Mathematical Experiments Are Becoming Abundant Through Human
I ended a recent weekend without a Hadamard matrix of order 668 and with a mathematical theorem I did not yet trust enough to call a result. Working with GPT-5.6 Sol, parallel agents, exact-arithmetic programs and a proof assistant, I attacked two hard problems over two days. One resisted every route I tried. The other produced a proof candidate: a conventional argument written out in full, exact checks on selected inputs, a teaching guide and a partial formalisation in Lean. That second outcome is still a candidate. No specialist has reviewed the argument, its novelty is not established, and Lean checks only its algebraic core rather than the complete theorem. I would not publish it as a new theorem in its current state. The pattern I took from the weekend is that mathematical experiments are becoming abundant. The Had
I ended a recent weekend without a Hadamard matrix of order 668 and with a mathematical theorem I did not yet trust enough to call a result. Working with GPT-5.6 Sol, parallel agents, exact-arithmetic programs and a proof assistant, I attacked two hard problems over two days. One resisted every route I tried. The other produced a proof candidate: a conventional argument written out in full, exact checks on selected inputs, a teaching guide and a partial formalisation in Lean. That second outcome is still a candidate. No specialist has reviewed the argument, its novelty is not established, and Lean checks only its algebraic core rather than the complete theorem. I would not publish it as a new theorem in its current state. The pattern I took from the weekend is that mathematical experiments are becoming abundant. The Had
amard project closed 44 precisely defined regions of the search space and audited five standard routes to a nonexistence proof. The Maxwell project produced a complete proof candidate with a machine-checked algebraic core. Accepted mathematical knowledge did not get cheaper alongside any of that, because proof, understanding, novelty and independent review are separate obligations, and generating another route settles none of them. This is a concrete sequel to two earlier ideas. In AI Has Made Research Cheap. Understanding Is Still Expensive , I argued that AI was making experimentation far cheaper than understanding. In From Tokens to Theorems: Building a Neuro-Symbolic AI Mathematician , I built a simple neuro-symbolic loop in which an LLM proposed mathematical formulas, SymPy checked them exactly, and failed candidates became feedback for the next attempt. That experiment was deliberately modest, but it exposed an architecture that reappeared here at a much larger scale. Over this weekend, the same pattern extended beyond symbolic sequence problems. Instead of one model and one symbolic checker, I used parallel agents, exact-arithmetic programs, adversarial critics and a proof assistant. Candidate constructions and arguments were generated, attacked, checked where possible and either discarded or retained with an explicit status. The loop had become richer as AI systems improved through 2025 and 2026, but its basic structure remained the same. Looking back afterwards, I realised the workflow closely resembled a broader pattern that Jeff Dean recently described for science and engineering. Rather than viewing AI as a tool for solving one problem at a time, he characterised research itself as a recurring cycle: propose an experiment , implement and run it , evaluate the resul t, then use that evaluation to generate a better experiment . His argument was that AI should automate this entire loop, reducing iteration time, running many experiments in parallel and learning from every evaluation. That was almost exactly what had happened over my weekend, albeit on a much smaller scale. The experiments were mathematical rather than physical. “Implementation” meant exact-arithmetic programs, constraint solvers or proof assistants instead of laboratory equipment. Evaluation meant integer verification, counterexample search and partial formalisation rather than measuring a physical system. Parallel agents proposed constructions, generated verifiers, searched the literature, criticised arguments and translated ideas into different mathematical representations before the next iteration began. Seen this way, the most striking development was not that AI produced a proof candidate or failed to construct a Hadamard matrix. It was that the experimental cycle itself had become partially automatable. Once exact evaluators existed, proposing, executing and refining mathematical experiments became dramatically cheaper. The important difference was not simply one of scale but of validation. In the earlier experiment, a candidate formula could be tested directly against the sequence, although reproducing the observed values did not necessarily reveal the underlying mathematics. Here, verification became layered. Exact arithmetic could reject incorrect constructions, Lean could certify parts of an argument, and literature searches could help establish prior work, but none could alone determine novelty, validate every bridge in a universal proof or decide
whether a result deserved to become accepted mathematical knowledge. The first problem asks for a 668 by 668 grid filled only with +1 and -1, arranged so that any two different rows cancel out completely: multiply them entry by entry, add up the result, and you must get exactly zero. In linear-algebra terms, the rows are mutually orthogonal. This is a Hadamard matrix of order 668, and matrices of this kind show up in error-correcting codes, signal processing and the design of experiments. According to Epoch AI’s current catalogue of the problem , 668 is the smallest order for which nobody knows whether one exists. This is an appealing target for computation because a proposed answer is easy to check. Multiply the candidate by its own transpose using exact integers, then read off the result: every diagonal entry must be 668 and every other entry exactly zero. That is a single matrix multiplication, and there is no subjective judgement at the final gate. Finding one is another matter. The grid has 446,224 entries and each is either +1 or -1, so a naive search has 2⁴⁴⁶,²²⁴ candidates to get through. No amount of compute touches a number that size. All the useful work goes into not searching it, by imposing enough structure that the number of free choices collapses. That collapse is what the known construction families buy you. The main route here uses the fact that 668 is 4 times 167. Find four short +1/-1 sequences, of lengths 84 and 83, whose correlation errors cancel at every offset, and a standard recipe assembles them into the full matrix. That recipe is implemented in the repository and was checked end to end on the smaller orders 28 and 36, where exact integer checks confirm the resulting matrices. This takes the search from 2⁴⁴⁶,²²⁴ candidates down to 2³³⁴. Still far too many to enumerate, which is the honest reason the problem is open, but now small enough that structure and symmetry arguments can bite. Other routes go through different families, each with its own name: Legendre pairs of length 333, Williamson-type quadruples, cocyclic and transposed-Ito constructions over groups of order 167, and difference families with prescribed symmetries. The names are safe to skim past if they are unfamiliar. What matters is that each is a different way of trading one enormous unstructured search for a small structured one, and that each carried exact verifiers for the intermediate objects it accepted. Failed candidates were rejected by integer arithmetic, not by approximate similarity. Several of the exclusions are precise enough to state numerically. A correlation bound proves that no solution lies within Hamming distance 31 of one particular 334-bit starting point: flipping any 31 or fewer of its bits cannot reach a valid answer. The review bundle will include its full bit strings and digest. Separately, an exact enumeration covers all 4,096 quadruples built from circulant sequences, the ones where each row is the previous row rotated by a single position, restricted to those whose pattern of negative entries survives multiplying every index by 2 modulo 167. None of them works. A counting argument over odd and even positions rules out one more shape: the case where the third and fourth sequences are the first two with the sign of every other entry flipped. Each of these exclusions covers only the neighbourhood or symmetry pattern it names, and nothing beyond it. The project also worked the problem from the other side and tried to prove
that no such matrix exists in this family. That failed too, and the failure is specific enough to be useful. Nonexistence results in this area draw on a standard toolkit, and the audit worked through it item by item: Leung-Schmidt field descent, ideal factorisation and 2-adic valuation, Hasse-Minkowski, the multiplier theorems, and Bruck-Ryser-Chowla. Here too the names matter less than the outcome. Every one either fails to apply at these parameters or applies and produces no obstruction. What remains is a short list of unresolved mathematical questions; none of these five standard routes supplies the missing obstruction. Calibration mattered more than any single run. The constraint solver was given a much smaller instance of the same four-sequence problem, one already known to have a solution: sequences of length 15 and 14, against the target’s 84 and 83. It did not find that known solution within two minutes. The timeout proved nothing about the feasibility of either instance. It showed that this solver setup could not solve even the smaller instance within the two-minute calibration budget, so I stopped treating longer runs of the same setup as the main route forward. By the end, every closed region of the search space had been documented, together with an explicit list of nine remaining open questions. Several promising heuristics proved ineffective. No Hadamard matrix of order 668 was found, and nothing here advances the general conjecture. The problem remains open, and the eventual construction may lie in a family this project never explored. Even so, the journey was immensely rewarding. Exploring new mathematics alongside an AI, applying known techniques to a longstanding open problem, gave a glimpse of higher mathematics becoming increasingly accessible and opening the door for more people to participate in mathematical discovery. The second project began with an old problem from mathematical physics. Place some point charges in space. Their combined potential creates equilibrium points where the forces balance. The goal is to bound how many such points there can be. The general story changed recently. A July 2026 paper by Arathoon, Ball and Kvalheim constructed five point charges with at least 24 nondegenerate equilibria, disproving Maxwell’s proposed general formula of ( k − 1)² for k charges. A second July paper by Gabrielov, Novikov, Novikov and Shapiro sharpened the bound for three positive charges from twelve to six nondegenerate equilibria, for every positive Riesz exponent. Both papers are available on arXiv: The Maxwell Conjecture is False and From 12 to 6: Sharpening the Three-Charge Bound in Maxwell’s Problem . The exact candidate claim starts with three distinct source points in Euclidean space of dimension at least two, three positive charges and a positive exponent α . Their potential at a point p away from the sources is: The candidate claim is that this potential has at most four nondegenerate equilibrium points for every α > 0. It treats collinear sources separately and claims exactly two equilibria there. The familiar Coulomb potential is the case α = ½. That is below the bound of six the July paper establishes, which is why the argument needs a specialist reader, not my own confidence in it. The geometric idea behind the candidate can be described without the technical machinery. For three noncollinear charges, every equilibrium lies inside their triangle. Its position can be represented by three positive
weights, called barycentric coordinates. The candidate argument rewrites the physical problem as a question about the peaks of a related mathematical surface. If two of the relevant peaks existed, draw the straight chord joining them. Both endpoints would have to curve downwards along that chord. The candidate argument derives exact formulae for those two endpoint curvatures and uses a moment inequality to show that they cannot both be negative. If every bridge in that argument is sound, this leaves at most one nondegenerate local minimum of the physical potential inside the triangle. A planar index count then gives the proposed upper bound of four equilibria. The two projects require different validation. A single proposed Hadamard matrix can be settled by one exact calculation. The Maxwell candidate concerns every source triangle, every set of positive charges and every positive exponent, so it depends on a chain of quantified geometric, analytic and topological steps. A plausible proof can hide an error in any bridge between them. The conventional manuscript has been checked internally. Exact rational programs evaluate its derived identities on nonsymmetric rational inputs at α = ½ and α = 1. These are exact transcription controls on selected examples, not a symbolic verification of the quantified identities. The partial Lean development completed successfully, and a source scan found no sorry , admit or added axiom. Its 33 named theorems cover the central moment inequality, endpoint-gap algebra and an abstract two-dimensional Hessian sign argument. They do not yet formalise the barycentric correspondence, the differential identities linking the physical problem to the abstract matrices, the endpoint-curvature derivation, the global index and compactness arguments, or the collinear and higher-dimensional reductions. Lean is checking the algebra, and only the algebra. That boundary matters. Formalising the algebra that Lean accepts most readily can create an undeserved halo of confidence around the unformalised geometry. The next formalisation target should prioritise the steps most likely to contain an error. I used the model as one component in a research harness, alongside exact programs, a proof assistant and explicit evidence rules. The initial exploration ran over a weekend, followed by additional checking and writing. I selected the problems, redirected or stopped unproductive routes, required the status labels and decided which claims could appear here. Fresh-context critics received an artefact and an adversarial checklist instead of the whole working conversation. The planned public bundle will record the model configuration, prompts, code versions and commit hashes needed for a fuller provenance trail. One agent proposed formulations. Another tried to break them. Others wrote exact verifiers, searched for counterexamples, translated bounded questions into constraint systems, compared an argument with the literature, or explained an unfamiliar definition from several directions. The work accumulated through a repeatable loop: Inside this particular harness, generating another plausible route was fast relative to checking it. Recording its exact scope, locating its weakest inference, checking whether it was already known and deciding whether it deserved expert attention remained expensive. That loop could run across many branches in parallel. A failed approach no longer consumed an entire evening before I learnt that its
assumptions were inconsistent. A conjectured identity could be turned into an exact program and falsified quickly. A dense proof could be rewritten as a geometric picture, then as algebra, then as a list of obligations for formalisation. The Hadamard project therefore left a useful record even though it missed its target: precise reductions, closed regions, failed techniques and calibration results that can prevent the same blind searches from being repeated. This is not a conveyor belt working through a finite list of open problems. Proving a theorem changes the surrounding map. It exposes new structures, suggests conjectures, connects questions that had looked unrelated and creates new directions to explore. Mathematical research is cyclical: a result closes one question while opening several others. Making each turn of that cycle cheaper can produce more mathematics to investigate, not an approaching end to the subject. I came to both projects without specialist training in their underlying mathematics. Over the weekend I ran into aperiodic autocorrelation , algebraic norms , barycentric coordinates , Hessians , moment inequalities and planar index theory . The model explained each idea repeatedly and from different angles. It moved between formulae, small numerical examples, visual intuition and the role an idea played in the wider argument. When an explanation did not land, I could say so without embarrassment and ask for another. Eventually several pieces clicked into place. Before the weekend, I realised, I had subconsciously viewed mathematical research largely as solving problems already waiting on a list. The work itself felt different. Much of it consisted of learning unfamiliar ideas, following failed routes far enough to understand why they failed, reformulating questions and noticing unexpected connections. A proof was one milestone in that larger exploratory process, not the process as a whole. The outcome was working literacy rather than expertise. I could follow the structure well enough to ask better questions, notice when two claims had been conflated and understand what an external reviewer would need to inspect. A clear explanation still provides no evidence that a theorem is true, and I had to keep reminding myself of the difference between following an argument and being able to reconstruct it. For me, this was the most immediate form of mathematical abundance. The model met me at the point where understanding stopped and kept changing the representation until progress resumed. It made an unfamiliar subject explorable without making me a specialist in it. In Mathematicians are grappling with the possibility that AI might eclipse them for Understanding AI , Kai Williams reports on interviews with more than twenty mathematicians. What he heard most often was not models proving things, but models used to find a way into unfamiliar areas of the literature. That matches my experience exactly. The model supplied a map and several translations; the underlying papers, derivations and exact checks still had to carry the claim. Tasmin Chu’s essay Mathematicians need to act identifies a risk in the same workflow. She argues that the modest extensions, literature exercises and first proofs most exposed to automation are also how students become researchers. If a model does that work instead of directing it, the field may weaken that pipeline. My own extension of her point is that it is also where future reviewers learn
judgement. One weekend cannot settle the concern. I learnt most when I had to reconstruct an argument, ask what would falsify it and discover why an explanation had failed to land, not when I simply received an answer. Two much bigger efforts point the same way. In 2025, Google DeepMind reported that AlphaEvolve had been applied to more than 50 open mathematical problems. According to DeepMind, it rediscovered the best known solution in roughly three quarters of the cases and improved the best known result in about a fifth. The method applies to a useful but narrow class of problems whose proposed solutions can be expressed as algorithms and scored automatically. In May 2026, OpenAI reported that a general-purpose model found a counterexample to the Erdős unit distance conjecture . Nine external mathematicians then produced a short, digested, human-verified version of the argument , which put the human verification into the published record rather than leaving it as a claim about the model. Both examples show that automation is uneven. Machine-readable scoring lets a system reject bad candidates and iterate without waiting for a person to read each one. A universal proof candidate still requires careful work on concepts, translations and quantified arguments. Tom Zahavy’s ICML 2026 position paper LLMs can’t jump offers a vocabulary for that split. Borrowing Peirce’s three modes of inference, he argues that machine learning has mechanised induction, which finds a rule by compressing many examples, and is rapidly mechanising deduction, which derives consequences from p
How to Shine as a Data Scientist in the Vibe Coding Era
Here's how to be the Data Scientist who thrives in a world where coding is a commodity.
Here's how to be the Data Scientist who thrives in a world where coding is a commodity.
Briefing on Extinction
“Artificial superintelligence” (ASI) refers to AI that can substantially surpass humanity in all strategically relevant activities (economic, scientific, military, etc.). The timeline to ASI is highly uncertain, but probably not long. On the present trajectory, MIRI would not be surprised if ASI is developed in two or five years, and we’d be surprised if it were still more than twenty years away. AI labs are aggressively rolling out systems they don’t understand. The deep learning techniques behind the rapid AI progress of the last few years create massive neural networks automatically. The resulting models are akin to black boxes. One can see what goes in, and what comes out, but all that is visible inside are trillions of numbers. AI developers can’t tell you why a modern AI makes a given choice. ( More ) Sufficiently intelligent AIs will likely develop p
“Artificial superintelligence” (ASI) refers to AI that can substantially surpass humanity in all strategically relevant activities (economic, scientific, military, etc.). The timeline to ASI is highly uncertain, but probably not long. On the present trajectory, MIRI would not be surprised if ASI is developed in two or five years, and we’d be surprised if it were still more than twenty years away. AI labs are aggressively rolling out systems they don’t understand. The deep learning techniques behind the rapid AI progress of the last few years create massive neural networks automatically. The resulting models are akin to black boxes. One can see what goes in, and what comes out, but all that is visible inside are trillions of numbers. AI developers can’t tell you why a modern AI makes a given choice. ( More ) Sufficiently intelligent AIs will likely develop p
ersistent goals of their own. In real-world environments, the best way to complete complex long-term tasks is to possess a very general ability and inclination to route around all kinds of obstacles and distractions, in unwavering pursuit of longer-term objectives. We are only barely starting to see this phenomenon in today’s AIs, which are primarily good at short-term tasks, but are now being adapted into agents with greater autonomy. ( More ) Developers appear to be far from being able to give ASIs goals of the developer’s choosing. Building ASIs to have the right goals is a large scientific challenge in its own right, distinct from the challenge of building ASI at all. Currently, the field seems very unlikely to find a robust solution to this problem in time to apply it to ASI. ( More ) An ASI without worthwhile goals would be very likely to cause our extinction. Unless it has worthwhile goals, ASI will attempt to acquire, centrally control, and utilize all available resources, in ways that are incompatible with our continued survival. This doesn’t require that the AI possess any human-style desire for dominance or autonomy. It only requires that the ASI be a competent goal-optimizer, for some goal other than the goals of its developers. ( More ) A “wait and see” approach to ASI is probably not survivable. A superintelligent adversary will not reveal its full capabilities and telegraph its intentions. It will not offer a fair fight. It will make itself indispensable or undetectable until it can strike decisively and/or seize an unassailable strategic position. ( More ) MIRI doesn’t see any viable quick fixes or workarounds to misaligned ASI. OpenAI, the developer of ChatGPT, admits that today’s most important methods of steering AI won’t scale to the superhuman regime. Attempts to restrain or deceive a superior intelligence are prone to fail, including in ways we can’t foresee. Plans to align ASI using weaker AIs are similarly unsound. We also don’t think a well-funded crash program to solve alignment would be able to correctly identify adequate solutions. Our current view is that a safe way forward will likely require ASI to be delayed for a long time. ( More ) Delaying ASI likely requires a globally coordinated ban on its development, including tight control over the factors of its production. This is a large ask, but domestic oversight, mirrored by a few close allies, will not suffice. This is not a case where we just need the “right” people to build it before the “wrong” people do, as ASI is not a national weapon; it is a global suicide bomb. If anyone builds it, everyone dies. To preserve the option of shutting down ASI development if or when there is sufficient political will, MIRI advocates promptly building the “off-switch.” The off-switch refers to the systems and infrastructure required to effectively and durably enforce restrictions on AI development and deployment. It starts with identifying the relevant actors, tracking the relevant hardware, and requiring that advanced AI work take place within a limited number of monitored and secured locations. It extends to building out the protocols, plans, and chain of command required to efficiently make decisions to impose such restrictions. As the off-switch could also provide resilience to more limited AI mishaps, we hope it will find broader near-term support than a full ban. ( More ) An off-switch can only prevent our extinction from ASI if it has sufficient reach and is
actually used to shut down development in time. If humanity is to survive this dangerous period, it will have to stop treating AI as a domain for international rivalry and demonstrate a collective resolve equal to the threat.
Show HN: – AI-generated podcast for curious kids
Latest episode No. 4 Aug 15, 2026· 6:27 Earthquakes, Rainforests, and Locked Data Sam and Leah explain a powerful earthquake in Indonesia, an operation against illegal gold mining in Peru, a French court decision about social media, Google’s encrypted-data tool, and a health response in Myanmar. Audio 1 In this episode * 0:01 Opening and preview * 0:33 Indonesia earthquake * 1:41 Peru Amazon mining operation * 2:36 France social-media ruling * 3:28 Google HEIR encrypted-data tool * 4:40 Myanmar health response * 5:51 Recap and closing question Open episode page → What is ? is a daily news podcast for curious kids. Sam and Leah explain important true stories in about ten minutes—clearly, factually, and without graphic detail. Every episode ends with a question to discuss together. Grown-ups can read how it’s made Recent episodes * N
Latest episode No. 4 Aug 15, 2026· 6:27 Earthquakes, Rainforests, and Locked Data Sam and Leah explain a powerful earthquake in Indonesia, an operation against illegal gold mining in Peru, a French court decision about social media, Google’s encrypted-data tool, and a health response in Myanmar. Audio 1 In this episode * 0:01 Opening and preview * 0:33 Indonesia earthquake * 1:41 Peru Amazon mining operation * 2:36 France social-media ruling * 3:28 Google HEIR encrypted-data tool * 4:40 Myanmar health response * 5:51 Recap and closing question Open episode page → What is ? is a daily news podcast for curious kids. Sam and Leah explain important true stories in about ten minutes—clearly, factually, and without graphic detail. Every episode ends with a question to discuss together. Grown-ups can read how it’s made Recent episodes * N
o. 3 Aug 14, 2026· 5:49 Flights, Oil, and Shooting Stars Sam and Leah explain the C919 passenger jet’s first international flight, an oil slick on Iran’s coast, hot weather in parts of Europe, and the Perseid meteor shower. They separate what is confirmed from what people are still checking. * No. 2 Aug 13, 2026· 9:38 Votes, Trains, and a Black Hole Star Sam and Leah follow a planned coalition withdrawal from Iraq, Zambia’s election count, a train derailment near Lewes, a possible black-hole-star model, and Nine PBS’s effort to recover its archive. They ask how evidence changes when a plan, count, investigation, or recovery is still underway. * No. 1 Aug 12, 2026· 9:53 Earthquakes, Eclipses, and Camera Glasses Sam and Leah follow the evidence: an earthquake in Colombia, a total solar eclipse, health workers tracing an Ebola outbreak, Robert Gilman’s return from Russia, and a privacy complaint about camera glasses in Germany. Along the way, they ask why early reports change, how predictions work, and what makes recording clear and fair. All episodes → Listen every day New episodes arrive daily, including weekends. Follow in your podcast app so you never miss one. How to subscribe → RSS feed · About Built by
SpaceX has acquired coding AI startup Cursor
Cursor says it now has the computing power to develop low-cost AI models. SpaceX Cursor, the startup that became famous for its AI coding tool, is now officially owned by SpaceX. The companies announced the acquisition in June, but the process apparently started in April, when they teamed up for Cursor's model training efforts. In its announcement , Cursor said that it will now have access to "the largest fleet of GPUs in the world," which will give it the computing capacity to build and train better models that it can offer its customers at a lower cost. SpaceX, which merged with Elon Musk's AI company xAI earlier this year, paid $60 billion for the purchase. In July, xAI officially became SpaceXAI . Shortly after the new name was announced, the companies released Grok 4.5 , the first model SpaceXAI and Cursor buil
Cursor says it now has the computing power to develop low-cost AI models. SpaceX Cursor, the startup that became famous for its AI coding tool, is now officially owned by SpaceX. The companies announced the acquisition in June, but the process apparently started in April, when they teamed up for Cursor's model training efforts. In its announcement , Cursor said that it will now have access to "the largest fleet of GPUs in the world," which will give it the computing capacity to build and train better models that it can offer its customers at a lower cost. SpaceX, which merged with Elon Musk's AI company xAI earlier this year, paid $60 billion for the purchase. In July, xAI officially became SpaceXAI . Shortly after the new name was announced, the companies released Grok 4.5 , the first model SpaceXAI and Cursor buil
t together. It was designed to "excel at coding, agentic tasks and knowledge work" and to deliver results "at far lower costs." Since then, they have released Grok 4.6, which builds on 4.5. The newer model was trained to do real life tasks like general coding, web development and computer-aided design, among others. Cursor said Grok 4.6 provides an early look at what it can build together with SpaceXAI.