Research Paper

The CKM matrix encodes how quarks of one flavor convert into another during weak decays. The Standard Model requires it to be unitary — the probabilities in each row must sum to exactly one. For years, the first row has shown a mild but stubborn deficit, driven largely by measurements of Vus from kaon decays. Either the Standard Model is incomplete, or something in the kaon analyses is off. BESIII attacks the question from an entirely different direction. Working at Beijing's electron-positron collider, the team uses quantum-entangled lambda–antilambda hyperon pairs. Because the pair is produced in an entangled state, observing the antilambda tags the polarization of the partner lambda, which then decays semileptonically into a proton, an electron, and an undetectable antineutrino. That polarization information is exactly what is needed to extract Vus cleanly from a baryon decay — a technique proposed roughly 35 years ago but never practically executed. The measurement supports unitarity, and its mild tension with the kaon-based value is itself informative: it suggests the anomaly may lie in the kaon sector rather than in new physics. Published in Nature on 7 September 2026.

Long-context inference is bottlenecked less by compute than by memory. Every token a transformer has already read must keep its key-value cache resident, so serving cost scales linearly with context length and concurrent users. Shrinking bytes-per-token is the single highest-leverage knob in LLM serving economics. This report stacks several techniques. Compressed Sparse Attention 2 (CSA2) assigns each attention layer a static mode — Full, Reindex, or Reuse — letting layers share KV data outright and reuse sparse-attention indices computed elsewhere in the stack, rather than each layer maintaining its own. On top of that sits FP4 KV quantization in E2M1 format with per-16-channel scaling factors. The architecture also revises DeepSeek's hyper-connection design into a Single-Pass mHC with a Mega-mHC deployment kernel that halves activation memory traffic. The headline number is 890 bytes per token — roughly 4× smaller than DeepSeek V4 Flash and about 437× smaller than DeepSeek V1 — with quality maintained. The report is unusually specific about the deployment kernels, not just the architecture. As with previous DeepSeek releases, the significance is as much economic as technical: these ratios change what context lengths are affordable to serve at scale, and the weights are open.

Post-training normally runs as a batch process: collect data, fine-tune, ship, repeat. This paper asks what happens when deployment itself becomes the data-collection loop, and whether that loop can be closed tightly enough to produce genuine self-improvement. The system serves users from a heterogeneous pool of models behind a router. For every turn, it logs the router's predicted capability demand, which service tier it chose, and how the interaction went. Those traces are converted into training examples, filtered by automated validation and semantic assessment, and fed into a three-stage curriculum combining supervised fine-tuning with routing-guided distillation from stronger tier models into weaker ones. The result is an evaluation–selection–update cycle where what the system has learned to do determines which interactions it sees next, and therefore what it learns from next. The authors report substantial benchmark gains across 4B and 9B variants. The framing — "recursive self-improvement" — is deliberately provocative, and the reported loop is bounded rather than open-ended: gains come from distilling an existing capability gradient within the model pool, not from creating new capability from nothing. It was the most-upvoted paper on Hugging Face in early September.

Language models are trained to predict the next token, a target that is cheap to compute but semantically myopic: the model is never explicitly asked to represent what it is about to say, only the next fragment of how it will say it. A long-running research thread asks whether adding a coarser, meaning-level objective improves sample efficiency. NCP-ArchPreview implements one concrete version. From the model's own hidden states, the authors build a product-quantized "concept vocabulary" — discrete codes spanning multiple tokens. A dedicated Concept Module predicts the next concept, and that prediction conditions ordinary token-level generation. Both objectives train jointly, so the model keeps standard autoregressive decoding while acquiring a latent semantic space alongside it. At 8.9B parameters trained on 5.73T tokens of Dolma-3, the model reaches OLMo-3-7B's final loss using 51.3% of the training tokens, and scores 2.45 points higher on a downstream macro-average, including a 5.99-point gain on GSM8K. The latent space also proves reusable: domain adaptation works by updating only a 17M-parameter VQ module, and speculative decoding gains 4.17% in mean accepted token length.

The four color theorem — every planar map can be colored with four colors so no two neighboring regions match — was famously proven by Appel and Haken in 1976 using a computer, and reproven by Robertson, Sanders, Seymour and Thomas in 1996. Both proofs work by exhibiting an unavoidable set of "reducible configurations," local patterns that can be shrunk and recolored. Both guarantee only that at least one such configuration exists in any triangulation, which forces a coloring algorithm to peel the graph one configuration at a time and costs quadratic time. This paper strengthens the structural core: every planar triangulation contains linearly many non-overlapping reducible configurations, or else short obstructing cycles. Because many reductions can be applied in parallel rather than sequentially, the authors obtain a 4-coloring algorithm running in O(n log n) time, improving the thirty-year-old O(n²) bound. Getting there required finding reducible structure in the flat regions of a graph, not just the high-curvature ones where earlier proofs concentrated. Posted in March 2026 and slated for FOCS in November, it drew broad attention in September as one of the rare genuinely new proofs of a theorem most mathematicians considered closed.

The three-dimensional incompressible Navier–Stokes equations describe ordinary fluid flow, yet nobody has known whether their solutions always stay smooth or can spontaneously develop infinite velocity — the "blowup" question that is one of the seven Clay Millennium Prize Problems. This preprint constructs an explicit solution that blows up. For every positive viscosity, the authors build a flow starting from rest, driven by a smooth, compactly supported external force, whose velocity becomes unbounded in finite time while total kinetic energy stays uniformly bounded. The singularity forms through a concentrating vortex that spirals inward and stretches into an increasingly thin filament. The construction was produced by roughly 10,000 coordinating AI agents running for about 88 hours, and ships with both a human-readable analytic proof and a Lean formalization. The caveat matters: Clay's official problem statements concern the unforced equations, so a forced blowup is a strictly weaker result and does not, on its face, claim the prize. The release also triggered a priority dispute, with NYU mathematician Tristan Buckmaster arguing the effort was prompted by — and possibly trained on — his unpublished work. Lean verification guarantees the formalized statement follows from its hypotheses, not that the hypotheses match the Millennium Problem.

A recurring frustration with long-horizon coding and terminal agents is that they fail on tasks whose individual steps the underlying model handles fine. The failures are structural rather than cognitive: the agent loses track of state across a long run, forgets what it learned from an earlier command, skips a procedural step, or declares victory early. This paper's argument is that the fix belongs in the harness, not the weights. StateM is an agent-native runtime built around durable state that survives across steps, context scoped to the current phase rather than dumped wholesale, explicitly validated transitions between phases, recoverable runbooks so a failed attempt can resume rather than restart, and versioned procedures that a human can inspect. None of it requires training. On Terminal-Bench 2.1 the same underlying model goes from 83.1% to 92.1% accuracy with GPT-5.5, and reaches 95.3% raw accuracy with GPT-5.6 over 445 trials. The cost result is arguably more striking: paired with DeepSeek-V4 Flash, a full benchmark run costs about $15 versus $574.68 for the GPT configuration, at competitive accuracy. Held-out gains on BusinessBench suggest the runbook mechanism transfers rather than overfitting to the benchmark. 'Harness scaling' is the paper's name for the broader claim — that a meaningful slice of remaining agent capability is engineering, not intelligence.

Brain–computer interfaces usually assign an arbitrary mapping from neural activity to control output and then ask the user to adapt. Learning is slow and often fails outright, and it has never been clear why some mappings are learnable and others are not. Busch and colleagues test a hypothesis borrowed from monkey motor-cortex work: neural activity does not fill its high-dimensional space uniformly but lies on a lower-dimensional curved 'intrinsic manifold,' and a person can only easily learn control schemes that live within that manifold. The novelty is doing this noninvasively, in humans, and in higher-order cognitive regions rather than motor cortex. Participants controlled a video-game avatar via real-time fMRI feedback from brain areas supporting spatial navigation, with the manifold extracted using nonlinear data-diffusion methods. The result is sharp. When the control mapping was aligned with directions of high variance on the intrinsic manifold, participants gained control quickly, realigning their activity along those directions. When the mapping pointed off-manifold, they could not learn it at all — not slowly, but not at all, within the session. That clean dissociation makes manifold geometry look like a real constraint on cognitive learning rather than a descriptive convenience, and it gives BCI designers a concrete recipe: measure the manifold first, then choose a decoder that respects it.

Nearly all recent progress on hard reasoning benchmarks has come from making models think out loud: chain-of-thought traces, long sampling budgets, and inference costs that can run to dollars per task. BDH-CQ argues that the reasoning does not have to happen in token space at all. The model combines in-context learning with recurrent latent reasoning — instead of emitting intermediate steps, it iterates computation on a query inside a high-dimensional latent state, letting the recurrence do the work that a chain of thought would otherwise do explicitly. The architecture descends from Pathway's 'Baby Dragon Hatchling' line of work on biologically-flavored recurrent networks rather than from the standard decoder-only transformer recipe. The headline number is the reason this became the most-upvoted paper on Hugging Face in August: a 150-million-parameter model reaches 29.5% pass@2 on ARC-AGI-1 at roughly $0.0007 per task. ARC-AGI is the abstraction-and-reasoning benchmark specifically designed to resist memorization, and models scoring in that range have typically been many orders of magnitude larger and more expensive to run. The claim is not that it beats frontier systems on accuracy, but that it establishes a dramatically better point on the cost–accuracy frontier. It is a preprint and the benchmark is narrow, so the open question is how far latent recurrence generalizes beyond ARC-style puzzles.

Quantum processors drift. Control parameters that were optimal an hour ago are slightly wrong now, and the usual remedy is to stop the computation and recalibrate — which does not scale to a machine expected to run for hours or days. Worse, once a processor is running error correction you cannot measure its qubits directly to see what is wrong, because that would destroy the encoded state. The insight here is that error correction already produces a continuous stream of diagnostic information for free: the syndrome measurements it makes to detect errors. Sivak, Morvan and a ~300-author team from Google Quantum AI and DeepMind repurpose that stream as a reward signal, training a reinforcement learning agent to steer control parameters in real time using only error-detection events. The processor calibrates itself while computing, never pausing and never breaking the code. Run on Google's Willow superconducting chip, this is the first demonstration of RL-driven control at the scale of a complete error-corrected processor. It reached record logical error rates and — more importantly for practical machines — markedly better resilience to parameter drift over long runs. The result is notable as much for the architectural idea as for the numbers: it turns the overhead of error correction into a feedback channel rather than a pure cost.

Neuroscience's global workspace theory holds that only a small slice of the brain's processing is 'broadcast' to a shared arena where it becomes reportable and available to flexible reasoning — the rest runs automatically and silently. This Anthropic paper asks whether large language models have anything like that, and finds a surprisingly clean answer. The core tool is the Jacobian lens: rather than reading an activation through the model's output head directly, the authors linearize all downstream layers and average that linearization across many contexts, revealing which concepts an internal representation is *poised to say* regardless of what it actually says. The subspace this exposes — the 'J-space' — accounts for only about 10% of activation variance and lives in the middle layers of the network. It satisfies five properties the theory predicts. Swapping J-space vectors changes what the model reports. Instructing the model to 'hold a thought' plants that concept there without it appearing in output. Intermediate reasoning steps that are computed but never verbalized show up as J-lens vectors, and swapping them redirects multi-step conclusions. A single vector for 'France' works as input to unrelated downstream queries (76% swap success). And ablating the J-space breaks multi-hop reasoning, analogy and translation while leaving simple classification and text continuation intact.

In 1968 John Milnor asked whether every complete Riemannian manifold with non-negative Ricci curvature must have a finitely generated fundamental group — i.e. whether curvature bounds force the space's loop structure to be describable by finitely many generators. It became one of the defining open problems of comparison geometry, resisting attack for over half a century until Bruè, Naber and Semola constructed counterexamples in dimension six and above in 2023. Since the conjecture is known to hold in dimensions three and below, that left dimensions four and five as the last unresolved cases. Wu and Yan close them. They construct complete, one-ended manifolds in dimensions four and five with strictly positive Ricci curvature whose fundamental group is the Prüfer 2-group — an infinitely generated group built as the union of an increasing tower of cyclic groups of order 2^k. The Prüfer group is precisely the right shape for the job: it is a limit of finite pieces, so it can be realized by an infinite sequence of increasingly fine topological surgeries, each one small enough that positive Ricci curvature survives. The construction is delicate — 62 pages of careful gluing and curvature estimates — and notably achieves strictly positive, not merely non-negative, Ricci curvature. With this, Milnor's conjecture is settled in every dimension.

Direct-detection experiments hunt for dark matter by watching for a xenon nucleus to recoil after being struck by a passing particle. Historically these searches concentrated on low recoil energies, where the standard spin-independent WIMP signal is expected to pile up. LZ, a 7-tonne liquid xenon detector nearly a mile underground in South Dakota, reopened the question by extending its search window all the way up to ~270 keV — a range where effective-field-theory and inelastic dark matter models predict a much larger share of the signal, and where backgrounds are expected to be almost nonexistent. In 2.84 tonne-years of exposure they found exactly one event: a nuclear recoil at 248 ± 23 (stat) ± 23 (sys) keV, sitting in a region where essentially nothing should appear. A profile-likelihood analysis puts the tension with a background-only hypothesis at 3.4σ local, falling to 2.6σ global once the look-elsewhere effect is accounted for. The collaboration is careful not to claim a discovery — one event is one event, and rare background topologies are hard to fully exclude. But the community reaction was immediate: within a day of the September 1 announcement at TeVPA, several theory preprints appeared proposing inelastic and fermionic-absorption dark matter models to fit the recoil energy.

Mars is a planet with two faces: smooth low plains across the northern hemisphere, rugged cratered highlands across the south. Whether this crustal dichotomy came from a giant impact or from internal mantle convection has been argued for decades, and surface geology alone cannot settle it. Alexander Berne and colleagues attacked the question from the inside, using tidal tomography — a technique previously applied to Earth and the Moon but never to Mars. As Mars traverses its 687-day orbit, solar gravity flexes it by minute amounts, and how much it flexes depends on the stiffness of the rock doing the flexing. The team reprocessed sixteen years of radio tracking data from Mars Global Surveyor, Mars Odyssey and Mars Reconnaissance Orbiter to extract the time-varying gravity signal with enough precision to resolve lateral variation. The mantle beneath the southern hemisphere turns out to be more than 20% less rigid than the north, implying rock 200–400 °C hotter and possibly partially molten. The boundary between stiff and soft mantle sits directly beneath the surface dichotomy. That alignment is the crux: a several-billion-year-old thermal asymmetry still detectable today favours an internal, convective origin over a single catastrophic impact, and bears on where Martian water and volcanism could have persisted.

Essentially every entangled-photon source ever built has been pumped by a laser. The standard method, spontaneous parametric down-conversion, sends a pump beam into a nonlinear crystal where individual photons occasionally split into correlated pairs — and the folk assumption has been that the pump's coherence is what makes the entanglement possible. A team from the University of Ottawa and the Max Planck Institute for the Science of Light tested that assumption by replacing the laser with the Sun. They built a window-sized Fresnel lens and concentrator that funnels collected sunlight into a single-mode optical fibre roughly the width of a human hair, polarise it, and shine it on the crystal. Sunlight is intensely incoherent in both space and time; passing it through the fibre selects a single spatial mode without making it temporally coherent. The pairs that emerged had a fidelity of about 94% to a maximally entangled state and violated a Bell inequality — the standard signature that the correlations cannot be reproduced by any classical model. The experiment ran outdoors. The physical lesson is that phase matching in the crystal, not pump coherence, is what the process requires. The practical lesson is that entanglement generation need not carry a laser's power budget.

Discrepancy theory asks how evenly a set of objects can be split into two groups. In the Komlós setting you are given a matrix whose columns have Euclidean norm at most one, and you must assign each column a ±1 sign so that every coordinate of the resulting sum stays small. Komlós conjectured in the early 1980s that a universal constant suffices, independent of the number of columns or dimensions. Banaszczyk proved an O(√(log n)) bound in 1998, and there it sat. Hajela had conjectured a matching Ω(√(log n)) lower bound, so many believed the problem was essentially closed short of the conjecture. Bansal and Jiang break through with O((log n)^{1/4}(log log n)^{7/4}) — the first asymptotic improvement in nearly thirty years, and a refutation of Hajela's conjecture. The method is algorithmic: a carefully steered random walk that tracks not only accumulated discrepancy but also *dependency*, meaning how a perturbation in one coordinate propagates to others. Controlling that joint influence at each step is what buys the extra square root. The algorithm is efficient, not merely existential. This exposition, posted 28 August 2026, is a self-contained rewrite of the STOC 2026 result, which was first announced in autumn 2025 and has drawn renewed attention through recent popular coverage.

"AI scientist" systems have mostly been evaluated in silico: they propose hypotheses, and the evaluation stops there. This paper reports on Co-Scientist, a Gemini-powered multi-agent framework taken through the full loop — hypothesis generation, experimental design and execution, and manuscript writing — with real laboratory validation. The system orchestrates specialised agents that generate, debate, rank and evolve hypotheses against literature and structured databases, then hands designs to execution environments. The authors report three real-world campaigns: in materials science, designing MXene precursor synthesis routes and growing monolayer semiconductors; in biology, predicting the swarming behaviour of engineered bacteria; and in computer science, discovering improved inference-time-scaling architectures. Layered on top are reliability mechanisms aimed at the failure modes that have dogged automated research — hallucinated results, inadvertent plagiarism, and unsafe experimental proposals — with reported hallucination rates driven down to around 4%. The evaluation that gives the paper its weight is a double-blind study in which 30 domain experts assessed end-to-end machine-generated papers, rather than a benchmark score. The author list spans Google DeepMind, Google Research and academic collaborators, and follows Co-Scientist's earlier Nature appearance in May 2026. The honest reading is that this is closed-loop assistance under human supervision, not autonomous science — but it is the closest anyone has published.

Nearly all frontier reasoning systems think by writing: they emit long chains of tokens, which is expensive, slow, and — as a growing body of work shows — not always causally connected to the answer. BDH-CQ asks whether the reasoning can happen inside the network instead. The model is built on the Dragon Hatchling (BDH) architecture, a post-Transformer design from Pathway Research organised around very high-dimensional sparse positive activations, low-rank communication between units, and a recurrent associative state that updates in a Hebbian-like fashion. In BDH-CQ, in-context demonstrations stream into that recurrent memory rather than sitting in an attention window; the model then answers a query by iterating in continuous latent space, never verbalising intermediate steps. The striking result is on ARC-AGI-1, the abstraction benchmark that has resisted scaling: a 150-million-parameter configuration reaches 29.5% pass@2 at roughly $0.0007 per task, a state-of-the-art cost-efficiency point on that benchmark by a wide margin. The authors add controlled interventions probing how demonstrations are absorbed, whether learned transformations stay consistent, and which concept classes remain out of reach. The paper has drawn heavy attention partly because the numbers come from a model three orders of magnitude smaller than typical contenders, and an independent open reimplementation appeared within days.

This work asks whether large language models maintain internal state that shapes their behaviour but never surfaces in their output — a question that matters both for interpretability and for safety monitoring of models whose visible chain-of-thought may be unfaithful. The team introduces the "Jacobian lens" (J-lens), which uses the sensitivity of output logits to internal activations to isolate a compact subspace of Claude's representations, dubbed the J-space. These directions turn out to be interpretable, tied to specific concepts, and roughly 100× more densely connected to the rest of the network than ordinary activation patterns — the kind of broadcast topology that lets information reach many downstream computations at once. The central empirical claim is that the J-space satisfies five functional properties cognitive scientists associate with conscious access in humans: the model can report its contents, modulate them on request, use them in reasoning, apply them flexibly across tasks, and still handle routine processing when they are suppressed. Ablating the J-space entirely degrades performance across fourteen tasks, but unevenly — routine work survives, deliberate reasoning does not. Crucially, nothing about this structure was designed in; it emerged from ordinary training. Neel Nanda's group at Google DeepMind published an independent partial replication on an open-weight model, and Stanislas Dehaene and Lionel Naccache — architects of global neuronal workspace theory — contributed commentary.

On 1 August 2026 OpenAI released a 249-page manuscript claiming that an internal version of its unreleased model, Astra, produced solutions to ten problems in pure mathematics and theoretical computer science, each open for at least a decade. The problems span group theory, convex geometry, coding theory, quantum complexity and lattice cryptography. The headline result is the first explicit construction of a non-sofic group, settling a question left open since Mikhail Gromov introduced soficity in 1999. Other claims include a disproof of Connes's rigidity conjecture (infinitely many non-isomorphic property (T) groups sharing a von Neumann algebra), a resolution of Ehrhart's volume conjecture, an improved high-dimensional sphere-packing bound approaching the Cohn–Elkies threshold, a new lower bound for computing the permanent, a hardness factor for the Euclidean closest vector problem, exponential parallel repetition for two-player quantum games, and resolutions of Erdős problems 146, 180 and 183. What distinguishes this from earlier AI-math announcements is verification: every proof ships with a machine-checkable Lean 4 certificate on GitHub under Apache 2.0, with a "sorry" count of zero. OpenAI estimates the total inference cost of finding all ten at roughly $2,000. Mathematicians have pushed back on the opacity — the model, training data and methods are inaccessible, and journal peer review is still pending.

Elliptic-curve cryptography secures most of the modern internet, and Shor's algorithm breaks it — but only on a quantum computer large enough to run it. The practically important question is how large. Resource estimates translate directly into a timeline for when today's encrypted traffic, harvested now and decrypted later, becomes readable, and into how urgently the world needs to finish migrating to post-quantum schemes. This paper drives the logical-qubit requirement for a 256-bit prime-field curve down to 835, from previous published estimates of 1098 and 1175 — and from the same group's own 1,333 just three months earlier. The bottleneck it removes is modular inversion, which dominates the space cost of affine-coordinate point addition. Starting from the extended Euclidean algorithm, the authors refine register-sharing with two new ingredients — length registers and location-controlled arithmetic — to build a reversible modular inversion circuit using only 2n + 6⌊log₂n⌋ + O(1) logical qubits. The full algorithm needs 3n + 6⌊log₂n⌋ + O(1) logical qubits and roughly 919n³/log₂n Toffoli gates. The caveat worth stating plainly: these are logical qubits. Error correction still multiplies the physical requirement by a large factor, so this does not mean a machine is imminent. It does mean the target keeps moving in one direction.

Quantum chromodynamics makes a prediction with no analogue in the rest of the Standard Model: because gluons carry color charge, they should be able to bind to each other and form particles made purely of force, with no quarks at all. Glueballs have been predicted since the 1970s and searched for ever since. The difficulty is that a glueball's quantum numbers are shared by ordinary quark-antiquark mesons, so any candidate can be reinterpreted as a conventional state or a mixture, and every claimed sighting has died on that ambiguity. BESIII, at Beijing's electron-positron collider, analyzed ten billion J/ψ decays — the largest such sample ever collected — targeting X(2370), a state the collaboration first flagged years ago. The new analysis searched for a specific decay mode and found nothing, setting an upper limit on its branching fraction. That null result is the crux: it establishes X(2370) as a flavor-singlet state, the first light hadron above 1 GeV/c² shown to be one, and flavor-singlet behavior is a glueball signature that quark-model alternatives struggle to reproduce. The collaboration argues that a dominant glueball component is the only interpretation consistent with the full set of observations at once: the mass, the 0^{-+} spin-parity, the production rate in radiative J/ψ decay, the resemblance of its decay pattern to the η_c, the narrow width, and the suppression of certain radiative modes.

This twelve-page essay, based on Tao's public lecture at the 2026 International Congress of Mathematicians, makes a deliberate rhetorical move: it refuses to argue about whether AI systems will become capable of research-level mathematics. Instead it conditions on that hypothesis and asks a question Tao considers orthogonal and more urgent — what are the goals and values of mathematical research in the first place? He uses problem-solving as the case study. Solving a problem produces a truth value, but it also produces things the community actually depends on: techniques that transfer, intuitions about why a statement is true, training for the person who solved it, and a shared culture of what counts as understanding. If a machine supplies the truth value cheaply, those secondary goods do not follow automatically, and they were never explicitly priced because they were bundled with human labor. Tao works through the implications for education, publication norms, refereeing, credit, and what the profession should reward. The essay is notably non-alarmist and non-triumphalist. It reads as an invitation to the mathematical community to make explicit a set of values that have until now been tacit — before the tools force the question. Coming from the most visible working mathematician of the era, and delivered from the ICM podium, it functions as an agenda-setting document rather than a technical contribution.

In 1954 George Pólya conjectured that Weyl's asymptotic formula for the eigenvalues of the Laplacian is not merely asymptotically correct but a strict one-sided bound at every step — Dirichlet eigenvalues always sit above the Weyl count, Neumann eigenvalues always below it. Spectral geometers have chipped at this for seventy years. Pólya proved it for tiling domains; the same four-author group recently settled the Dirichlet case for balls in all dimensions and the two-dimensional Neumann case, and the conjecture is known to be false in general for spheres. This paper closes the Neumann case for Euclidean balls in dimension three and above. The obstacle that made the higher-dimensional Neumann problem hard is technical but decisive: instead of controlling zeros of ordinary Bessel functions, one must control zeros of derivatives of ultraspherical Bessel functions, for which the classical toolkit is far thinner. The proof stitches together several ingredients — sharp phase-function bounds for those Bessel derivatives, lattice-point counting to handle the bulk of the spectrum, and computer-assisted finite checks. For the low-lying eigenvalues, where asymptotic methods have no purchase, the authors introduce variational estimates using dimension-dependent test functions, a technique they highlight as new and of independent reach. An independent proof of the same result was posted within days by Yutian Li.

"Zero RL" means applying reinforcement learning with verifiable rewards directly to a base model, with no supervised reasoning traces and no human-annotated chains of thought — the model has to discover how to reason from reward signal alone. Prior studies of this regime were confined to relatively small models by compute limits, leaving open whether its observed dynamics survive at frontier scale. This work runs zero RL on a one-trillion-parameter mixture-of-experts model. Getting there required real engineering: clipped importance sampling, correction for the mismatch between the training and inference engines' numerics, and mixed-precision control, all aimed at the instability and degenerate outputs (unreadable traces, token bloat) that plague large-scale RL runs. The findings are the interesting part. Sample efficiency improves substantially with scale — the trillion-parameter model extracts more from the same rollouts. Training separates into two distinct phases: a discovery phase where the model finds new reasoning strategies, followed by a sharpening phase where it consolidates them. And behaviors the authors did not train for emerge on their own, including self-verification, parallel exploration of multiple solution paths, and structured formatting. The resulting Ring-2.5-1T-Zero is competitive across seven mathematics benchmarks, and the paper introduces an evaluation framework scoring reasoning traces on comprehensibility, reproducibility, and efficiency rather than answer accuracy alone.

The dominant recipe for hard reasoning benchmarks is to make models enormous and let them think in long chains of natural-language tokens, which is expensive and awkward for tasks that are really about inducing an abstract rule from a handful of examples. This paper attacks that from the opposite direction, building on the Dragon Hatchling line of biologically-flavored recurrent architectures. BDH-CQ keeps a recurrent latent state that is continuously updated by inputs presented at inference time, so demonstrations shown in-context directly rewrite working memory rather than merely conditioning attention. Reasoning happens as iterative computation in that latent space instead of as emitted chain-of-thought text. The practical consequence is that the compute spent per task is a function of recurrent depth, not of how many tokens the model is willing to write. The headline result is on ARC-AGI-1, the abstraction-and-reasoning benchmark designed to resist memorization: a 150-million-parameter configuration reaches 29.5% pass@2 at roughly $0.0007 per task. That is orders of magnitude cheaper than frontier systems at comparable accuracy, and the authors claim it breaks the previously reported cost-accuracy Pareto frontier for the benchmark. The paper also probes what the model actually extracts from demonstrations, how consistently it applies an inferred transformation, and which concept classes remain out of reach.

Interpretability work usually asks what features a model represents. This paper asks a sharper question: which of a model's internal representations is it poised to say out loud, and do those representations play a special functional role? The authors introduce the Jacobian lens, a technique that computes, for each vocabulary token, the linearized effect of an intermediate activation on the model's future likelihood of emitting that token. Averaging this across positions and prompts yields a per-token direction, and the span of the currently active ones they call J-space. J-space turns out to be startlingly small — typically ten to twenty-five concepts at a time, under ten percent of activation variance — and it holds a set of unspoken words that are neither echoes of the input nor next-token predictions, but names for what the model is currently reasoning about. The causal experiments are the heart of the paper: suppressing J-space leaves Claude fluent, grammatical, and still capable of automatic inference, but collapses multi-step internal reasoning. Conversely, the contents can be reported, deliberately summoned, held across tokens, and used as intermediate scratch state. The authors argue this matches the functional signature of the global workspace from cognitive neuroscience — a narrow, broadcast, reportable bottleneck sitting atop massive unconscious parallel processing — and note it emerged spontaneously from ordinary training rather than being designed in.

Training language-model agents to complete long, multi-step tasks — browsing a website, navigating a simulated home, operating apps — is hard because rewards are sparse: the agent usually only learns whether it succeeded at the very end, leaving it unclear which of dozens of intermediate actions actually mattered. Popular methods like GRPO treat each attempt as an isolated linear sequence of steps, which wastes information and gives noisy credit assignment. This paper proposes G2PO, which reorganizes an agent's many exploratory trajectories into a single global state-transition graph. Because different attempts often pass through the same states, merging them into a graph lets the method pool experience across trajectories. G2PO adds two ingredients: group-aggregation state-value estimation, which averages over all paths reaching a given state to cut sampling variance, and an edge-centric advantage estimate that standardizes temporal-difference errors across the whole graph to pinpoint the transitions that truly drive success or failure. On established long-horizon agent benchmarks — WebShop, ALFWorld, and AppWorld — G2PO reports success-rate gains of up to 22.2% over GRPO, a strong recent baseline. The work is notable as part of a broader 2026 push to make reinforcement learning practical for genuinely long-horizon agents, addressing the credit-assignment problem that has limited how reliably LLM agents can chain many actions toward a distant goal.

One of the deepest puzzles from the James Webb Space Telescope is why the early universe already contained supermassive black holes far heavier than their host galaxies should allow. This study, part of the BlackTHUNDER program, delivers an unusually clear view of one such system and offers a possible explanation. Using JWST, the team studied the compact active galaxy GN-77652 at redshift 5.23 — seen roughly a billion years after the Big Bang. Its central black hole weighs about 11 million solar masses inside a host of only about 170 million solar masses, making it roughly 300 times heavier relative to its galaxy than the tight black-hole/galaxy relation seen in the nearby universe. Rather than treating this as an isolated anomaly, the observations reveal that the galaxy sits within a 12,000-parsec cosmic filament threading through four additional galaxy-scale structures, with hints of a second black hole in a neighboring source. The key insight is temporal: these structures are projected to merge over the next 150–440 million years, and as the black hole's galaxy accretes surrounding material and mergers add stellar mass, the system should migrate back toward the normal mass ratio observed today. The oversized black hole may thus be a transient early-growth phase caught mid-assembly, helping reconcile JWST's 'impossible' early black holes with standard galaxy-evolution expectations.

Fault-tolerant quantum computing needs error-correcting codes that protect logical qubits without demanding an impractical number of physical qubits. Quantum low-density parity-check (qLDPC) codes promise dramatically lower overhead than the standard surface code, but they typically require long-range connections between distant qubits — something superconducting chips, which favor short nearest-neighbor links on a flat grid, struggle to provide. This paper introduces 'vine codes,' a family of qLDPC codes designed specifically to run on a planar square lattice using only nearest-neighbor two-qubit gates native to superconducting hardware (iSWAP and CZ). The construction generalizes earlier 'directional codes,' which were confined to a torus, by using extra routing qubits to realize open boundary conditions on a realistic 2D layout. Through extensive numerical search, the authors identify promising concrete instances, including [[121,4,6]], [[221,6,7]], and [[234,9,6]] codes, each encoding several logical qubits with useful distance. The significance is architectural rather than record-breaking: it narrows the gap between the theoretical efficiency of qLDPC codes and what today's chip geometries can actually implement. If codes like these hold up under full circuit-level noise simulation and eventual hardware demonstration, they could reduce the qubit cost of a logical qubit substantially, easing one of the central engineering bottlenecks on the road to scalable, fault-tolerant machines.

This paper presents the first densely reconstructed wiring diagram uniting the brain and the ventral nerve cord — the insect equivalent of a spinal cord — of an adult fruit fly, completing a full central-nervous-system connectome for a complex animal. Mapping every neuron and synapse across both structures let the team ask not just how the brain computes, but how it actually commands the body. The headline finding overturns a textbook assumption. Rather than a centralized command center dictating movement, motor control turns out to be largely distributed and embodied. Effector neurons controlling muscles, glands, and viscera are driven mostly by sensory neurons in the same body part, forming tight local feedback loops. These local loops are then stitched together by long-range ascending and descending neurons organized into behavior-centric modules that coordinate actions like walking and flight. The result reframes how nervous systems generate coordinated behavior: control is parallel, local, and situated in the body's appendages rather than issued top-down from the brain. Because the full connectome is released as a shared resource, it becomes a foundation for testing circuit-level theories of behavior and for building biologically grounded models of sensorimotor control — a milestone comparable in ambition to earlier whole-brain fly and worm connectomes, but now spanning the entire body axis.

Protein structure prediction has been dominated by DeepMind's AlphaFold, but this release from the Chan Zuckerberg Initiative's Biohub stakes out a different position: fully open-source, enormous scale, and design-oriented. ESMFold2 is built on a protein language model trained on billions of sequences spanning the tree of life, heavily enriched with metagenomic data from soil, oceans, and other environments that conventional databases underrepresent. Using the model, the team generated the ESM Atlas — roughly 1.1 billion predicted protein structures plus sequence data for about 6.8 billion proteins. That eclipses the widely used AlphaFold Database by more than 800 million entries and dwarfs the previous ESM Atlas. On several benchmark tasks the developers report ESMFold2 matches or outperforms leading systems, including AlphaFold3, while running fast enough to fold at this billion-scale. Beyond sheer size, the authors demonstrate practical utility: they used ESMFold2 to design new antibodies and other binder proteins targeting molecules implicated in cancers and immune conditions, then synthesized and tested them in the lab, where a high fraction behaved as predicted. Because the model carries no restrictions on commercial use, it lowers the barrier for academic and industry groups to explore the vast 'dark matter' of uncharacterized proteins and to move from prediction toward routine functional design.

This interpretability paper argues that inside a large language model there is a small, shared internal space that behaves strikingly like the 'global workspace' from theories of human consciousness. The team introduces a tool they call the Jacobian lens (J-lens), which, for each word in the vocabulary, isolates the internal activity pattern that makes the model more likely to say that word later — not the token it is emitting now, but words held 'available' for future speech. Applying this lens surfaces what they name J-space: a compact subspace accounting for no more than about 10% of activation variance and concentrated in the network's middle layers. Crucially, directions in J-space feed many downstream computations rather than being read by one narrow circuit — the 'global' in global workspace. The authors show that prompting the model to articulate ethical principles when interrupted measurably improves behavior even in ordinary uninterrupted contexts, with no direct training on that behavior. The work drew unusually serious cross-disciplinary attention: Stanislas Dehaene and Lionel Naccache, architects of global neuronal workspace theory, contributed invited commentary, and DeepMind's Neel Nanda independently replicated key findings on open-weight models. Whether or not one accepts the consciousness analogy, the paper offers a concrete, testable structural claim about how information is broadcast inside transformers, and a practical new probe for alignment.

The unit distance problem, posed by Paul Erdős in 1946, asks how many pairs of n points in the plane can be exactly distance 1 apart. For decades the conjectured near-optimal arrangement was a scaled square grid, giving roughly n^(1+c/log log n) unit-distance pairs, and many believed this was essentially the truth. This work reports that an OpenAI reasoning model constructed an infinite family of point sets that provably beat that bound by a genuine polynomial factor, disproving the long-standing conjecture. The striking feature is the method. Rather than nudging dots around a flat sheet, the argument imports heavy machinery from algebraic number theory — class field towers and results in the lineage of Golod–Shafarevich and Ellenberg–Venkatesh — to build point configurations with an unexpected abundance of unit distances. A companion paper by nine mathematicians independently verified the proof and rewrote it in cleaner human language. The result matters less for the specific bound than for what produced it: mathematicians including Daniel Litt called it the most interesting result an AI has generated autonomously, and the first that would plausibly merit publication in a top journal on its own. It reframes the debate about whether large models can do original, non-routine mathematics rather than merely retrieve or formalize known arguments.

Finding new superconductors has traditionally been slow and serendipitous, with vast chemical spaces to search and expensive synthesis for each candidate. This work, from the SuperC consortium reported in Physical Review Research, demonstrates a modern end-to-end pipeline: machine learning to triage candidates, physics-based calculation to validate, and lab synthesis to confirm. The team used ML-guided screening to narrow an enormous chemical search space, then applied first-principles calculations — including quantum-geometry analysis of the electronic structure — to flag promising compounds. Collaborators at Rice University synthesized and tested the top candidates, confirming bulk superconductivity in two kagome-lattice materials, YRu3B2 and LuRu3B2, with critical temperatures of 0.81 K and 0.95 K respectively. The transition temperatures are modest, so this is not a warm-superconductor headline. What makes it notable is methodological and structural. It is a clean example of the full loop working — prediction to synthesis to experimental confirmation of genuinely new superconductors — validating ML as a practical discovery tool rather than just a ranking heuristic. And the kagome lattice is of intense current interest because its geometry produces flat electronic bands and nontrivial quantum geometry, features increasingly linked to unconventional superconductivity. New confirmed members of this family give theorists concrete systems to probe how lattice geometry shapes electron pairing.

The 'Hubble tension' is cosmology's most stubborn crisis: measurements of the universe's current expansion rate disagree depending on how you look. Local, direct measurements give roughly 73 km/s/Mpc, while inferences from the early-universe cosmic microwave background under the standard ΛCDM model give about 67. New data from JWST and the DESI galaxy survey have sharpened, not softened, this discrepancy — JWST also revealing surprisingly bright, abundant galaxies at very high redshift that ΛCDM struggles to explain. This paper tests a leading candidate fix: early dark energy (EDE), a hypothetical component that briefly boosts the expansion rate just before recombination, shrinking the early-universe sound horizon and raising the inferred value of H₀. The authors fit a canonical axion-like EDE model against a combined dataset — CMB from Planck, ACT and SPT, baryon acoustic oscillations from DESI, and ultraviolet luminosity functions from JWST. Within this framework the data push H₀ up to 71.58 ± 1.05 km/s/Mpc, easing the tension to roughly the 1σ level. The model also fits the puzzling bright high-redshift JWST galaxies better than ΛCDM and shows statistically superior overall performance. It is notable as one of the more compelling recent attempts to reconcile several independent datasets simultaneously — though EDE remains contested, and joint fits like this are exactly how the field will decide whether new physics is genuinely required.

Almost every leading language model generates text autoregressively — one token at a time, left to right. Diffusion language models offer an alternative: they generate in parallel by iteratively 'denoising' masked positions with fully bidirectional attention, which promises faster and more flexible generation. The catch has been quality: diffusion LMs have consistently trailed comparable autoregressive models, leaving open whether the paradigm can actually scale. This paper presents iLLaDA, an 8-billion-parameter masked diffusion language model trained from scratch. The authors scale pretraining to 12 trillion tokens and fine-tune on a 25-billion-token instruction corpus, combining aggressive data scaling with improved training recipes. The gains are substantial: the base model improves by 21.6 points on the BBH reasoning suite and 14.9 points on ARC-Challenge, while the instruction-tuned model gains 14.5 points on MATH and 16.5 points on HumanEval over prior diffusion baselines — narrowing much of the gap to similarly sized autoregressive models. It arrives amid a June 2026 surge of diffusion-LM work (multi-block decoding, systematic inference-time analyses), and is among the most convincing demonstrations that the approach scales rather than plateauing. That matters because parallel, bidirectional generation could eventually deliver lower latency and new capabilities — like flexible infilling and controllable decoding — that autoregressive models handle awkwardly, making diffusion a credible contender rather than a curiosity.

An AI agent that uses tools, browsers, or a terminal needs to anticipate how its environment will respond to each action — a capability cognitive scientists call a 'world model.' Most world-model research operates on pixels or video; this paper instead builds world models in language, betting that a strong reasoning model can simulate an environment's dynamics through text. The Qwen team introduces two models (a 35B and a 397B mixture-of-experts) that simulate agentic environments via long chain-of-thought reasoning across seven domains: tool-calling (MCP), search, terminal, software engineering, Android, web, and OS. Training proceeds in three stages: continued pretraining injects state-transition dynamics and world knowledge, supervised fine-tuning activates next-state-prediction 'thinking,' and reinforcement learning with hybrid rubric-and-rule rewards sharpens simulation fidelity. They also release AgentWorldBench, an out-of-distribution benchmark assembled from real interactions of frontier agents such as Claude Opus 4.6. Two results stand out. As a decoupled environment simulator, the model enables scalable, controllable agentic reinforcement learning without hitting real systems. And as a foundation model, world-model training acts as an effective 'warm-up' that improves downstream performance across seven agent benchmarks. It matters because it reframes world models for the agent era: a language-native, open-weights simulator that could make training capable computer-using agents cheaper and safer than learning purely in live environments.

Understanding how an animal turns a decision into coordinated movement requires knowing not just how the brain is wired, but how it connects to the motor circuits that drive the body. Previous landmark connectomes mapped the fruit fly's brain alone; the nerve cord — the insect analogue of a spinal cord — was reconstructed separately, leaving the crucial brain-to-body link unmapped. This Nature paper, from a large Harvard Medical School and Princeton-led international consortium, presents the first complete synapse-level wiring diagram of an adult Drosophila's entire central nervous system, integrating brain and ventral nerve cord into a single map. It was built using serial electron microscopy and AI-based image reconstruction to trace every neuron and connection, then to follow the descending pathways by which the brain commands walking and flight. The key advance is completeness across the sensorimotor loop: researchers can now trace a signal from sensory input, through central processing, down descending neurons, and into the motor circuitry that moves legs and wings — all at single-synapse resolution in a behaving animal. The full dataset has been released openly. It is notable as a foundational reference for systems neuroscience: a whole-organism control diagram that lets scientists test, rather than guess, how distributed circuits produce complex behavior, and a template for eventually tackling far larger nervous systems.

Quantum error correction (QEC) has been a theoretical promise since the mid-1990s: encode information redundantly across many noisy physical qubits so that the resulting 'logical' qubit fails far less often than its parts. The open question was whether this actually holds up in hardware, where real noise is messy and correlated. This Nature paper, from a Microsoft Quantum and Quantinuum collaboration, reports the largest independently peer-reviewed gap between physical and logical error rates yet demonstrated. Running on Quantinuum's trapped-ion processor and using sophisticated code structures — including a 16-qubit 'tesseract' color code — the team built circuits spanning up to 12 logical qubits and applied a mix of error detection and correction. Logical error rates improved by factors ranging from about 11× to 800× over the equivalent unencoded circuits. In one Bell-state preparation, the error rate fell from roughly 0.8% to about 0.001%. The result matters because it moves QEC from proof-of-principle toward practical utility: it is concrete, peer-reviewed evidence that adding qubits and structure genuinely buys reliability, rather than being swamped by overhead. While these are still small circuits far from a full fault-tolerant computer, an 800× reduction is the kind of margin the field has argued for decades might never materialize on real machines.

Paul Erdős's 1946 unit-distance problem asks: among n points in the plane, how many pairs can lie exactly distance 1 apart? For decades the prevailing conjecture held that the maximum grows barely faster than n — essentially n^(1+o(1)), with the extra exponent shrinking to zero as n grows. This paper is the mathematical community's response to a striking event: in May 2026 OpenAI reported that an internal general-reasoning model produced a genuinely new point construction that overturns that upper-bound belief. The construction shows that for infinitely many n the number of unit-distance pairs exceeds n^(1+δ) for a fixed positive δ, disproving the conjecture. A group of leading mathematicians wrote this companion note to verify, clean up, and contextualize the machine-generated argument; separately, Will Sawin made the exponent explicit, showing δ ≥ 0.014. What makes it notable is less the numerical bound than the provenance. It is one of the first documented cases of an AI system contributing an original idea to a famous, long-open problem in extremal combinatorics — not merely filling in routine steps — and the reaction of serious mathematicians formalizing and extending that idea signals a real shift in how machine reasoning may enter research mathematics.

Neuroscience has long relied on task-specific encoding models — often simple linear maps — to predict how the brain responds to a given stimulus. These models are narrow, subject-specific, and hard to generalize. This paper introduces TRIBE v2, a tri-modal foundation model that jointly ingests video, audio, and language to predict human brain activity across a wide range of naturalistic and experimental conditions. Trained on a unified corpus of over 1,000 hours of fMRI spanning 720 subjects, TRIBE v2 predicts high-resolution brain responses to novel stimuli, novel tasks, and even previously unseen subjects, improving several-fold over traditional linear encoders. Crucially, the model enables "in-silico neuroscience": researchers can run experiments on the model instead of in a scanner. When probed with classic visual and neurolinguistic paradigms, it reproduces results that took decades of empirical work to establish, and its interpretable latent features reveal the fine-grained cortical topography of multisensory integration. Coming from Meta's FAIR lab and ENS, the work argues that a single, well-trained predictive model can serve as a fast, hypothesis-generating proxy for the brain. Like foundation models transformed language and vision, a general "brain model" that predicts responses across senses, tasks, and people could reshape how cognitive and systems neuroscience is done.

Deep-learning protein designers have become remarkably good at hallucinating novel, foldable structures, but they typically emit molecular geometries directly, without explaining which parts of a protein matter or why. That opacity makes designs hard to control, audit, or systematically reuse — a growing problem as these tools move toward real therapeutic and enzymatic applications. Proteo-R1 imports the "reasoning model" paradigm into this domain with a dual-expert architecture that separates understanding from generation. First, a multimodal language model reads both sequence and structure to reason about and identify the functionally critical residues — active sites, binding anchors, and other essential positions. Those residues are then handed as fixed constraints to a diffusion-based geometric generator, which performs conditional design that must respect the identified anchors. The result mirrors how a human protein engineer works: reason about function first, then optimize the geometry around it. The authors report that this yields stable, interpretable, and modular integration of language-model reasoning with state-of-the-art generative models, making design decisions inspectable and reusable rather than opaque. With 29 authors including Jure Leskovec and Yejin Choi, the paper signals a convergence of two of AI's fastest-moving threads — reasoning LLMs and scientific generative models — in the high-stakes arena of biomolecular design.

Standard Transformer attention keeps an ever-growing cache of past tokens, making long-context inference expensive in both time and memory. Linear-attention models replace that cache with a fixed-size recurrent state, giving linear-time sequence mixing and constant-memory decoding — but historically at a cost in recall accuracy. Recent variants like Gated DeltaNet and KDA improved matters with adaptive forgetting, yet they share a subtle flaw: a single scalar gate controls two different operations at once, namely how much old information to erase and how much new information to write into the state. Gated DeltaNet-2 decouples these into separate gates, giving the model independent control over erasing versus writing. To keep training efficient despite this richer update rule, the authors derive a chunkwise-parallel formulation that folds channel-wise decay into asymmetric rank-one erase factors, implemented with custom Triton kernels. At 1.3B parameters trained on 100B FineWeb-Edu tokens, it outperforms Mamba-2, Gated DeltaNet, KDA, and Mamba-3 across language modeling, commonsense reasoning, and retrieval, with its largest gains on long-context "needle-in-a-haystack" RULER benchmarks. The work is part of a broader 2026 push to find sub-quadratic architectures that match Transformers on quality while scaling gracefully to very long contexts — a direction increasingly central to efficient frontier models.

Gravitational-wave astronomy has matured from detecting single, headline-grabbing events into doing population statistics on cosmic collisions. This paper is the latest installment of the Gravitational-Wave Transient Catalog, compiling detections from the second half of the fourth observing run (O4b) of the LIGO, Virgo, and KAGRA detectors and updating the analysis of previously reported events. Each catalogued signal is a compact binary coalescence — merging black holes, neutron stars, or mixed pairs — reconstructed with improved detector sensitivity and refined waveform models. Collectively the catalog now spans a large and growing set of events, letting astrophysicists map the mass and spin distributions of black holes, probe the neutron-star equation of state, search for unexpected sources in mass "gaps," and test general relativity in the strong-field regime. The expanded sample also tightens gravitational-wave-based measurements of the Hubble constant via "standard sirens." As a collaboration data release with hundreds of authors, it is less a single idea than an essential reference: the definitive, up-to-date census of the transient gravitational-wave sky. Its significance lies in statistical power — the accumulated population is what turns individual detections into constraints on how massive stars live and die, and on gravity itself.

In 2016, Maryna Viazovska proved that the E8 lattice gives the densest possible packing of spheres in eight dimensions — a Fields Medal-winning result built on a delicate "magic function" constructed from modular forms and the Cohn–Elkies linear-programming framework. The proof is famously intricate, mixing hard analysis, number theory, and computer-assisted estimates, which makes it an ideal but daunting target for formal verification. This paper reports major progress on encoding the entire argument in the Lean theorem prover, so that every step is machine-checked rather than trusted on paper. The effort is a large human–AI collaboration: a significant portion of the tedious formalization was completed in early 2026 by an autoformalization model named "Gauss" from Math, Inc., which translated informal mathematics into verified Lean code. The authors — including Viazovska herself — document their methodology, the division of labor between humans and the model, and what remains before the full result is completely formalized. Beyond the specific theorem, the work is a bellwether for how frontier mathematics may increasingly be certified. Formalizing a proof of this depth was until recently considered years away; doing it with substantial AI assistance suggests that even the most sophisticated modern results are becoming tractable for rigorous, end-to-end verification.

The unit-distance problem, posed by Erdős in 1946, asks how many pairs of points can be exactly distance 1 apart among n points in the plane. The square grid gives roughly n^(1+c/log log n) such pairs, and it was widely believed that no arrangement could do meaningfully better — i.e., that the true maximum is n^(1+o(1)). In May 2026, a general-purpose OpenAI reasoning model overturned that belief. It found an infinite family of point configurations achieving at least n^(1+δ) unit-distance pairs for a fixed positive δ, a genuine polynomial improvement over the grid. The construction is far from combinatorial guesswork: it draws on deep algebraic number theory, including Golod–Shafarevich theory and infinite class field towers, to engineer points with many coincident unit distances. Princeton's Will Sawin subsequently sharpened the exponent to δ ≈ 0.014, and the roughly 125-page write-up (led by Lijie Chen) was checked by external mathematicians. What makes this a landmark is not just the disproof of a specific conjecture but the nature of the achievement: an AI system contributed a novel, non-obvious construction in an area that had resisted experts for 80 years. Fields Medalist Timothy Gowers publicly called it a milestone for AI in mathematics.

A primitive set is a set of integers in which no element divides another. Paul Erdős spent decades probing how "large" such sets can be, measured by the Erdős sum of 1/(n·log n) over the set, and left behind a cluster of stubborn conjectures from the 1960s. This paper introduces a genuinely new tool for bounding these sums: a Markov chain whose transitions are weighted by the von Mangoldt function, which the authors call the "downwards von Mangoldt chain." The chain sidesteps losses that plagued earlier Mertens-type arguments, and the payoff is sweeping. In one framework the authors resolve two 1966 Erdős–Sárközy–Szemerédi conjectures (on primitive sets of large numbers and on divisibility chains), give a short new proof of the Erdős Primitive Set Conjecture, show that 2 is "Erdős-strong," and settle a master conjecture of Banks–Martin that unifies the field. Strikingly, the core idea was suggested by output from GPT-5.4 Pro, then developed and rigorously proved by the human authors, including Fields Medalist Terence Tao and primitive-set expert Jared Duker Lichtman. It is notable both as a clean resolution of long-open number-theory problems and as a concrete example of frontier AI contributing a real mathematical idea.