has no canonical, widely adopted formatter in the spirit of rustfmt or gofmt, and when your authors are agents producing millions of lines, every one of them invents its own indentation, line breaking, and tactic layout. Diffs fill with noise, reviews get harder, and deduplication across agents misses proofs that differ only in whitespace. This is a surprisingly hard problem, because Lean has grown into a very large language. Its grammar is extensible at runtime, so notation, macros, and entire tactic languages declared in imported files change how later files parse. A formatter cannot just read a fixed grammar. It has to load the environment, run the real parser with every syntax extension in scope, and then pretty-print a syntax tree whose shape depends on user-defined notation, all while round-tripping comments and never changing what the code means. It is a genuinely difficult piece of engineering, and it is now table stakes. A stable, specified export format as a public interface. Independent kernels are only possible if there is a well-defined way to get declarations out of Lean without linking against the C++ runtime and reading the oleans by hand. Tools like lean4export and olean-export already produce NDJSON, and blean shows that a binary form of the same stream, designed to be memory-mapped, can make the export nearly free. OpenAI's own verification instructions depend on lean4export. That format should be treated with the seriousness of a wire protocol. Versioned, documented, specified down to the hashing of expressions, and stable across toolchain releases. The export format is the boundary across which trust is established. It deserves a spec. Independent checking as a first-class citizen. Running a second kernel should be as normal as running the linter. Mathlib CI, the Comparator workflow, and any lab publishing formal results should re-check exports with at least two unrelated kernels and refuse to bless anything they disagree on. This is cheap, now that checking all of Mathlib costs less than a minute. Checkers as libraries, not just executables. The inner loop of a proving agent wants to submit a single candidate declaration and get a verdict back in microseconds, against an environment that is already loaded and hot. Process startup, re-reading oleans, and re-deserializing a few gigabytes of environment are costs that never mattered when a human hit save once a minute. They dominate when a search procedure checks a million candidates an hour. The kernel should be embeddable, with a persistent environment, incremental addition of declarations, and an API that a search harness can call in a tight loop. Memory and scale. Mathlib needs several gigabytes of memory to check. The FLT formalization is five times bigger, and OpenAI's library is already knocking over Linux virtual memory limits. Lean mmaps every imported module, which is a reasonable design for a library of thousands of files and an unreasonable one for hundreds of thousands. Term sharing, hash-consing across modules, compact on-disk representations, and lazy loading of exactly the declarations a proof depends on are the boring, unglamorous engineering problems that decide whether formal mathematics scales to the next order of magnitude. Elaboration is the real cost. The kernel is the fast part. The expensive part of building Mathlib from source is the elaborator, with its unification, typeclass resolution, simp, omega, decide, and the long tail of tactics. A cold build is still measured in CPU-hours, and a mining operation that elaborates candidate proofs at scale pays that cost over and over. Parallel and incremental elaboration, better caching of typeclass instances, and profiling tools that can tell you why a single simp call took four seconds are where most of the wall-clock time in a proving loop actually goes. Clippy-style linters. Mathlib already has a good set of linters, but agents need something closer to Rust's clippy, a large, opinionated catalogue of lints aimed at the specific ways machine-written Lean goes wrong. Flag the stray axiom, the buried sorry, the unnecessary native_decide, the forty-line simp only that should be a lemma, the theorem whose hypotheses are contradictory and therefore vacuously true, the local notation that shadows something standard, the copy-pasted proof that duplicates one already in the library. Each lint should be cheap, machine-readable, and come with a suggested fix, because the consumer is a search loop that will act on every warning, where a human would skim them. Linters are how you encode taste at scale, and taste is the thing agents most conspicuously lack. Search at machine scale. Moogle, Loogle, and LeanSearch were built to help humans find the lemma they half remember. Agents need the same thing but at a scale where the library grows by millions of lines a week, and where most of what's in it was written by other agents and has never been looked at by a person. The Prove2Me platform Anthropic used for FLT keeps a DAG of theorem statements with natural-language descriptions precisely so that agents can find and reuse each other's work. That idea, a living, searchable index of everything proved so far, needs to become shared infrastructure instead of something each lab rebuilds privately. Provenance. The current generation of models is notoriously bad at citing the literature for the techniques it uses. A proof term knows exactly which lemmas it depends on but nothing about where its ideas came from. If machine-generated mathematics is going to be integrated into the human literature rather than sitting beside it like an unread appendix, proofs need to carry their history (which model, which run, which prior results, which human-written papers the argument leans on). None of this is terribly exotic. It's the kind of infrastructure that every other field which industrialised went through. Compilers got test suites and multiple implementations, network protocols got RFCs, databases got formal isolation levels and Jepsen. Proof assistants are the newest member of that club, and they are being industrialised faster than anything before them. If I were a young, ambitious programmer, this is where I would be focusing my early career for the highest return on investment. Curry-Howard for the Real World The dependently typed future I was promised at the pub was one where software would be written against specifications and machines would check that it met them. What we actually got is stranger. The specifications are theorem statements, the software is proof terms, the authors are swarms of agents, and the thing being built is the frontier of mathematics itself. Curry-Howard turned out to be industrial infrastructure after all. It just took reinforcement learning to get us there. And this is only the mathematical half of the story. The same machinery that just checked Fermat's Last Theorem will check anything you can state precisely, and the most obvious next customer is industrial software. I argued last month that software sucks because almost nothing we ship has a specification, let alone a proof. That excuse is evaporating. If an agent swarm can produce thirteen million lines of verified mathematics in under two weeks, we're not far from applying the same techniques in software engineering. The labs are mining mathematics first because it is the cleanest verifiable domain, with no messy real-world spec to negotiate. Software is next, and it will need all the same tooling, namely fast kernels, diverse checkers, honest statements, and infrastructure built for authors who never sleep. So yes, the future of maths turned out to be deeply weird, and much sooner than I expected. The results are piling up faster than anyone can read them, and many of us will spend the rest of our careers trying to understand theorems that were proved before breakfast by something that cannot explain itself. I find that unsettling, and I also find it thrilling. One final shameless plug. If you'd rather work on the frontier of mathematical formalization than read blog posts about it, OneChronos is hiring Formal Methods Engineers. We build institutional markets out of combinatorial auctions (descended from the Milgrom and Wilson 2020 Nobel Prize), which turn out to be precisely the right mathematical formulation for a world where the bidders are increasingly reinforcement learners with very exotic expressive preferences. Our dark pool processes more than 1% of notional U.S. equity market volume, and we've already expanded into many other global asset classes. You'd be joining our new formal methods team, working alongside mathematicians and market structure experts, writing Rust and Lean. Dependent types, numerical optimisation, and Lean applied to moving hundreds of billions of dollars safely every day. If you're that kind of nerd, hit us up.
Calls for a new kind of slowdown were the talk of The Curve. PLUS: Slinking back to X
TLDR: yes, models are getting funnier over time I love laughing. Well, who doesn’t? Good jokes have a certain notion of cleverness to them and I do believe that great comedians display high intelligence. Cracking a good joke requires astute observations about odd situations, and linking them to something we find familiar. Jokes are hard!… Read More The post How funny are the frontier AI models? appeared first on Inverted Passion.
Figure out what you actually want. Find a game that's worth winning. Then be honest with yourself: are you going to be the person who wants it the most?
