arXiv · 17 Feb 2026 · Combinatorics
Huy Tuan Pham and Lisa Sauermann proved that, for any prime p, a set of nonzero residues can always be listed in an order where every running total is different, a question Ronald Graham posed in 1971. Their paper handles sets up to a power-law fraction of the prime’s size, and combined with earlier work by Müyesser, Pokrovskiy, Kravitz, Bedert and colleagues, this settles the conjecture for all sufficiently large primes. The argument leans on randomness together with anti-concentration estimates and Fourier analysis. It was posted in February and is a preprint without a journal reference, but it reached wider attention on 28 September 2026 when Quanta Magazine profiled the full line of work, with Princeton’s Noga Alon among the experts commenting on it.
]]>arXiv · 23 Oct 2025 · Graph theory
Natalie Behague, Daniel Iľkovič and Richard Montgomery proved a 2004 conjecture of Jeong Han Kim and Van Vu, showing that every random d-regular graph can be sandwiched between two ordinary binomial random graphs with closely matched edge probabilities, extending earlier work that only covered much larger degrees. The proof builds the regular graph edge by edge using weighted coin flips that guarantee it contains a binomial graph, then reverses the process to complete the sandwich from the other side, fully analyzing a coupling technique earlier attempts had left incomplete. The preprint drew little notice outside specialists until Quanta Magazine profiled it on 18 September 2026, where Tel Aviv University’s Michael Krivelevich described finally seeing the finished proof as “some kind of relief.” Because binomial random graphs are far better understood than regular ones, the result lets researchers automatically transfer known properties across to regular graphs, a structure that shows up throughout network theory.
]]>arXiv · 25 Mar 2026 · Graph theory
Mikkel Thorup, Carsten Thomassen, Ken-ichi Kawarabayashi, Bojan Mohar and two graduate students spent nearly a decade building a genuinely different proof of the 1976 four-color theorem, organized around 8,202 small interchangeable map patterns rather than the 1,482 used in the standard proof. Because the new patterns also cover flatter regions of a map instead of only the sharply curved ones the older approach relied on, many of them can be resolved in parallel rather than one at a time. That parallelism turns the widely used 1996 quadratic-time algorithm for four-coloring a planar map into a near-linear, O(n log n) one, a real complexity gain rather than a rephrasing of the existing theorem, and the result is due to be presented at the Foundations of Computer Science conference in November. The preprint drew little notice beyond a few blogs after it was posted in March, until Quanta Magazine profiled it this month, prompting Inria’s Georges Gonthier, who formalized the original four-color proof in a computer proof assistant, to call it “really cool to see a real result for once.”
]]>NYU · 7 Sep 2026 · Partial differential equations
Levent Alpöge (Anthropic) and Tristan Buckmaster (NYU) constructed smooth initial data and smooth, compactly supported forcing under which solutions to the three-dimensional incompressible Euler equations, the two-dimensional Boussinesq system, and the incompressible porous-medium equation all become singular in finite time, extending a multiscale layering construction pioneered by Diego Córdoba and Luis Martínez-Zoroa. The Euler and porous-medium proofs were formally checked in the Lean proof assistant, and the authors say large language models, including Claude and OpenAI’s models, assisted heavily in developing the arguments. Fields medalist Terence Tao, who had no role in the work, called it “a remarkable achievement” and a genuine stepping stone toward the far harder unforced blowup problem for Navier-Stokes, while noting substantial technical obstacles remain before that Millennium Prize problem itself is reached. A separate, unpublished claim of an AI-generated proof for forced Navier-Stokes from OpenAI has not been made available for independent scrutiny and is not covered here.
arXiv · 3 Mar 2026 · Probability theory
Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion proved that on any infinite transitive graph, past the critical probability for percolation, the chance a given point lies in a large finite cluster decays exponentially in the graph’s isoperimetric profile. The result settles the “supercritical” half of a sharpness conjecture whose “subcritical” half was solved in 2007, closing out a research program Itai Benjamini and Oded Schramm proposed decades ago, using methods described as strikingly simple for how broadly they apply. The preprint drew little notice outside probability circles until Quanta profiled it on 31 August 2026, where probabilists including Tel Aviv University’s Asaf Nachmias praised the proof as stunning.
]]>Anthropic · 10 Aug 2026 · Analytic number theory
Anthropic reports that an unreleased Claude model improved the proven lower bound on the share of the Riemann zeta function’s nontrivial zeros known to lie on the critical line from 41.6% to 67.2%, combining recent zero-density estimates with a 2000 theorem of Enrico Bombieri after testing around 650 approaches across roughly 60 coordinated subagents. The result does not touch the Riemann hypothesis itself, which would require the bound to reach 100%, but two independent number theorists, Brian Conrey and Dan Goldston, examined the paper on short notice, and Oxford’s James Maynard described it as “a new real idea, which this new result seems to provide.” Anthropic’s own mathematicians also validated and helped formalize the argument.
arXiv · 5 Aug 2025 · Combinatorics
Nikhil Bansal and Haotian Jiang proved an Õ(log^1/4 n) bound for the Komlós conjecture, a problem about how evenly a set of vectors can be split into two balanced groups, improving on Banaszczyk’s O(√log n) bound and edging close to the conjectured constant bound; the same paper also fully resolves the related Beck-Fiala conjecture for vectors with at least log² n nonzero entries. The preprint drew little public notice for a year until Quanta Magazine profiled it on 21 August 2026, where discrepancy theorists Aleksandar Nikolov and Daniel Spielman called it “a huge step forward” and “a beautiful result” respectively, and Quanta described it as the first major advance on the Komlós problem in close to thirty years. What changed this month is the result becoming widely known and expert-vetted, not the underlying mathematics.
Physical Review Letters · 28 Aug 2026 · Particle physics
The LUX-ZEPLIN (LZ) collaboration reports 4.5-sigma evidence of coherent elastic scattering between boron-8 solar neutrinos and xenon nuclei inside its dark-matter detector, drawn from a 5.7 tonne-year exposure recorded between March 2023 and April 2025. A companion measurement from the XENONnT collaboration, using the same scattering process, published in the same Physical Review Letters issue. Together the two results show these detectors, built to search for dark-matter particles between 3 and 9 GeV, are now sensitive enough to do precision electroweak physics as a side effect, since the same low-energy neutrino background that limits dark-matter searches can also be used to measure Standard Model parameters like the weak mixing angle.
]]>arXiv · 31 Jul 2026 · Algebraic geometry
Mathematicians have disproved the Jacobian conjecture, the 87-year-old question of whether every polynomial self-map of complex space with constant nonzero Jacobian determinant must have a polynomial inverse, in every dimension above two. The refutation began on 19 July when Levent Alpöge posted a three-dimensional counterexample found with the help of an AI system, and a geometric explanation of why it works followed within days from other mathematicians, including Terence Tao, who wrote up his own account of the mechanism on his blog. This preprint, by Shuhong Gao, gives a rigorous, self-contained account of the underlying tangent-sweep construction and extends it to five new explicit counterexamples spanning three, four and five dimensions, with every algebraic identity checked in exact rational arithmetic. The original two-dimensional form of the conjecture remains open.
Max Planck Institute for Nuclear Physics · 4 Aug 2026 · Particle physics
The Double Chooz collaboration has measured antineutrinos still emitted by a nuclear reactor’s spent fuel and residual fission products months after shutdown, the first direct experimental confirmation of a long-standing theoretical prediction. Over 17.2 days of observation at the Chooz plant in France, the detector recorded roughly 100 candidate antineutrino events whose rate matched simulations of long-lived fission-product decay. Because the signal persists independent of active fission, the result was published in Physical Review Letters and points to a way of remotely checking whether a shut-down reactor’s fuel inventory matches what has been declared, relevant to nuclear-safeguards monitoring.
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